itg2addnclem

Description: An alternate expression for the S.2 integral that includes an arbitrarily small but strictly positive "buffer zone" wherever the simple function is nonzero. (Contributed by Brendan Leahy, 10-Oct-2017) (Revised by Brendan Leahy, 10-Mar-2018)

		Ref	Expression
	Hypothesis	itg2addnclem.1	$⊢ L = \{x \| \exists g \in dom \int^{1} (\exists y \in ℝ^{+} (z \in ℝ ⟼ if (g (z) = 0, 0, g (z) + y)) \leq_{f} F \land x = \int^{1} (g))\}$
	Assertion	itg2addnclem	$⊢ F : ℝ ⟶ [0, +∞] \to \int^{2} (F) = sup (L ℝ^{*} <)$

Proof

Step	Hyp	Ref	Expression
1		itg2addnclem.1	$⊢ L = \{x \| \exists g \in dom \int^{1} (\exists y \in ℝ^{+} (z \in ℝ ⟼ if (g (z) = 0, 0, g (z) + y)) \leq_{f} F \land x = \int^{1} (g))\}$
2		eqid	$⊢ \{x \| \exists f \in dom \int^{1} (f \leq_{f} F \land x = \int^{1} (f))\} = \{x \| \exists f \in dom \int^{1} (f \leq_{f} F \land x = \int^{1} (f))\}$
3	2	itg2val	$⊢ F : ℝ ⟶ [0, +∞] \to \int^{2} (F) = sup (\{x \| \exists f \in dom \int^{1} (f \leq_{f} F \land x = \int^{1} (f))\} ℝ^{*} <)$
4	1	supeq1i	$⊢ sup (L ℝ^{} <) = sup (\{x \| \exists g \in dom \int^{1} (\exists y \in ℝ^{+} (z \in ℝ ⟼ if (g (z) = 0, 0, g (z) + y)) \leq_{f} F \land x = \int^{1} (g))\} ℝ^{} <)$
5		xrltso	$⊢ < Or ℝ^{*}$
6	5	a1i	$⊢ F : ℝ ⟶ [0, +∞] \to < Or ℝ^{*}$
7		simprr	$⊢ (f \in dom \int^{1} \land (f \leq_{f} F \land x = \int^{1} (f))) \to x = \int^{1} (f)$
8		itg1cl	$⊢ f \in dom \int^{1} \to \int^{1} (f) \in ℝ$
9	8	rexrd	$⊢ f \in dom \int^{1} \to \int^{1} (f) \in ℝ^{*}$
10	9	adantr	$⊢ (f \in dom \int^{1} \land (f \leq_{f} F \land x = \int^{1} (f))) \to \int^{1} (f) \in ℝ^{*}$
11	7 10	eqeltrd	$⊢ (f \in dom \int^{1} \land (f \leq_{f} F \land x = \int^{1} (f))) \to x \in ℝ^{*}$
12	11	rexlimiva	$⊢ \exists f \in dom \int^{1} (f \leq_{f} F \land x = \int^{1} (f)) \to x \in ℝ^{*}$
13	12	abssi	$⊢ \{x \| \exists f \in dom \int^{1} (f \leq_{f} F \land x = \int^{1} (f))\} \subseteq ℝ^{*}$
14		supxrcl	$⊢ \{x \| \exists f \in dom \int^{1} (f \leq_{f} F \land x = \int^{1} (f))\} \subseteq ℝ^{} \to sup (\{x \| \exists f \in dom \int^{1} (f \leq_{f} F \land x = \int^{1} (f))\} ℝ^{} <) \in ℝ^{*}$
15	13 14	mp1i	$⊢ F : ℝ ⟶ [0, +∞] \to sup (\{x \| \exists f \in dom \int^{1} (f \leq_{f} F \land x = \int^{1} (f))\} ℝ^{} <) \in ℝ^{}$
16		fveq1	$⊢ g = f \to g (z) = f (z)$
17	16	eqeq1d	$⊢ g = f \to (g (z) = 0 \leftrightarrow f (z) = 0)$
18	16	oveq1d	$⊢ g = f \to g (z) + y = f (z) + y$
19	17 18	ifbieq2d	$⊢ g = f \to if (g (z) = 0, 0, g (z) + y) = if (f (z) = 0, 0, f (z) + y)$
20	19	mpteq2dv	$⊢ g = f \to (z \in ℝ ⟼ if (g (z) = 0, 0, g (z) + y)) = (z \in ℝ ⟼ if (f (z) = 0, 0, f (z) + y))$
21	20	breq1d	$⊢ g = f \to ((z \in ℝ ⟼ if (g (z) = 0, 0, g (z) + y)) \leq_{f} F \leftrightarrow (z \in ℝ ⟼ if (f (z) = 0, 0, f (z) + y)) \leq_{f} F)$
22	21	rexbidv	$⊢ g = f \to (\exists y \in ℝ^{+} (z \in ℝ ⟼ if (g (z) = 0, 0, g (z) + y)) \leq_{f} F \leftrightarrow \exists y \in ℝ^{+} (z \in ℝ ⟼ if (f (z) = 0, 0, f (z) + y)) \leq_{f} F)$
23		fveq2	$⊢ g = f \to \int^{1} (g) = \int^{1} (f)$
24	23	eqeq2d	$⊢ g = f \to (x = \int^{1} (g) \leftrightarrow x = \int^{1} (f))$
25	22 24	anbi12d	$⊢ g = f \to ((\exists y \in ℝ^{+} (z \in ℝ ⟼ if (g (z) = 0, 0, g (z) + y)) \leq_{f} F \land x = \int^{1} (g)) \leftrightarrow (\exists y \in ℝ^{+} (z \in ℝ ⟼ if (f (z) = 0, 0, f (z) + y)) \leq_{f} F \land x = \int^{1} (f)))$
26	25	cbvrexvw	$⊢ \exists g \in dom \int^{1} (\exists y \in ℝ^{+} (z \in ℝ ⟼ if (g (z) = 0, 0, g (z) + y)) \leq_{f} F \land x = \int^{1} (g)) \leftrightarrow \exists f \in dom \int^{1} (\exists y \in ℝ^{+} (z \in ℝ ⟼ if (f (z) = 0, 0, f (z) + y)) \leq_{f} F \land x = \int^{1} (f))$
27		breq2	$⊢ 0 = if (f (z) = 0, 0, f (z) + y) \to (f (z) \leq 0 \leftrightarrow f (z) \leq if (f (z) = 0, 0, f (z) + y))$
28		breq2	$⊢ f (z) + y = if (f (z) = 0, 0, f (z) + y) \to (f (z) \leq f (z) + y \leftrightarrow f (z) \leq if (f (z) = 0, 0, f (z) + y))$
29		id	$⊢ f (z) = 0 \to f (z) = 0$
30		0le0	$⊢ 0 \leq 0$
31	29 30	eqbrtrdi	$⊢ f (z) = 0 \to f (z) \leq 0$
32	31	adantl	$⊢ (((f \in dom \int^{1} \land y \in ℝ^{+}) \land z \in ℝ) \land f (z) = 0) \to f (z) \leq 0$
33		rpge0	$⊢ y \in ℝ^{+} \to 0 \leq y$
34	33	ad2antlr	$⊢ ((f \in dom \int^{1} \land y \in ℝ^{+}) \land z \in ℝ) \to 0 \leq y$
35		i1ff	$⊢ f \in dom \int^{1} \to f : ℝ ⟶ ℝ$
36	35	ffvelcdmda	$⊢ (f \in dom \int^{1} \land z \in ℝ) \to f (z) \in ℝ$
37	36	adantlr	$⊢ ((f \in dom \int^{1} \land y \in ℝ^{+}) \land z \in ℝ) \to f (z) \in ℝ$
38		rpre	$⊢ y \in ℝ^{+} \to y \in ℝ$
39	38	ad2antlr	$⊢ ((f \in dom \int^{1} \land y \in ℝ^{+}) \land z \in ℝ) \to y \in ℝ$
40	37 39	addge01d	$⊢ ((f \in dom \int^{1} \land y \in ℝ^{+}) \land z \in ℝ) \to (0 \leq y \leftrightarrow f (z) \leq f (z) + y)$
41	34 40	mpbid	$⊢ ((f \in dom \int^{1} \land y \in ℝ^{+}) \land z \in ℝ) \to f (z) \leq f (z) + y$
42	41	adantr	$⊢ (((f \in dom \int^{1} \land y \in ℝ^{+}) \land z \in ℝ) \land \neg f (z) = 0) \to f (z) \leq f (z) + y$
43	27 28 32 42	ifbothda	$⊢ ((f \in dom \int^{1} \land y \in ℝ^{+}) \land z \in ℝ) \to f (z) \leq if (f (z) = 0, 0, f (z) + y)$
44	43	adantlll	$⊢ (((F : ℝ ⟶ [0, +∞] \land f \in dom \int^{1}) \land y \in ℝ^{+}) \land z \in ℝ) \to f (z) \leq if (f (z) = 0, 0, f (z) + y)$
45	35	ad2antlr	$⊢ ((F : ℝ ⟶ [0, +∞] \land f \in dom \int^{1}) \land y \in ℝ^{+}) \to f : ℝ ⟶ ℝ$
46	45	ffvelcdmda	$⊢ (((F : ℝ ⟶ [0, +∞] \land f \in dom \int^{1}) \land y \in ℝ^{+}) \land z \in ℝ) \to f (z) \in ℝ$
47	46	rexrd	$⊢ (((F : ℝ ⟶ [0, +∞] \land f \in dom \int^{1}) \land y \in ℝ^{+}) \land z \in ℝ) \to f (z) \in ℝ^{*}$
48		0re	$⊢ 0 \in ℝ$
49	38	ad2antlr	$⊢ (((F : ℝ ⟶ [0, +∞] \land f \in dom \int^{1}) \land y \in ℝ^{+}) \land z \in ℝ) \to y \in ℝ$
50	46 49	readdcld	$⊢ (((F : ℝ ⟶ [0, +∞] \land f \in dom \int^{1}) \land y \in ℝ^{+}) \land z \in ℝ) \to f (z) + y \in ℝ$
51		ifcl	$⊢ (0 \in ℝ \land f (z) + y \in ℝ) \to if (f (z) = 0, 0, f (z) + y) \in ℝ$
52	48 50 51	sylancr	$⊢ (((F : ℝ ⟶ [0, +∞] \land f \in dom \int^{1}) \land y \in ℝ^{+}) \land z \in ℝ) \to if (f (z) = 0, 0, f (z) + y) \in ℝ$
53	52	rexrd	$⊢ (((F : ℝ ⟶ [0, +∞] \land f \in dom \int^{1}) \land y \in ℝ^{+}) \land z \in ℝ) \to if (f (z) = 0, 0, f (z) + y) \in ℝ^{*}$
54		iccssxr	$⊢ [0, +∞] \subseteq ℝ^{*}$
55		fss	$⊢ (F : ℝ ⟶ [0, +∞] \land [0, +∞] \subseteq ℝ^{}) \to F : ℝ ⟶ ℝ^{}$
56	54 55	mpan2	$⊢ F : ℝ ⟶ [0, +∞] \to F : ℝ ⟶ ℝ^{*}$
57	56	ad2antrr	$⊢ ((F : ℝ ⟶ [0, +∞] \land f \in dom \int^{1}) \land y \in ℝ^{+}) \to F : ℝ ⟶ ℝ^{*}$
58	57	ffvelcdmda	$⊢ (((F : ℝ ⟶ [0, +∞] \land f \in dom \int^{1}) \land y \in ℝ^{+}) \land z \in ℝ) \to F (z) \in ℝ^{*}$
59		xrletr	$⊢ (f (z) \in ℝ^{} \land if (f (z) = 0, 0, f (z) + y) \in ℝ^{} \land F (z) \in ℝ^{*}) \to ((f (z) \leq if (f (z) = 0, 0, f (z) + y) \land if (f (z) = 0, 0, f (z) + y) \leq F (z)) \to f (z) \leq F (z))$
60	47 53 58 59	syl3anc	$⊢ (((F : ℝ ⟶ [0, +∞] \land f \in dom \int^{1}) \land y \in ℝ^{+}) \land z \in ℝ) \to ((f (z) \leq if (f (z) = 0, 0, f (z) + y) \land if (f (z) = 0, 0, f (z) + y) \leq F (z)) \to f (z) \leq F (z))$
61	44 60	mpand	$⊢ (((F : ℝ ⟶ [0, +∞] \land f \in dom \int^{1}) \land y \in ℝ^{+}) \land z \in ℝ) \to (if (f (z) = 0, 0, f (z) + y) \leq F (z) \to f (z) \leq F (z))$
62	61	ralimdva	$⊢ ((F : ℝ ⟶ [0, +∞] \land f \in dom \int^{1}) \land y \in ℝ^{+}) \to (\forall z \in ℝ if (f (z) = 0, 0, f (z) + y) \leq F (z) \to \forall z \in ℝ f (z) \leq F (z))$
63		reex	$⊢ ℝ \in V$
64	63	a1i	$⊢ ((F : ℝ ⟶ [0, +∞] \land f \in dom \int^{1}) \land y \in ℝ^{+}) \to ℝ \in V$
65		eqidd	$⊢ ((F : ℝ ⟶ [0, +∞] \land f \in dom \int^{1}) \land y \in ℝ^{+}) \to (z \in ℝ ⟼ if (f (z) = 0, 0, f (z) + y)) = (z \in ℝ ⟼ if (f (z) = 0, 0, f (z) + y))$
66		id	$⊢ F : ℝ ⟶ [0, +∞] \to F : ℝ ⟶ [0, +∞]$
67	66	feqmptd	$⊢ F : ℝ ⟶ [0, +∞] \to F = (z \in ℝ ⟼ F (z))$
68	67	ad2antrr	$⊢ ((F : ℝ ⟶ [0, +∞] \land f \in dom \int^{1}) \land y \in ℝ^{+}) \to F = (z \in ℝ ⟼ F (z))$
69	64 52 58 65 68	ofrfval2	$⊢ ((F : ℝ ⟶ [0, +∞] \land f \in dom \int^{1}) \land y \in ℝ^{+}) \to ((z \in ℝ ⟼ if (f (z) = 0, 0, f (z) + y)) \leq_{f} F \leftrightarrow \forall z \in ℝ if (f (z) = 0, 0, f (z) + y) \leq F (z))$
70	35	feqmptd	$⊢ f \in dom \int^{1} \to f = (z \in ℝ ⟼ f (z))$
71	70	ad2antlr	$⊢ ((F : ℝ ⟶ [0, +∞] \land f \in dom \int^{1}) \land y \in ℝ^{+}) \to f = (z \in ℝ ⟼ f (z))$
72	64 46 58 71 68	ofrfval2	$⊢ ((F : ℝ ⟶ [0, +∞] \land f \in dom \int^{1}) \land y \in ℝ^{+}) \to (f \leq_{f} F \leftrightarrow \forall z \in ℝ f (z) \leq F (z))$
73	62 69 72	3imtr4d	$⊢ ((F : ℝ ⟶ [0, +∞] \land f \in dom \int^{1}) \land y \in ℝ^{+}) \to ((z \in ℝ ⟼ if (f (z) = 0, 0, f (z) + y)) \leq_{f} F \to f \leq_{f} F)$
74	73	rexlimdva	$⊢ (F : ℝ ⟶ [0, +∞] \land f \in dom \int^{1}) \to (\exists y \in ℝ^{+} (z \in ℝ ⟼ if (f (z) = 0, 0, f (z) + y)) \leq_{f} F \to f \leq_{f} F)$
75	74	anim1d	$⊢ (F : ℝ ⟶ [0, +∞] \land f \in dom \int^{1}) \to ((\exists y \in ℝ^{+} (z \in ℝ ⟼ if (f (z) = 0, 0, f (z) + y)) \leq_{f} F \land x = \int^{1} (f)) \to (f \leq_{f} F \land x = \int^{1} (f)))$
76	75	reximdva	$⊢ F : ℝ ⟶ [0, +∞] \to (\exists f \in dom \int^{1} (\exists y \in ℝ^{+} (z \in ℝ ⟼ if (f (z) = 0, 0, f (z) + y)) \leq_{f} F \land x = \int^{1} (f)) \to \exists f \in dom \int^{1} (f \leq_{f} F \land x = \int^{1} (f)))$
77	26 76	biimtrid	$⊢ F : ℝ ⟶ [0, +∞] \to (\exists g \in dom \int^{1} (\exists y \in ℝ^{+} (z \in ℝ ⟼ if (g (z) = 0, 0, g (z) + y)) \leq_{f} F \land x = \int^{1} (g)) \to \exists f \in dom \int^{1} (f \leq_{f} F \land x = \int^{1} (f)))$
78	77	ss2abdv	$⊢ F : ℝ ⟶ [0, +∞] \to \{x \| \exists g \in dom \int^{1} (\exists y \in ℝ^{+} (z \in ℝ ⟼ if (g (z) = 0, 0, g (z) + y)) \leq_{f} F \land x = \int^{1} (g))\} \subseteq \{x \| \exists f \in dom \int^{1} (f \leq_{f} F \land x = \int^{1} (f))\}$
79	78	sseld	$⊢ F : ℝ ⟶ [0, +∞] \to (b \in \{x \| \exists g \in dom \int^{1} (\exists y \in ℝ^{+} (z \in ℝ ⟼ if (g (z) = 0, 0, g (z) + y)) \leq_{f} F \land x = \int^{1} (g))\} \to b \in \{x \| \exists f \in dom \int^{1} (f \leq_{f} F \land x = \int^{1} (f))\})$
80		simp3r	$⊢ (F : ℝ ⟶ [0, +∞] \land f \in dom \int^{1} \land (f \leq_{f} F \land x = \int^{1} (f))) \to x = \int^{1} (f)$
81	9	3ad2ant2	$⊢ (F : ℝ ⟶ [0, +∞] \land f \in dom \int^{1} \land (f \leq_{f} F \land x = \int^{1} (f))) \to \int^{1} (f) \in ℝ^{*}$
82	80 81	eqeltrd	$⊢ (F : ℝ ⟶ [0, +∞] \land f \in dom \int^{1} \land (f \leq_{f} F \land x = \int^{1} (f))) \to x \in ℝ^{*}$
83	82	rexlimdv3a	$⊢ F : ℝ ⟶ [0, +∞] \to (\exists f \in dom \int^{1} (f \leq_{f} F \land x = \int^{1} (f)) \to x \in ℝ^{*})$
84	83	abssdv	$⊢ F : ℝ ⟶ [0, +∞] \to \{x \| \exists f \in dom \int^{1} (f \leq_{f} F \land x = \int^{1} (f))\} \subseteq ℝ^{*}$
85		xrsupss	$⊢ \{x \| \exists f \in dom \int^{1} (f \leq_{f} F \land x = \int^{1} (f))\} \subseteq ℝ^{} \to \exists a \in ℝ^{} (\forall b \in \{x \| \exists f \in dom \int^{1} (f \leq_{f} F \land x = \int^{1} (f))\} \neg a < b \land \forall b \in ℝ^{*} (b < a \to \exists s \in \{x \| \exists f \in dom \int^{1} (f \leq_{f} F \land x = \int^{1} (f))\} b < s))$
86	84 85	syl	$⊢ F : ℝ ⟶ [0, +∞] \to \exists a \in ℝ^{} (\forall b \in \{x \| \exists f \in dom \int^{1} (f \leq_{f} F \land x = \int^{1} (f))\} \neg a < b \land \forall b \in ℝ^{} (b < a \to \exists s \in \{x \| \exists f \in dom \int^{1} (f \leq_{f} F \land x = \int^{1} (f))\} b < s))$
87	6 86	supub	$⊢ F : ℝ ⟶ [0, +∞] \to (b \in \{x \| \exists f \in dom \int^{1} (f \leq_{f} F \land x = \int^{1} (f))\} \to \neg sup (\{x \| \exists f \in dom \int^{1} (f \leq_{f} F \land x = \int^{1} (f))\} ℝ^{*} <) < b)$
88	79 87	syld	$⊢ F : ℝ ⟶ [0, +∞] \to (b \in \{x \| \exists g \in dom \int^{1} (\exists y \in ℝ^{+} (z \in ℝ ⟼ if (g (z) = 0, 0, g (z) + y)) \leq_{f} F \land x = \int^{1} (g))\} \to \neg sup (\{x \| \exists f \in dom \int^{1} (f \leq_{f} F \land x = \int^{1} (f))\} ℝ^{*} <) < b)$
89	88	imp	$⊢ (F : ℝ ⟶ [0, +∞] \land b \in \{x \| \exists g \in dom \int^{1} (\exists y \in ℝ^{+} (z \in ℝ ⟼ if (g (z) = 0, 0, g (z) + y)) \leq_{f} F \land x = \int^{1} (g))\}) \to \neg sup (\{x \| \exists f \in dom \int^{1} (f \leq_{f} F \land x = \int^{1} (f))\} ℝ^{*} <) < b$
90		supxrlub	$⊢ (\{x \| \exists f \in dom \int^{1} (f \leq_{f} F \land x = \int^{1} (f))\} \subseteq ℝ^{} \land b \in ℝ^{}) \to (b < sup (\{x \| \exists f \in dom \int^{1} (f \leq_{f} F \land x = \int^{1} (f))\} ℝ^{*} <) \leftrightarrow \exists s \in \{x \| \exists f \in dom \int^{1} (f \leq_{f} F \land x = \int^{1} (f))\} b < s)$
91	13 90	mpan	$⊢ b \in ℝ^{} \to (b < sup (\{x \| \exists f \in dom \int^{1} (f \leq_{f} F \land x = \int^{1} (f))\} ℝ^{} <) \leftrightarrow \exists s \in \{x \| \exists f \in dom \int^{1} (f \leq_{f} F \land x = \int^{1} (f))\} b < s)$
92	91	adantl	$⊢ (F : ℝ ⟶ [0, +∞] \land b \in ℝ^{}) \to (b < sup (\{x \| \exists f \in dom \int^{1} (f \leq_{f} F \land x = \int^{1} (f))\} ℝ^{} <) \leftrightarrow \exists s \in \{x \| \exists f \in dom \int^{1} (f \leq_{f} F \land x = \int^{1} (f))\} b < s)$
93		simprrr	$⊢ ((F : ℝ ⟶ [0, +∞] \land b \in ℝ^{*}) \land (f \in dom \int^{1} \land (f \leq_{f} F \land s = \int^{1} (f)))) \to s = \int^{1} (f)$
94	93	breq2d	$⊢ ((F : ℝ ⟶ [0, +∞] \land b \in ℝ^{*}) \land (f \in dom \int^{1} \land (f \leq_{f} F \land s = \int^{1} (f)))) \to (b < s \leftrightarrow b < \int^{1} (f))$
95		simplll	$⊢ (((F : ℝ ⟶ [0, +∞] \land b \in ℝ^{*}) \land (f \in dom \int^{1} \land (f \leq_{f} F \land s = \int^{1} (f)))) \land b < \int^{1} (f)) \to F : ℝ ⟶ [0, +∞]$
96		i1f0	$⊢ ℝ \times \{0\} \in dom \int^{1}$
97		2rp	$⊢ 2 \in ℝ^{+}$
98	97	ne0ii	$⊢ ℝ^{+} \neq \emptyset$
99		ffvelcdm	$⊢ (F : ℝ ⟶ [0, +∞] \land z \in ℝ) \to F (z) \in [0, +∞]$
100		elxrge0	$⊢ F (z) \in [0, +∞] \leftrightarrow (F (z) \in ℝ^{*} \land 0 \leq F (z))$
101	99 100	sylib	$⊢ (F : ℝ ⟶ [0, +∞] \land z \in ℝ) \to (F (z) \in ℝ^{*} \land 0 \leq F (z))$
102	101	simprd	$⊢ (F : ℝ ⟶ [0, +∞] \land z \in ℝ) \to 0 \leq F (z)$
103	102	ralrimiva	$⊢ F : ℝ ⟶ [0, +∞] \to \forall z \in ℝ 0 \leq F (z)$
104	63	a1i	$⊢ F : ℝ ⟶ [0, +∞] \to ℝ \in V$
105		c0ex	$⊢ 0 \in V$
106	105	a1i	$⊢ (F : ℝ ⟶ [0, +∞] \land z \in ℝ) \to 0 \in V$
107		eqidd	$⊢ F : ℝ ⟶ [0, +∞] \to (z \in ℝ ⟼ 0) = (z \in ℝ ⟼ 0)$
108	104 106 99 107 67	ofrfval2	$⊢ F : ℝ ⟶ [0, +∞] \to ((z \in ℝ ⟼ 0) \leq_{f} F \leftrightarrow \forall z \in ℝ 0 \leq F (z))$
109	103 108	mpbird	$⊢ F : ℝ ⟶ [0, +∞] \to (z \in ℝ ⟼ 0) \leq_{f} F$
110	109	ralrimivw	$⊢ F : ℝ ⟶ [0, +∞] \to \forall y \in ℝ^{+} (z \in ℝ ⟼ 0) \leq_{f} F$
111		r19.2z	$⊢ (ℝ^{+} \neq \emptyset \land \forall y \in ℝ^{+} (z \in ℝ ⟼ 0) \leq_{f} F) \to \exists y \in ℝ^{+} (z \in ℝ ⟼ 0) \leq_{f} F$
112	98 110 111	sylancr	$⊢ F : ℝ ⟶ [0, +∞] \to \exists y \in ℝ^{+} (z \in ℝ ⟼ 0) \leq_{f} F$
113		fveq2	$⊢ g = ℝ \times \{0\} \to \int^{1} (g) = \int^{1} (ℝ \times \{0\})$
114		itg10	$⊢ \int^{1} (ℝ \times \{0\}) = 0$
115	113 114	eqtr2di	$⊢ g = ℝ \times \{0\} \to 0 = \int^{1} (g)$
116	115	biantrud	$⊢ g = ℝ \times \{0\} \to (\exists y \in ℝ^{+} (z \in ℝ ⟼ if (g (z) = 0, 0, g (z) + y)) \leq_{f} F \leftrightarrow (\exists y \in ℝ^{+} (z \in ℝ ⟼ if (g (z) = 0, 0, g (z) + y)) \leq_{f} F \land 0 = \int^{1} (g)))$
117		fveq1	$⊢ g = ℝ \times \{0\} \to g (z) = (ℝ \times \{0\}) (z)$
118	105	fvconst2	$⊢ z \in ℝ \to (ℝ \times \{0\}) (z) = 0$
119	117 118	sylan9eq	$⊢ (g = ℝ \times \{0\} \land z \in ℝ) \to g (z) = 0$
120		iftrue	$⊢ g (z) = 0 \to if (g (z) = 0, 0, g (z) + y) = 0$
121	119 120	syl	$⊢ (g = ℝ \times \{0\} \land z \in ℝ) \to if (g (z) = 0, 0, g (z) + y) = 0$
122	121	mpteq2dva	$⊢ g = ℝ \times \{0\} \to (z \in ℝ ⟼ if (g (z) = 0, 0, g (z) + y)) = (z \in ℝ ⟼ 0)$
123	122	breq1d	$⊢ g = ℝ \times \{0\} \to ((z \in ℝ ⟼ if (g (z) = 0, 0, g (z) + y)) \leq_{f} F \leftrightarrow (z \in ℝ ⟼ 0) \leq_{f} F)$
124	123	rexbidv	$⊢ g = ℝ \times \{0\} \to (\exists y \in ℝ^{+} (z \in ℝ ⟼ if (g (z) = 0, 0, g (z) + y)) \leq_{f} F \leftrightarrow \exists y \in ℝ^{+} (z \in ℝ ⟼ 0) \leq_{f} F)$
125	116 124	bitr3d	$⊢ g = ℝ \times \{0\} \to ((\exists y \in ℝ^{+} (z \in ℝ ⟼ if (g (z) = 0, 0, g (z) + y)) \leq_{f} F \land 0 = \int^{1} (g)) \leftrightarrow \exists y \in ℝ^{+} (z \in ℝ ⟼ 0) \leq_{f} F)$
126	125	rspcev	$⊢ (ℝ \times \{0\} \in dom \int^{1} \land \exists y \in ℝ^{+} (z \in ℝ ⟼ 0) \leq_{f} F) \to \exists g \in dom \int^{1} (\exists y \in ℝ^{+} (z \in ℝ ⟼ if (g (z) = 0, 0, g (z) + y)) \leq_{f} F \land 0 = \int^{1} (g))$
127	96 112 126	sylancr	$⊢ F : ℝ ⟶ [0, +∞] \to \exists g \in dom \int^{1} (\exists y \in ℝ^{+} (z \in ℝ ⟼ if (g (z) = 0, 0, g (z) + y)) \leq_{f} F \land 0 = \int^{1} (g))$
128		id	$⊢ b = −∞ \to b = −∞$
129		mnflt	$⊢ 0 \in ℝ \to −∞ < 0$
130	48 129	mp1i	$⊢ b = −∞ \to −∞ < 0$
131	128 130	eqbrtrd	$⊢ b = −∞ \to b < 0$
132		eqeq1	$⊢ a = 0 \to (a = \int^{1} (g) \leftrightarrow 0 = \int^{1} (g))$
133	132	anbi2d	$⊢ a = 0 \to ((\exists y \in ℝ^{+} (z \in ℝ ⟼ if (g (z) = 0, 0, g (z) + y)) \leq_{f} F \land a = \int^{1} (g)) \leftrightarrow (\exists y \in ℝ^{+} (z \in ℝ ⟼ if (g (z) = 0, 0, g (z) + y)) \leq_{f} F \land 0 = \int^{1} (g)))$
134	133	rexbidv	$⊢ a = 0 \to (\exists g \in dom \int^{1} (\exists y \in ℝ^{+} (z \in ℝ ⟼ if (g (z) = 0, 0, g (z) + y)) \leq_{f} F \land a = \int^{1} (g)) \leftrightarrow \exists g \in dom \int^{1} (\exists y \in ℝ^{+} (z \in ℝ ⟼ if (g (z) = 0, 0, g (z) + y)) \leq_{f} F \land 0 = \int^{1} (g)))$
135		breq2	$⊢ a = 0 \to (b < a \leftrightarrow b < 0)$
136	134 135	anbi12d	$⊢ a = 0 \to ((\exists g \in dom \int^{1} (\exists y \in ℝ^{+} (z \in ℝ ⟼ if (g (z) = 0, 0, g (z) + y)) \leq_{f} F \land a = \int^{1} (g)) \land b < a) \leftrightarrow (\exists g \in dom \int^{1} (\exists y \in ℝ^{+} (z \in ℝ ⟼ if (g (z) = 0, 0, g (z) + y)) \leq_{f} F \land 0 = \int^{1} (g)) \land b < 0))$
137	105 136	spcev	$⊢ (\exists g \in dom \int^{1} (\exists y \in ℝ^{+} (z \in ℝ ⟼ if (g (z) = 0, 0, g (z) + y)) \leq_{f} F \land 0 = \int^{1} (g)) \land b < 0) \to \exists a (\exists g \in dom \int^{1} (\exists y \in ℝ^{+} (z \in ℝ ⟼ if (g (z) = 0, 0, g (z) + y)) \leq_{f} F \land a = \int^{1} (g)) \land b < a)$
138	127 131 137	syl2an	$⊢ (F : ℝ ⟶ [0, +∞] \land b = −∞) \to \exists a (\exists g \in dom \int^{1} (\exists y \in ℝ^{+} (z \in ℝ ⟼ if (g (z) = 0, 0, g (z) + y)) \leq_{f} F \land a = \int^{1} (g)) \land b < a)$
139	95 138	sylan	$⊢ ((((F : ℝ ⟶ [0, +∞] \land b \in ℝ^{*}) \land (f \in dom \int^{1} \land (f \leq_{f} F \land s = \int^{1} (f)))) \land b < \int^{1} (f)) \land b = −∞) \to \exists a (\exists g \in dom \int^{1} (\exists y \in ℝ^{+} (z \in ℝ ⟼ if (g (z) = 0, 0, g (z) + y)) \leq_{f} F \land a = \int^{1} (g)) \land b < a)$
140		simp-4r	$⊢ ((((F : ℝ ⟶ [0, +∞] \land b \in ℝ^{}) \land (f \in dom \int^{1} \land (f \leq_{f} F \land s = \int^{1} (f)))) \land b < \int^{1} (f)) \land b \neq −∞) \to b \in ℝ^{}$
141	8	adantr	$⊢ (f \in dom \int^{1} \land (f \leq_{f} F \land s = \int^{1} (f))) \to \int^{1} (f) \in ℝ$
142	141	ad3antlr	$⊢ ((((F : ℝ ⟶ [0, +∞] \land b \in ℝ^{*}) \land (f \in dom \int^{1} \land (f \leq_{f} F \land s = \int^{1} (f)))) \land b < \int^{1} (f)) \land b \neq −∞) \to \int^{1} (f) \in ℝ$
143		simpllr	$⊢ (((F : ℝ ⟶ [0, +∞] \land b \in ℝ^{}) \land (f \in dom \int^{1} \land (f \leq_{f} F \land s = \int^{1} (f)))) \land b < \int^{1} (f)) \to b \in ℝ^{}$
144		ngtmnft	$⊢ b \in ℝ^{*} \to (b = −∞ \leftrightarrow \neg −∞ < b)$
145	144	biimprd	$⊢ b \in ℝ^{*} \to (\neg −∞ < b \to b = −∞)$
146	145	necon1ad	$⊢ b \in ℝ^{*} \to (b \neq −∞ \to −∞ < b)$
147	146	imp	$⊢ (b \in ℝ^{*} \land b \neq −∞) \to −∞ < b$
148	143 147	sylan	$⊢ ((((F : ℝ ⟶ [0, +∞] \land b \in ℝ^{*}) \land (f \in dom \int^{1} \land (f \leq_{f} F \land s = \int^{1} (f)))) \land b < \int^{1} (f)) \land b \neq −∞) \to −∞ < b$
149		simpr	$⊢ (F : ℝ ⟶ [0, +∞] \land b \in ℝ^{}) \to b \in ℝ^{}$
150	9	adantr	$⊢ (f \in dom \int^{1} \land (f \leq_{f} F \land s = \int^{1} (f))) \to \int^{1} (f) \in ℝ^{*}$
151	149 150	anim12i	$⊢ ((F : ℝ ⟶ [0, +∞] \land b \in ℝ^{}) \land (f \in dom \int^{1} \land (f \leq_{f} F \land s = \int^{1} (f)))) \to (b \in ℝ^{} \land \int^{1} (f) \in ℝ^{*})$
152		xrltle	$⊢ (b \in ℝ^{} \land \int^{1} (f) \in ℝ^{}) \to (b < \int^{1} (f) \to b \leq \int^{1} (f))$
153	152	imp	$⊢ ((b \in ℝ^{} \land \int^{1} (f) \in ℝ^{}) \land b < \int^{1} (f)) \to b \leq \int^{1} (f)$
154	151 153	sylan	$⊢ (((F : ℝ ⟶ [0, +∞] \land b \in ℝ^{*}) \land (f \in dom \int^{1} \land (f \leq_{f} F \land s = \int^{1} (f)))) \land b < \int^{1} (f)) \to b \leq \int^{1} (f)$
155	154	adantr	$⊢ ((((F : ℝ ⟶ [0, +∞] \land b \in ℝ^{*}) \land (f \in dom \int^{1} \land (f \leq_{f} F \land s = \int^{1} (f)))) \land b < \int^{1} (f)) \land b \neq −∞) \to b \leq \int^{1} (f)$
156		xrre	$⊢ ((b \in ℝ^{*} \land \int^{1} (f) \in ℝ) \land (−∞ < b \land b \leq \int^{1} (f))) \to b \in ℝ$
157	140 142 148 155 156	syl22anc	$⊢ ((((F : ℝ ⟶ [0, +∞] \land b \in ℝ^{*}) \land (f \in dom \int^{1} \land (f \leq_{f} F \land s = \int^{1} (f)))) \land b < \int^{1} (f)) \land b \neq −∞) \to b \in ℝ$
158	127	ad3antrrr	$⊢ (((F : ℝ ⟶ [0, +∞] \land (f \in dom \int^{1} \land (f \leq_{f} F \land s = \int^{1} (f)))) \land (b < \int^{1} (f) \land b \in ℝ)) \land {vol}^{*} (f^{-1} [ran f ∖ \{0\}]) = 0) \to \exists g \in dom \int^{1} (\exists y \in ℝ^{+} (z \in ℝ ⟼ if (g (z) = 0, 0, g (z) + y)) \leq_{f} F \land 0 = \int^{1} (g))$
159		simplrl	$⊢ (((F : ℝ ⟶ [0, +∞] \land (f \in dom \int^{1} \land (f \leq_{f} F \land s = \int^{1} (f)))) \land (b < \int^{1} (f) \land b \in ℝ)) \land {vol}^{*} (f^{-1} [ran f ∖ \{0\}]) = 0) \to b < \int^{1} (f)$
160		simplrl	$⊢ ((F : ℝ ⟶ [0, +∞] \land (f \in dom \int^{1} \land (f \leq_{f} F \land s = \int^{1} (f)))) \land (b < \int^{1} (f) \land b \in ℝ)) \to f \in dom \int^{1}$
161		simpl	$⊢ (f \in dom \int^{1} \land {vol}^{*} (f^{-1} [ran f ∖ \{0\}]) = 0) \to f \in dom \int^{1}$
162		cnvimass	$⊢ f^{-1} [ran f ∖ \{0\}] \subseteq dom f$
163	162 35	fssdm	$⊢ f \in dom \int^{1} \to f^{-1} [ran f ∖ \{0\}] \subseteq ℝ$
164	163	adantr	$⊢ (f \in dom \int^{1} \land {vol}^{*} (f^{-1} [ran f ∖ \{0\}]) = 0) \to f^{-1} [ran f ∖ \{0\}] \subseteq ℝ$
165		simpr	$⊢ (f \in dom \int^{1} \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) = 0) \to {vol}^{} (f^{-1} [ran f ∖ \{0\}]) = 0$
166		fdm	$⊢ f : ℝ ⟶ ℝ \to dom f = ℝ$
167	166	eqcomd	$⊢ f : ℝ ⟶ ℝ \to ℝ = dom f$
168		ffun	$⊢ f : ℝ ⟶ ℝ \to Fun f$
169		difpreima	$⊢ Fun f \to f^{-1} [ran f ∖ \{0\}] = f^{-1} [ran f] ∖ f^{-1} [\{0\}]$
170	168 169	syl	$⊢ f : ℝ ⟶ ℝ \to f^{-1} [ran f ∖ \{0\}] = f^{-1} [ran f] ∖ f^{-1} [\{0\}]$
171		cnvimarndm	$⊢ f^{-1} [ran f] = dom f$
172	171	difeq1i	$⊢ f^{-1} [ran f] ∖ f^{-1} [\{0\}] = dom f ∖ f^{-1} [\{0\}]$
173	170 172	eqtrdi	$⊢ f : ℝ ⟶ ℝ \to f^{-1} [ran f ∖ \{0\}] = dom f ∖ f^{-1} [\{0\}]$
174	167 173	difeq12d	$⊢ f : ℝ ⟶ ℝ \to ℝ ∖ f^{-1} [ran f ∖ \{0\}] = dom f ∖ (dom f ∖ f^{-1} [\{0\}])$
175		cnvimass	$⊢ f^{-1} [\{0\}] \subseteq dom f$
176		dfss4	$⊢ f^{-1} [\{0\}] \subseteq dom f \leftrightarrow dom f ∖ (dom f ∖ f^{-1} [\{0\}]) = f^{-1} [\{0\}]$
177	175 176	mpbi	$⊢ dom f ∖ (dom f ∖ f^{-1} [\{0\}]) = f^{-1} [\{0\}]$
178	174 177	eqtrdi	$⊢ f : ℝ ⟶ ℝ \to ℝ ∖ f^{-1} [ran f ∖ \{0\}] = f^{-1} [\{0\}]$
179	178	eleq2d	$⊢ f : ℝ ⟶ ℝ \to (z \in (ℝ ∖ f^{-1} [ran f ∖ \{0\}]) \leftrightarrow z \in f^{-1} [\{0\}])$
180		ffn	$⊢ f : ℝ ⟶ ℝ \to f Fn ℝ$
181		fniniseg	$⊢ f Fn ℝ \to (z \in f^{-1} [\{0\}] \leftrightarrow (z \in ℝ \land f (z) = 0))$
182		simpr	$⊢ (z \in ℝ \land f (z) = 0) \to f (z) = 0$
183	181 182	biimtrdi	$⊢ f Fn ℝ \to (z \in f^{-1} [\{0\}] \to f (z) = 0)$
184	180 183	syl	$⊢ f : ℝ ⟶ ℝ \to (z \in f^{-1} [\{0\}] \to f (z) = 0)$
185	179 184	sylbid	$⊢ f : ℝ ⟶ ℝ \to (z \in (ℝ ∖ f^{-1} [ran f ∖ \{0\}]) \to f (z) = 0)$
186	35 185	syl	$⊢ f \in dom \int^{1} \to (z \in (ℝ ∖ f^{-1} [ran f ∖ \{0\}]) \to f (z) = 0)$
187	186	imp	$⊢ (f \in dom \int^{1} \land z \in (ℝ ∖ f^{-1} [ran f ∖ \{0\}])) \to f (z) = 0$
188	187	adantlr	$⊢ ((f \in dom \int^{1} \land {vol}^{*} (f^{-1} [ran f ∖ \{0\}]) = 0) \land z \in (ℝ ∖ f^{-1} [ran f ∖ \{0\}])) \to f (z) = 0$
189	161 164 165 188	itg10a	$⊢ (f \in dom \int^{1} \land {vol}^{*} (f^{-1} [ran f ∖ \{0\}]) = 0) \to \int^{1} (f) = 0$
190	160 189	sylan	$⊢ (((F : ℝ ⟶ [0, +∞] \land (f \in dom \int^{1} \land (f \leq_{f} F \land s = \int^{1} (f)))) \land (b < \int^{1} (f) \land b \in ℝ)) \land {vol}^{*} (f^{-1} [ran f ∖ \{0\}]) = 0) \to \int^{1} (f) = 0$
191	159 190	breqtrd	$⊢ (((F : ℝ ⟶ [0, +∞] \land (f \in dom \int^{1} \land (f \leq_{f} F \land s = \int^{1} (f)))) \land (b < \int^{1} (f) \land b \in ℝ)) \land {vol}^{*} (f^{-1} [ran f ∖ \{0\}]) = 0) \to b < 0$
192	158 191 137	syl2anc	$⊢ (((F : ℝ ⟶ [0, +∞] \land (f \in dom \int^{1} \land (f \leq_{f} F \land s = \int^{1} (f)))) \land (b < \int^{1} (f) \land b \in ℝ)) \land {vol}^{*} (f^{-1} [ran f ∖ \{0\}]) = 0) \to \exists a (\exists g \in dom \int^{1} (\exists y \in ℝ^{+} (z \in ℝ ⟼ if (g (z) = 0, 0, g (z) + y)) \leq_{f} F \land a = \int^{1} (g)) \land b < a)$
193		simprl	$⊢ (F : ℝ ⟶ [0, +∞] \land (f \in dom \int^{1} \land (f \leq_{f} F \land s = \int^{1} (f)))) \to f \in dom \int^{1}$
194		simpr	$⊢ (b < \int^{1} (f) \land b \in ℝ) \to b \in ℝ$
195	193 194	anim12i	$⊢ ((F : ℝ ⟶ [0, +∞] \land (f \in dom \int^{1} \land (f \leq_{f} F \land s = \int^{1} (f)))) \land (b < \int^{1} (f) \land b \in ℝ)) \to (f \in dom \int^{1} \land b \in ℝ)$
196	63	a1i	$⊢ ((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{*} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to ℝ \in V$
197		fvex	$⊢ f (u) \in V$
198	197	a1i	$⊢ (((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{*} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \land u \in ℝ) \to f (u) \in V$
199		ovex	$⊢ \frac{\int^{1} (f) - b}{2 {vol}^{*} (f^{-1} [ran f ∖ \{0\}])} \in V$
200	199 105	ifex	$⊢ if (u \in f^{-1} [ran f ∖ \{0\}], \frac{\int^{1} (f) - b}{2 {vol}^{*} (f^{-1} [ran f ∖ \{0\}])}, 0) \in V$
201	200	a1i	$⊢ (((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \land u \in ℝ) \to if (u \in f^{-1} [ran f ∖ \{0\}], \frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}, 0) \in V$
202	35	feqmptd	$⊢ f \in dom \int^{1} \to f = (u \in ℝ ⟼ f (u))$
203	202	ad2antrr	$⊢ ((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{*} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to f = (u \in ℝ ⟼ f (u))$
204		eqidd	$⊢ ((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to (u \in ℝ ⟼ if (u \in f^{-1} [ran f ∖ \{0\}], \frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}, 0)) = (u \in ℝ ⟼ if (u \in f^{-1} [ran f ∖ \{0\}], \frac{\int^{1} (f) - b}{2 {vol}^{*} (f^{-1} [ran f ∖ \{0\}])}, 0))$
205	196 198 201 203 204	offval2	$⊢ ((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to f -_{f} (u \in ℝ ⟼ if (u \in f^{-1} [ran f ∖ \{0\}], \frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}, 0)) = (u \in ℝ ⟼ f (u) - if (u \in f^{-1} [ran f ∖ \{0\}], \frac{\int^{1} (f) - b}{2 {vol}^{*} (f^{-1} [ran f ∖ \{0\}])}, 0))$
206		ovif2	$⊢ f (u) - if (u \in f^{-1} [ran f ∖ \{0\}], \frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}, 0) = if (u \in f^{-1} [ran f ∖ \{0\}], f (u) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), f (u) - 0)$
207	171 166	eqtrid	$⊢ f : ℝ ⟶ ℝ \to f^{-1} [ran f] = ℝ$
208	207	difeq1d	$⊢ f : ℝ ⟶ ℝ \to f^{-1} [ran f] ∖ f^{-1} [\{0\}] = ℝ ∖ f^{-1} [\{0\}]$
209	170 208	eqtrd	$⊢ f : ℝ ⟶ ℝ \to f^{-1} [ran f ∖ \{0\}] = ℝ ∖ f^{-1} [\{0\}]$
210	209	eleq2d	$⊢ f : ℝ ⟶ ℝ \to (u \in f^{-1} [ran f ∖ \{0\}] \leftrightarrow u \in (ℝ ∖ f^{-1} [\{0\}]))$
211	35 210	syl	$⊢ f \in dom \int^{1} \to (u \in f^{-1} [ran f ∖ \{0\}] \leftrightarrow u \in (ℝ ∖ f^{-1} [\{0\}]))$
212	211	ad3antrrr	$⊢ (((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{*} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \land u \in ℝ) \to (u \in f^{-1} [ran f ∖ \{0\}] \leftrightarrow u \in (ℝ ∖ f^{-1} [\{0\}]))$
213		simpr	$⊢ (((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{*} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \land u \in ℝ) \to u \in ℝ$
214	213	biantrurd	$⊢ (((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{*} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \land u \in ℝ) \to (\neg u \in f^{-1} [\{0\}] \leftrightarrow (u \in ℝ \land \neg u \in f^{-1} [\{0\}]))$
215		eldif	$⊢ u \in (ℝ ∖ f^{-1} [\{0\}]) \leftrightarrow (u \in ℝ \land \neg u \in f^{-1} [\{0\}])$
216	214 215	bitr4di	$⊢ (((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{*} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \land u \in ℝ) \to (\neg u \in f^{-1} [\{0\}] \leftrightarrow u \in (ℝ ∖ f^{-1} [\{0\}]))$
217	212 216	bitr4d	$⊢ (((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{*} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \land u \in ℝ) \to (u \in f^{-1} [ran f ∖ \{0\}] \leftrightarrow \neg u \in f^{-1} [\{0\}])$
218	217	con2bid	$⊢ (((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{*} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \land u \in ℝ) \to (u \in f^{-1} [\{0\}] \leftrightarrow \neg u \in f^{-1} [ran f ∖ \{0\}])$
219		fniniseg	$⊢ f Fn ℝ \to (u \in f^{-1} [\{0\}] \leftrightarrow (u \in ℝ \land f (u) = 0))$
220	35 180 219	3syl	$⊢ f \in dom \int^{1} \to (u \in f^{-1} [\{0\}] \leftrightarrow (u \in ℝ \land f (u) = 0))$
221	220	ad3antrrr	$⊢ (((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{*} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \land u \in ℝ) \to (u \in f^{-1} [\{0\}] \leftrightarrow (u \in ℝ \land f (u) = 0))$
222	218 221	bitr3d	$⊢ (((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{*} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \land u \in ℝ) \to (\neg u \in f^{-1} [ran f ∖ \{0\}] \leftrightarrow (u \in ℝ \land f (u) = 0))$
223		oveq1	$⊢ f (u) = 0 \to f (u) - 0 = 0 - 0$
224		0m0e0	$⊢ 0 - 0 = 0$
225	223 224	eqtrdi	$⊢ f (u) = 0 \to f (u) - 0 = 0$
226	225	adantl	$⊢ (u \in ℝ \land f (u) = 0) \to f (u) - 0 = 0$
227	222 226	biimtrdi	$⊢ (((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{*} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \land u \in ℝ) \to (\neg u \in f^{-1} [ran f ∖ \{0\}] \to f (u) - 0 = 0)$
228	227	imp	$⊢ ((((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{*} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \land u \in ℝ) \land \neg u \in f^{-1} [ran f ∖ \{0\}]) \to f (u) - 0 = 0$
229	228	ifeq2da	$⊢ (((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \land u \in ℝ) \to if (u \in f^{-1} [ran f ∖ \{0\}], f (u) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), f (u) - 0) = if (u \in f^{-1} [ran f ∖ \{0\}], f (u) - (\frac{\int^{1} (f) - b}{2 {vol}^{*} (f^{-1} [ran f ∖ \{0\}])}), 0)$
230	206 229	eqtrid	$⊢ (((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \land u \in ℝ) \to f (u) - if (u \in f^{-1} [ran f ∖ \{0\}], \frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}, 0) = if (u \in f^{-1} [ran f ∖ \{0\}], f (u) - (\frac{\int^{1} (f) - b}{2 {vol}^{*} (f^{-1} [ran f ∖ \{0\}])}), 0)$
231	230	mpteq2dva	$⊢ ((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to (u \in ℝ ⟼ f (u) - if (u \in f^{-1} [ran f ∖ \{0\}], \frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}, 0)) = (u \in ℝ ⟼ if (u \in f^{-1} [ran f ∖ \{0\}], f (u) - (\frac{\int^{1} (f) - b}{2 {vol}^{*} (f^{-1} [ran f ∖ \{0\}])}), 0))$
232	205 231	eqtrd	$⊢ ((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to f -_{f} (u \in ℝ ⟼ if (u \in f^{-1} [ran f ∖ \{0\}], \frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}, 0)) = (u \in ℝ ⟼ if (u \in f^{-1} [ran f ∖ \{0\}], f (u) - (\frac{\int^{1} (f) - b}{2 {vol}^{*} (f^{-1} [ran f ∖ \{0\}])}), 0))$
233		simpll	$⊢ ((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{*} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to f \in dom \int^{1}$
234	199	a1i	$⊢ (((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \land u \in ℝ) \to \frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])} \in V$
235		1ex	$⊢ 1 \in V$
236	235 105	ifex	$⊢ if (u \in f^{-1} [ran f ∖ \{0\}], 1, 0) \in V$
237	236	a1i	$⊢ (((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{*} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \land u \in ℝ) \to if (u \in f^{-1} [ran f ∖ \{0\}], 1, 0) \in V$
238		fconstmpt	$⊢ ℝ \times \{\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}\} = (u \in ℝ ⟼ \frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])})$
239	238	a1i	$⊢ ((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to ℝ \times \{\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}\} = (u \in ℝ ⟼ \frac{\int^{1} (f) - b}{2 {vol}^{*} (f^{-1} [ran f ∖ \{0\}])})$
240		eqidd	$⊢ ((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{*} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to (u \in ℝ ⟼ if (u \in f^{-1} [ran f ∖ \{0\}], 1, 0)) = (u \in ℝ ⟼ if (u \in f^{-1} [ran f ∖ \{0\}], 1, 0))$
241	196 234 237 239 240	offval2	$⊢ ((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to (ℝ \times \{\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}\}) \times_{f} (u \in ℝ ⟼ if (u \in f^{-1} [ran f ∖ \{0\}], 1, 0)) = (u \in ℝ ⟼ (\frac{\int^{1} (f) - b}{2 {vol}^{*} (f^{-1} [ran f ∖ \{0\}])}) if (u \in f^{-1} [ran f ∖ \{0\}], 1, 0))$
242		ovif2	$⊢ (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) if (u \in f^{-1} [ran f ∖ \{0\}], 1, 0) = if (u \in f^{-1} [ran f ∖ \{0\}], (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \cdot 1, (\frac{\int^{1} (f) - b}{2 {vol}^{*} (f^{-1} [ran f ∖ \{0\}])}) \cdot 0)$
243		resubcl	$⊢ (\int^{1} (f) \in ℝ \land b \in ℝ) \to \int^{1} (f) - b \in ℝ$
244	8 243	sylan	$⊢ (f \in dom \int^{1} \land b \in ℝ) \to \int^{1} (f) - b \in ℝ$
245	244	adantr	$⊢ ((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{*} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to \int^{1} (f) - b \in ℝ$
246		2re	$⊢ 2 \in ℝ$
247		i1fima	$⊢ f \in dom \int^{1} \to f^{-1} [ran f ∖ \{0\}] \in dom vol$
248		mblvol	$⊢ f^{-1} [ran f ∖ \{0\}] \in dom vol \to vol (f^{-1} [ran f ∖ \{0\}]) = {vol}^{*} (f^{-1} [ran f ∖ \{0\}])$
249	247 248	syl	$⊢ f \in dom \int^{1} \to vol (f^{-1} [ran f ∖ \{0\}]) = {vol}^{*} (f^{-1} [ran f ∖ \{0\}])$
250		neldifsn	$⊢ \neg 0 \in (ran f ∖ \{0\})$
251		i1fima2	$⊢ (f \in dom \int^{1} \land \neg 0 \in (ran f ∖ \{0\})) \to vol (f^{-1} [ran f ∖ \{0\}]) \in ℝ$
252	250 251	mpan2	$⊢ f \in dom \int^{1} \to vol (f^{-1} [ran f ∖ \{0\}]) \in ℝ$
253	249 252	eqeltrrd	$⊢ f \in dom \int^{1} \to {vol}^{*} (f^{-1} [ran f ∖ \{0\}]) \in ℝ$
254		remulcl	$⊢ (2 \in ℝ \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \in ℝ) \to 2 {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \in ℝ$
255	246 253 254	sylancr	$⊢ f \in dom \int^{1} \to 2 {vol}^{*} (f^{-1} [ran f ∖ \{0\}]) \in ℝ$
256	255	ad2antrr	$⊢ ((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to 2 {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \in ℝ$
257		2cnd	$⊢ ((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{*} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to 2 \in ℂ$
258	253	ad2antrr	$⊢ ((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \in ℝ$
259	258	recnd	$⊢ ((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \in ℂ$
260		2ne0	$⊢ 2 \neq 0$
261	260	a1i	$⊢ ((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{*} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to 2 \neq 0$
262		simpr	$⊢ ((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0$
263	257 259 261 262	mulne0d	$⊢ ((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to 2 {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0$
264	245 256 263	redivcld	$⊢ ((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to \frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])} \in ℝ$
265	264	recnd	$⊢ ((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to \frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])} \in ℂ$
266	265	mulridd	$⊢ ((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \cdot 1 = \frac{\int^{1} (f) - b}{2 {vol}^{*} (f^{-1} [ran f ∖ \{0\}])}$
267	265	mul01d	$⊢ ((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \cdot 0 = 0$
268	266 267	ifeq12d	$⊢ ((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to if (u \in f^{-1} [ran f ∖ \{0\}], (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \cdot 1, (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \cdot 0) = if (u \in f^{-1} [ran f ∖ \{0\}], \frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}, 0)$
269	242 268	eqtrid	$⊢ ((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) if (u \in f^{-1} [ran f ∖ \{0\}], 1, 0) = if (u \in f^{-1} [ran f ∖ \{0\}], \frac{\int^{1} (f) - b}{2 {vol}^{*} (f^{-1} [ran f ∖ \{0\}])}, 0)$
270	269	mpteq2dv	$⊢ ((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to (u \in ℝ ⟼ (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) if (u \in f^{-1} [ran f ∖ \{0\}], 1, 0)) = (u \in ℝ ⟼ if (u \in f^{-1} [ran f ∖ \{0\}], \frac{\int^{1} (f) - b}{2 {vol}^{*} (f^{-1} [ran f ∖ \{0\}])}, 0))$
271	241 270	eqtrd	$⊢ ((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to (ℝ \times \{\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}\}) \times_{f} (u \in ℝ ⟼ if (u \in f^{-1} [ran f ∖ \{0\}], 1, 0)) = (u \in ℝ ⟼ if (u \in f^{-1} [ran f ∖ \{0\}], \frac{\int^{1} (f) - b}{2 {vol}^{*} (f^{-1} [ran f ∖ \{0\}])}, 0))$
272		eqid	$⊢ (u \in ℝ ⟼ if (u \in f^{-1} [ran f ∖ \{0\}], 1, 0)) = (u \in ℝ ⟼ if (u \in f^{-1} [ran f ∖ \{0\}], 1, 0))$
273	272	i1f1	$⊢ (f^{-1} [ran f ∖ \{0\}] \in dom vol \land vol (f^{-1} [ran f ∖ \{0\}]) \in ℝ) \to (u \in ℝ ⟼ if (u \in f^{-1} [ran f ∖ \{0\}], 1, 0)) \in dom \int^{1}$
274	247 252 273	syl2anc	$⊢ f \in dom \int^{1} \to (u \in ℝ ⟼ if (u \in f^{-1} [ran f ∖ \{0\}], 1, 0)) \in dom \int^{1}$
275	274	ad2antrr	$⊢ ((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{*} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to (u \in ℝ ⟼ if (u \in f^{-1} [ran f ∖ \{0\}], 1, 0)) \in dom \int^{1}$
276	275 264	i1fmulc	$⊢ ((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to (ℝ \times \{\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}\}) \times_{f} (u \in ℝ ⟼ if (u \in f^{-1} [ran f ∖ \{0\}], 1, 0)) \in dom \int^{1}$
277	271 276	eqeltrrd	$⊢ ((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to (u \in ℝ ⟼ if (u \in f^{-1} [ran f ∖ \{0\}], \frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}, 0)) \in dom \int^{1}$
278		i1fsub	$⊢ (f \in dom \int^{1} \land (u \in ℝ ⟼ if (u \in f^{-1} [ran f ∖ \{0\}], \frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}, 0)) \in dom \int^{1}) \to f -_{f} (u \in ℝ ⟼ if (u \in f^{-1} [ran f ∖ \{0\}], \frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}, 0)) \in dom \int^{1}$
279	233 277 278	syl2anc	$⊢ ((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to f -_{f} (u \in ℝ ⟼ if (u \in f^{-1} [ran f ∖ \{0\}], \frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}, 0)) \in dom \int^{1}$
280	232 279	eqeltrrd	$⊢ ((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to (u \in ℝ ⟼ if (u \in f^{-1} [ran f ∖ \{0\}], f (u) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0)) \in dom \int^{1}$
281		iftrue	$⊢ (0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]) \to if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) = f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])})$
282		iftrue	$⊢ z \in f^{-1} [ran f ∖ \{0\}] \to if (z \in f^{-1} [ran f ∖ \{0\}], f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) = f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])})$
283	282	breq2d	$⊢ z \in f^{-1} [ran f ∖ \{0\}] \to (0 \leq if (z \in f^{-1} [ran f ∖ \{0\}], f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) \leftrightarrow 0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}))$
284	283 282	ifbieq1d	$⊢ z \in f^{-1} [ran f ∖ \{0\}] \to if (0 \leq if (z \in f^{-1} [ran f ∖ \{0\}], f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0), if (z \in f^{-1} [ran f ∖ \{0\}], f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0), 0) = if (0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0)$
285		iftrue	$⊢ 0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \to if (0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) = f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])})$
286	284 285	sylan9eqr	$⊢ (0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]) \to if (0 \leq if (z \in f^{-1} [ran f ∖ \{0\}], f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0), if (z \in f^{-1} [ran f ∖ \{0\}], f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0), 0) = f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])})$
287	281 286	eqtr4d	$⊢ (0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]) \to if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) = if (0 \leq if (z \in f^{-1} [ran f ∖ \{0\}], f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0), if (z \in f^{-1} [ran f ∖ \{0\}], f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{*} (f^{-1} [ran f ∖ \{0\}])}), 0), 0)$
288		iffalse	$⊢ \neg (0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]) \to if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{*} (f^{-1} [ran f ∖ \{0\}])}), 0) = 0$
289		ianor	$⊢ \neg (0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]) \leftrightarrow (\neg 0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \lor \neg z \in f^{-1} [ran f ∖ \{0\}])$
290	283	ifbid	$⊢ z \in f^{-1} [ran f ∖ \{0\}] \to if (0 \leq if (z \in f^{-1} [ran f ∖ \{0\}], f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0), if (z \in f^{-1} [ran f ∖ \{0\}], f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0), 0) = if (0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), if (z \in f^{-1} [ran f ∖ \{0\}], f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0), 0)$
291		iffalse	$⊢ \neg 0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \to if (0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), if (z \in f^{-1} [ran f ∖ \{0\}], f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{*} (f^{-1} [ran f ∖ \{0\}])}), 0), 0) = 0$
292	290 291	sylan9eqr	$⊢ (\neg 0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]) \to if (0 \leq if (z \in f^{-1} [ran f ∖ \{0\}], f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0), if (z \in f^{-1} [ran f ∖ \{0\}], f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{*} (f^{-1} [ran f ∖ \{0\}])}), 0), 0) = 0$
293	292	ex	$⊢ \neg 0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \to (z \in f^{-1} [ran f ∖ \{0\}] \to if (0 \leq if (z \in f^{-1} [ran f ∖ \{0\}], f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0), if (z \in f^{-1} [ran f ∖ \{0\}], f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{*} (f^{-1} [ran f ∖ \{0\}])}), 0), 0) = 0)$
294		iffalse	$⊢ \neg z \in f^{-1} [ran f ∖ \{0\}] \to if (z \in f^{-1} [ran f ∖ \{0\}], f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{*} (f^{-1} [ran f ∖ \{0\}])}), 0) = 0$
295		eqid	$⊢ 0 = 0$
296		eqeq1	$⊢ if (z \in f^{-1} [ran f ∖ \{0\}], f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) = if (0 \leq if (z \in f^{-1} [ran f ∖ \{0\}], f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0), if (z \in f^{-1} [ran f ∖ \{0\}], f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0), 0) \to (if (z \in f^{-1} [ran f ∖ \{0\}], f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) = 0 \leftrightarrow if (0 \leq if (z \in f^{-1} [ran f ∖ \{0\}], f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0), if (z \in f^{-1} [ran f ∖ \{0\}], f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0), 0) = 0)$
297		eqeq1	$⊢ 0 = if (0 \leq if (z \in f^{-1} [ran f ∖ \{0\}], f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0), if (z \in f^{-1} [ran f ∖ \{0\}], f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0), 0) \to (0 = 0 \leftrightarrow if (0 \leq if (z \in f^{-1} [ran f ∖ \{0\}], f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0), if (z \in f^{-1} [ran f ∖ \{0\}], f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0), 0) = 0)$
298	296 297	ifboth	$⊢ (if (z \in f^{-1} [ran f ∖ \{0\}], f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) = 0 \land 0 = 0) \to if (0 \leq if (z \in f^{-1} [ran f ∖ \{0\}], f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0), if (z \in f^{-1} [ran f ∖ \{0\}], f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{*} (f^{-1} [ran f ∖ \{0\}])}), 0), 0) = 0$
299	294 295 298	sylancl	$⊢ \neg z \in f^{-1} [ran f ∖ \{0\}] \to if (0 \leq if (z \in f^{-1} [ran f ∖ \{0\}], f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0), if (z \in f^{-1} [ran f ∖ \{0\}], f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0), 0) = 0$
300	293 299	pm2.61d1	$⊢ \neg 0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \to if (0 \leq if (z \in f^{-1} [ran f ∖ \{0\}], f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0), if (z \in f^{-1} [ran f ∖ \{0\}], f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{*} (f^{-1} [ran f ∖ \{0\}])}), 0), 0) = 0$
301	300 299	jaoi	$⊢ (\neg 0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \lor \neg z \in f^{-1} [ran f ∖ \{0\}]) \to if (0 \leq if (z \in f^{-1} [ran f ∖ \{0\}], f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0), if (z \in f^{-1} [ran f ∖ \{0\}], f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{*} (f^{-1} [ran f ∖ \{0\}])}), 0), 0) = 0$
302	289 301	sylbi	$⊢ \neg (0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]) \to if (0 \leq if (z \in f^{-1} [ran f ∖ \{0\}], f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0), if (z \in f^{-1} [ran f ∖ \{0\}], f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{*} (f^{-1} [ran f ∖ \{0\}])}), 0), 0) = 0$
303	288 302	eqtr4d	$⊢ \neg (0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]) \to if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) = if (0 \leq if (z \in f^{-1} [ran f ∖ \{0\}], f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0), if (z \in f^{-1} [ran f ∖ \{0\}], f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{*} (f^{-1} [ran f ∖ \{0\}])}), 0), 0)$
304	287 303	pm2.61i	$⊢ if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) = if (0 \leq if (z \in f^{-1} [ran f ∖ \{0\}], f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0), if (z \in f^{-1} [ran f ∖ \{0\}], f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0), 0)$
305		eleq1w	$⊢ u = z \to (u \in f^{-1} [ran f ∖ \{0\}] \leftrightarrow z \in f^{-1} [ran f ∖ \{0\}])$
306		fveq2	$⊢ u = z \to f (u) = f (z)$
307	306	oveq1d	$⊢ u = z \to f (u) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) = f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])})$
308	305 307	ifbieq1d	$⊢ u = z \to if (u \in f^{-1} [ran f ∖ \{0\}], f (u) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) = if (z \in f^{-1} [ran f ∖ \{0\}], f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0)$
309		eqid	$⊢ (u \in ℝ ⟼ if (u \in f^{-1} [ran f ∖ \{0\}], f (u) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0)) = (u \in ℝ ⟼ if (u \in f^{-1} [ran f ∖ \{0\}], f (u) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0))$
310		ovex	$⊢ f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{*} (f^{-1} [ran f ∖ \{0\}])}) \in V$
311	310 105	ifex	$⊢ if (z \in f^{-1} [ran f ∖ \{0\}], f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{*} (f^{-1} [ran f ∖ \{0\}])}), 0) \in V$
312	308 309 311	fvmpt	$⊢ z \in ℝ \to (u \in ℝ ⟼ if (u \in f^{-1} [ran f ∖ \{0\}], f (u) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0)) (z) = if (z \in f^{-1} [ran f ∖ \{0\}], f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0)$
313	312	breq2d	$⊢ z \in ℝ \to (0 \leq (u \in ℝ ⟼ if (u \in f^{-1} [ran f ∖ \{0\}], f (u) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0)) (z) \leftrightarrow 0 \leq if (z \in f^{-1} [ran f ∖ \{0\}], f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0))$
314	313 312	ifbieq1d	$⊢ z \in ℝ \to if (0 \leq (u \in ℝ ⟼ if (u \in f^{-1} [ran f ∖ \{0\}], f (u) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0)) (z), (u \in ℝ ⟼ if (u \in f^{-1} [ran f ∖ \{0\}], f (u) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0)) (z), 0) = if (0 \leq if (z \in f^{-1} [ran f ∖ \{0\}], f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0), if (z \in f^{-1} [ran f ∖ \{0\}], f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0), 0)$
315	304 314	eqtr4id	$⊢ z \in ℝ \to if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) = if (0 \leq (u \in ℝ ⟼ if (u \in f^{-1} [ran f ∖ \{0\}], f (u) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0)) (z), (u \in ℝ ⟼ if (u \in f^{-1} [ran f ∖ \{0\}], f (u) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0)) (z), 0)$
316	315	mpteq2ia	$⊢ (z \in ℝ ⟼ if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0)) = (z \in ℝ ⟼ if (0 \leq (u \in ℝ ⟼ if (u \in f^{-1} [ran f ∖ \{0\}], f (u) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0)) (z), (u \in ℝ ⟼ if (u \in f^{-1} [ran f ∖ \{0\}], f (u) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0)) (z), 0))$
317	316	i1fpos	$⊢ (u \in ℝ ⟼ if (u \in f^{-1} [ran f ∖ \{0\}], f (u) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0)) \in dom \int^{1} \to (z \in ℝ ⟼ if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{*} (f^{-1} [ran f ∖ \{0\}])}), 0)) \in dom \int^{1}$
318	280 317	syl	$⊢ ((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to (z \in ℝ ⟼ if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{*} (f^{-1} [ran f ∖ \{0\}])}), 0)) \in dom \int^{1}$
319	195 318	sylan	$⊢ (((F : ℝ ⟶ [0, +∞] \land (f \in dom \int^{1} \land (f \leq_{f} F \land s = \int^{1} (f)))) \land (b < \int^{1} (f) \land b \in ℝ)) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to (z \in ℝ ⟼ if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{*} (f^{-1} [ran f ∖ \{0\}])}), 0)) \in dom \int^{1}$
320	195 264	sylan	$⊢ (((F : ℝ ⟶ [0, +∞] \land (f \in dom \int^{1} \land (f \leq_{f} F \land s = \int^{1} (f)))) \land (b < \int^{1} (f) \land b \in ℝ)) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to \frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])} \in ℝ$
321	8	ad2antrl	$⊢ (F : ℝ ⟶ [0, +∞] \land (f \in dom \int^{1} \land (f \leq_{f} F \land s = \int^{1} (f)))) \to \int^{1} (f) \in ℝ$
322	321 194 243	syl2an	$⊢ ((F : ℝ ⟶ [0, +∞] \land (f \in dom \int^{1} \land (f \leq_{f} F \land s = \int^{1} (f)))) \land (b < \int^{1} (f) \land b \in ℝ)) \to \int^{1} (f) - b \in ℝ$
323	322	adantr	$⊢ (((F : ℝ ⟶ [0, +∞] \land (f \in dom \int^{1} \land (f \leq_{f} F \land s = \int^{1} (f)))) \land (b < \int^{1} (f) \land b \in ℝ)) \land {vol}^{*} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to \int^{1} (f) - b \in ℝ$
324	255	adantr	$⊢ (f \in dom \int^{1} \land (f \leq_{f} F \land s = \int^{1} (f))) \to 2 {vol}^{*} (f^{-1} [ran f ∖ \{0\}]) \in ℝ$
325	324	ad3antlr	$⊢ (((F : ℝ ⟶ [0, +∞] \land (f \in dom \int^{1} \land (f \leq_{f} F \land s = \int^{1} (f)))) \land (b < \int^{1} (f) \land b \in ℝ)) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to 2 {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \in ℝ$
326		simprl	$⊢ ((F : ℝ ⟶ [0, +∞] \land (f \in dom \int^{1} \land (f \leq_{f} F \land s = \int^{1} (f)))) \land (b < \int^{1} (f) \land b \in ℝ)) \to b < \int^{1} (f)$
327		simprr	$⊢ ((F : ℝ ⟶ [0, +∞] \land (f \in dom \int^{1} \land (f \leq_{f} F \land s = \int^{1} (f)))) \land (b < \int^{1} (f) \land b \in ℝ)) \to b \in ℝ$
328	141	ad2antlr	$⊢ ((F : ℝ ⟶ [0, +∞] \land (f \in dom \int^{1} \land (f \leq_{f} F \land s = \int^{1} (f)))) \land (b < \int^{1} (f) \land b \in ℝ)) \to \int^{1} (f) \in ℝ$
329	327 328	posdifd	$⊢ ((F : ℝ ⟶ [0, +∞] \land (f \in dom \int^{1} \land (f \leq_{f} F \land s = \int^{1} (f)))) \land (b < \int^{1} (f) \land b \in ℝ)) \to (b < \int^{1} (f) \leftrightarrow 0 < \int^{1} (f) - b)$
330	326 329	mpbid	$⊢ ((F : ℝ ⟶ [0, +∞] \land (f \in dom \int^{1} \land (f \leq_{f} F \land s = \int^{1} (f)))) \land (b < \int^{1} (f) \land b \in ℝ)) \to 0 < \int^{1} (f) - b$
331	330	adantr	$⊢ (((F : ℝ ⟶ [0, +∞] \land (f \in dom \int^{1} \land (f \leq_{f} F \land s = \int^{1} (f)))) \land (b < \int^{1} (f) \land b \in ℝ)) \land {vol}^{*} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to 0 < \int^{1} (f) - b$
332	253	adantr	$⊢ (f \in dom \int^{1} \land (f \leq_{f} F \land s = \int^{1} (f))) \to {vol}^{*} (f^{-1} [ran f ∖ \{0\}]) \in ℝ$
333	332	ad3antlr	$⊢ (((F : ℝ ⟶ [0, +∞] \land (f \in dom \int^{1} \land (f \leq_{f} F \land s = \int^{1} (f)))) \land (b < \int^{1} (f) \land b \in ℝ)) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \in ℝ$
334		mblss	$⊢ f^{-1} [ran f ∖ \{0\}] \in dom vol \to f^{-1} [ran f ∖ \{0\}] \subseteq ℝ$
335		ovolge0	$⊢ f^{-1} [ran f ∖ \{0\}] \subseteq ℝ \to 0 \leq {vol}^{*} (f^{-1} [ran f ∖ \{0\}])$
336	247 334 335	3syl	$⊢ f \in dom \int^{1} \to 0 \leq {vol}^{*} (f^{-1} [ran f ∖ \{0\}])$
337		ltlen	$⊢ (0 \in ℝ \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \in ℝ) \to (0 < {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \leftrightarrow (0 \leq {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0))$
338	48 253 337	sylancr	$⊢ f \in dom \int^{1} \to (0 < {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \leftrightarrow (0 \leq {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \land {vol}^{*} (f^{-1} [ran f ∖ \{0\}]) \neq 0))$
339	338	biimprd	$⊢ f \in dom \int^{1} \to ((0 \leq {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to 0 < {vol}^{*} (f^{-1} [ran f ∖ \{0\}]))$
340	336 339	mpand	$⊢ f \in dom \int^{1} \to ({vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0 \to 0 < {vol}^{} (f^{-1} [ran f ∖ \{0\}]))$
341	340	ad2antrl	$⊢ (F : ℝ ⟶ [0, +∞] \land (f \in dom \int^{1} \land (f \leq_{f} F \land s = \int^{1} (f)))) \to ({vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0 \to 0 < {vol}^{} (f^{-1} [ran f ∖ \{0\}]))$
342	341	imp	$⊢ ((F : ℝ ⟶ [0, +∞] \land (f \in dom \int^{1} \land (f \leq_{f} F \land s = \int^{1} (f)))) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to 0 < {vol}^{} (f^{-1} [ran f ∖ \{0\}])$
343	342	adantlr	$⊢ (((F : ℝ ⟶ [0, +∞] \land (f \in dom \int^{1} \land (f \leq_{f} F \land s = \int^{1} (f)))) \land (b < \int^{1} (f) \land b \in ℝ)) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to 0 < {vol}^{} (f^{-1} [ran f ∖ \{0\}])$
344		2pos	$⊢ 0 < 2$
345		mulgt0	$⊢ ((2 \in ℝ \land 0 < 2) \land ({vol}^{} (f^{-1} [ran f ∖ \{0\}]) \in ℝ \land 0 < {vol}^{} (f^{-1} [ran f ∖ \{0\}]))) \to 0 < 2 {vol}^{*} (f^{-1} [ran f ∖ \{0\}])$
346	246 344 345	mpanl12	$⊢ ({vol}^{} (f^{-1} [ran f ∖ \{0\}]) \in ℝ \land 0 < {vol}^{} (f^{-1} [ran f ∖ \{0\}])) \to 0 < 2 {vol}^{*} (f^{-1} [ran f ∖ \{0\}])$
347	333 343 346	syl2anc	$⊢ (((F : ℝ ⟶ [0, +∞] \land (f \in dom \int^{1} \land (f \leq_{f} F \land s = \int^{1} (f)))) \land (b < \int^{1} (f) \land b \in ℝ)) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to 0 < 2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])$
348	323 325 331 347	divgt0d	$⊢ (((F : ℝ ⟶ [0, +∞] \land (f \in dom \int^{1} \land (f \leq_{f} F \land s = \int^{1} (f)))) \land (b < \int^{1} (f) \land b \in ℝ)) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to 0 < \frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}$
349	320 348	elrpd	$⊢ (((F : ℝ ⟶ [0, +∞] \land (f \in dom \int^{1} \land (f \leq_{f} F \land s = \int^{1} (f)))) \land (b < \int^{1} (f) \land b \in ℝ)) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to \frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])} \in ℝ^{+}$
350		simprl	$⊢ (f \in dom \int^{1} \land (f \leq_{f} F \land s = \int^{1} (f))) \to f \leq_{f} F$
351	350	ad3antlr	$⊢ (((F : ℝ ⟶ [0, +∞] \land (f \in dom \int^{1} \land (f \leq_{f} F \land s = \int^{1} (f)))) \land (b < \int^{1} (f) \land b \in ℝ)) \land {vol}^{*} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to f \leq_{f} F$
352		ffn	$⊢ F : ℝ ⟶ [0, +∞] \to F Fn ℝ$
353	35 180	syl	$⊢ f \in dom \int^{1} \to f Fn ℝ$
354	353	adantr	$⊢ (f \in dom \int^{1} \land (f \leq_{f} F \land s = \int^{1} (f))) \to f Fn ℝ$
355		simpr	$⊢ (F Fn ℝ \land f Fn ℝ) \to f Fn ℝ$
356		simpl	$⊢ (F Fn ℝ \land f Fn ℝ) \to F Fn ℝ$
357	63	a1i	$⊢ (F Fn ℝ \land f Fn ℝ) \to ℝ \in V$
358		inidm	$⊢ ℝ \cap ℝ = ℝ$
359		eqidd	$⊢ ((F Fn ℝ \land f Fn ℝ) \land z \in ℝ) \to f (z) = f (z)$
360		eqidd	$⊢ ((F Fn ℝ \land f Fn ℝ) \land z \in ℝ) \to F (z) = F (z)$
361	355 356 357 357 358 359 360	ofrfval	$⊢ (F Fn ℝ \land f Fn ℝ) \to (f \leq_{f} F \leftrightarrow \forall z \in ℝ f (z) \leq F (z))$
362	352 354 361	syl2an	$⊢ (F : ℝ ⟶ [0, +∞] \land (f \in dom \int^{1} \land (f \leq_{f} F \land s = \int^{1} (f)))) \to (f \leq_{f} F \leftrightarrow \forall z \in ℝ f (z) \leq F (z))$
363	362	ad2antrr	$⊢ (((F : ℝ ⟶ [0, +∞] \land (f \in dom \int^{1} \land (f \leq_{f} F \land s = \int^{1} (f)))) \land (b < \int^{1} (f) \land b \in ℝ)) \land {vol}^{*} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to (f \leq_{f} F \leftrightarrow \forall z \in ℝ f (z) \leq F (z))$
364		simpl	$⊢ (f \in dom \int^{1} \land (f \leq_{f} F \land s = \int^{1} (f))) \to f \in dom \int^{1}$
365	364	anim2i	$⊢ (F : ℝ ⟶ [0, +∞] \land (f \in dom \int^{1} \land (f \leq_{f} F \land s = \int^{1} (f)))) \to (F : ℝ ⟶ [0, +∞] \land f \in dom \int^{1})$
366	365 194	anim12i	$⊢ ((F : ℝ ⟶ [0, +∞] \land (f \in dom \int^{1} \land (f \leq_{f} F \land s = \int^{1} (f)))) \land (b < \int^{1} (f) \land b \in ℝ)) \to ((F : ℝ ⟶ [0, +∞] \land f \in dom \int^{1}) \land b \in ℝ)$
367		breq1	$⊢ 0 = if (if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) = 0, 0, if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) + (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])})) \to (0 \leq F (z) \leftrightarrow if (if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) = 0, 0, if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) + (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])})) \leq F (z))$
368		breq1	$⊢ if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) + (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) = if (if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) = 0, 0, if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) + (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])})) \to (if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) + (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \leq F (z) \leftrightarrow if (if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) = 0, 0, if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) + (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])})) \leq F (z))$
369		simplll	$⊢ (((F : ℝ ⟶ [0, +∞] \land f \in dom \int^{1}) \land b \in ℝ) \land {vol}^{*} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to F : ℝ ⟶ [0, +∞]$
370	369	ffvelcdmda	$⊢ ((((F : ℝ ⟶ [0, +∞] \land f \in dom \int^{1}) \land b \in ℝ) \land {vol}^{*} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \land z \in ℝ) \to F (z) \in [0, +∞]$
371	370 100	sylib	$⊢ ((((F : ℝ ⟶ [0, +∞] \land f \in dom \int^{1}) \land b \in ℝ) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \land z \in ℝ) \to (F (z) \in ℝ^{} \land 0 \leq F (z))$
372	371	simprd	$⊢ ((((F : ℝ ⟶ [0, +∞] \land f \in dom \int^{1}) \land b \in ℝ) \land {vol}^{*} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \land z \in ℝ) \to 0 \leq F (z)$
373	372	ad2antrr	$⊢ ((((((F : ℝ ⟶ [0, +∞] \land f \in dom \int^{1}) \land b \in ℝ) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \land z \in ℝ) \land f (z) \leq F (z)) \land if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{*} (f^{-1} [ran f ∖ \{0\}])}), 0) = 0) \to 0 \leq F (z)$
374		oveq1	$⊢ f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) = if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) \to f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) + (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) = if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) + (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])})$
375	374	breq1d	$⊢ f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) = if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) \to (f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) + (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \leq F (z) \leftrightarrow if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) + (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \leq F (z))$
376		oveq1	$⊢ 0 = if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) \to 0 + (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) = if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) + (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])})$
377	376	breq1d	$⊢ 0 = if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) \to (0 + (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \leq F (z) \leftrightarrow if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) + (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \leq F (z))$
378	35	ad3antlr	$⊢ (((F : ℝ ⟶ [0, +∞] \land f \in dom \int^{1}) \land b \in ℝ) \land {vol}^{*} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to f : ℝ ⟶ ℝ$
379	378	ffvelcdmda	$⊢ ((((F : ℝ ⟶ [0, +∞] \land f \in dom \int^{1}) \land b \in ℝ) \land {vol}^{*} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \land z \in ℝ) \to f (z) \in ℝ$
380	379	recnd	$⊢ ((((F : ℝ ⟶ [0, +∞] \land f \in dom \int^{1}) \land b \in ℝ) \land {vol}^{*} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \land z \in ℝ) \to f (z) \in ℂ$
381	244	recnd	$⊢ (f \in dom \int^{1} \land b \in ℝ) \to \int^{1} (f) - b \in ℂ$
382	381	adantr	$⊢ ((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{*} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to \int^{1} (f) - b \in ℂ$
383	255	recnd	$⊢ f \in dom \int^{1} \to 2 {vol}^{*} (f^{-1} [ran f ∖ \{0\}]) \in ℂ$
384	383	ad2antrr	$⊢ ((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to 2 {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \in ℂ$
385	382 384 263	divcld	$⊢ ((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to \frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])} \in ℂ$
386	385	adantlll	$⊢ (((F : ℝ ⟶ [0, +∞] \land f \in dom \int^{1}) \land b \in ℝ) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to \frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])} \in ℂ$
387	386	adantr	$⊢ ((((F : ℝ ⟶ [0, +∞] \land f \in dom \int^{1}) \land b \in ℝ) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \land z \in ℝ) \to \frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])} \in ℂ$
388	380 387	npcand	$⊢ ((((F : ℝ ⟶ [0, +∞] \land f \in dom \int^{1}) \land b \in ℝ) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \land z \in ℝ) \to f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) + (\frac{\int^{1} (f) - b}{2 {vol}^{*} (f^{-1} [ran f ∖ \{0\}])}) = f (z)$
389	388	adantr	$⊢ (((((F : ℝ ⟶ [0, +∞] \land f \in dom \int^{1}) \land b \in ℝ) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \land z \in ℝ) \land f (z) \leq F (z)) \to f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) + (\frac{\int^{1} (f) - b}{2 {vol}^{*} (f^{-1} [ran f ∖ \{0\}])}) = f (z)$
390		simpr	$⊢ (((((F : ℝ ⟶ [0, +∞] \land f \in dom \int^{1}) \land b \in ℝ) \land {vol}^{*} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \land z \in ℝ) \land f (z) \leq F (z)) \to f (z) \leq F (z)$
391	389 390	eqbrtrd	$⊢ (((((F : ℝ ⟶ [0, +∞] \land f \in dom \int^{1}) \land b \in ℝ) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \land z \in ℝ) \land f (z) \leq F (z)) \to f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) + (\frac{\int^{1} (f) - b}{2 {vol}^{*} (f^{-1} [ran f ∖ \{0\}])}) \leq F (z)$
392	391	ad2antrr	$⊢ (((((((F : ℝ ⟶ [0, +∞] \land f \in dom \int^{1}) \land b \in ℝ) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \land z \in ℝ) \land f (z) \leq F (z)) \land \neg if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) = 0) \land (0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}])) \to f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) + (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \leq F (z)$
393	288	pm2.24d	$⊢ \neg (0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]) \to (\neg if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) = 0 \to 0 + (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \leq F (z))$
394	393	impcom	$⊢ (\neg if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) = 0 \land \neg (0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}])) \to 0 + (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \leq F (z)$
395	394	adantll	$⊢ (((((((F : ℝ ⟶ [0, +∞] \land f \in dom \int^{1}) \land b \in ℝ) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \land z \in ℝ) \land f (z) \leq F (z)) \land \neg if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) = 0) \land \neg (0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}])) \to 0 + (\frac{\int^{1} (f) - b}{2 {vol}^{*} (f^{-1} [ran f ∖ \{0\}])}) \leq F (z)$
396	375 377 392 395	ifbothda	$⊢ ((((((F : ℝ ⟶ [0, +∞] \land f \in dom \int^{1}) \land b \in ℝ) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \land z \in ℝ) \land f (z) \leq F (z)) \land \neg if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) = 0) \to if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) + (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \leq F (z)$
397	367 368 373 396	ifbothda	$⊢ (((((F : ℝ ⟶ [0, +∞] \land f \in dom \int^{1}) \land b \in ℝ) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \land z \in ℝ) \land f (z) \leq F (z)) \to if (if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) = 0, 0, if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) + (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])})) \leq F (z)$
398	397	ex	$⊢ ((((F : ℝ ⟶ [0, +∞] \land f \in dom \int^{1}) \land b \in ℝ) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \land z \in ℝ) \to (f (z) \leq F (z) \to if (if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) = 0, 0, if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) + (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])})) \leq F (z))$
399	366 398	sylanl1	$⊢ ((((F : ℝ ⟶ [0, +∞] \land (f \in dom \int^{1} \land (f \leq_{f} F \land s = \int^{1} (f)))) \land (b < \int^{1} (f) \land b \in ℝ)) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \land z \in ℝ) \to (f (z) \leq F (z) \to if (if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) = 0, 0, if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) + (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])})) \leq F (z))$
400	399	ralimdva	$⊢ (((F : ℝ ⟶ [0, +∞] \land (f \in dom \int^{1} \land (f \leq_{f} F \land s = \int^{1} (f)))) \land (b < \int^{1} (f) \land b \in ℝ)) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to (\forall z \in ℝ f (z) \leq F (z) \to \forall z \in ℝ if (if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) = 0, 0, if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) + (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])})) \leq F (z))$
401	363 400	sylbid	$⊢ (((F : ℝ ⟶ [0, +∞] \land (f \in dom \int^{1} \land (f \leq_{f} F \land s = \int^{1} (f)))) \land (b < \int^{1} (f) \land b \in ℝ)) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to (f \leq_{f} F \to \forall z \in ℝ if (if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) = 0, 0, if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) + (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])})) \leq F (z))$
402	351 401	mpd	$⊢ (((F : ℝ ⟶ [0, +∞] \land (f \in dom \int^{1} \land (f \leq_{f} F \land s = \int^{1} (f)))) \land (b < \int^{1} (f) \land b \in ℝ)) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to \forall z \in ℝ if (if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) = 0, 0, if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) + (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])})) \leq F (z)$
403		ovex	$⊢ if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) + (\frac{\int^{1} (f) - b}{2 {vol}^{*} (f^{-1} [ran f ∖ \{0\}])}) \in V$
404	105 403	ifex	$⊢ if (if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) = 0, 0, if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) + (\frac{\int^{1} (f) - b}{2 {vol}^{*} (f^{-1} [ran f ∖ \{0\}])})) \in V$
405	404	a1i	$⊢ (F : ℝ ⟶ [0, +∞] \land z \in ℝ) \to if (if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) = 0, 0, if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) + (\frac{\int^{1} (f) - b}{2 {vol}^{*} (f^{-1} [ran f ∖ \{0\}])})) \in V$
406		eqidd	$⊢ F : ℝ ⟶ [0, +∞] \to (z \in ℝ ⟼ if (if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) = 0, 0, if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) + (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}))) = (z \in ℝ ⟼ if (if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) = 0, 0, if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) + (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])})))$
407	104 405 99 406 67	ofrfval2	$⊢ F : ℝ ⟶ [0, +∞] \to ((z \in ℝ ⟼ if (if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) = 0, 0, if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) + (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}))) \leq_{f} F \leftrightarrow \forall z \in ℝ if (if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) = 0, 0, if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) + (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])})) \leq F (z))$
408	407	ad3antrrr	$⊢ (((F : ℝ ⟶ [0, +∞] \land (f \in dom \int^{1} \land (f \leq_{f} F \land s = \int^{1} (f)))) \land (b < \int^{1} (f) \land b \in ℝ)) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to ((z \in ℝ ⟼ if (if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) = 0, 0, if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) + (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}))) \leq_{f} F \leftrightarrow \forall z \in ℝ if (if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) = 0, 0, if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) + (\frac{\int^{1} (f) - b}{2 {vol}^{*} (f^{-1} [ran f ∖ \{0\}])})) \leq F (z))$
409	402 408	mpbird	$⊢ (((F : ℝ ⟶ [0, +∞] \land (f \in dom \int^{1} \land (f \leq_{f} F \land s = \int^{1} (f)))) \land (b < \int^{1} (f) \land b \in ℝ)) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to (z \in ℝ ⟼ if (if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) = 0, 0, if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) + (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}))) \leq_{f} F$
410		oveq2	$⊢ y = \frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])} \to if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) + y = if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) + (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])})$
411	410	ifeq2d	$⊢ y = \frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])} \to if (if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) = 0, 0, if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) + y) = if (if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) = 0, 0, if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) + (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}))$
412	411	mpteq2dv	$⊢ y = \frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])} \to (z \in ℝ ⟼ if (if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) = 0, 0, if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) + y)) = (z \in ℝ ⟼ if (if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) = 0, 0, if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) + (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])})))$
413	412	breq1d	$⊢ y = \frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])} \to ((z \in ℝ ⟼ if (if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) = 0, 0, if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) + y)) \leq_{f} F \leftrightarrow (z \in ℝ ⟼ if (if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) = 0, 0, if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) + (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}))) \leq_{f} F)$
414	413	rspcev	$⊢ (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])} \in ℝ^{+} \land (z \in ℝ ⟼ if (if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) = 0, 0, if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) + (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}))) \leq_{f} F) \to \exists y \in ℝ^{+} (z \in ℝ ⟼ if (if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) = 0, 0, if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) + y)) \leq_{f} F$
415	349 409 414	syl2anc	$⊢ (((F : ℝ ⟶ [0, +∞] \land (f \in dom \int^{1} \land (f \leq_{f} F \land s = \int^{1} (f)))) \land (b < \int^{1} (f) \land b \in ℝ)) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to \exists y \in ℝ^{+} (z \in ℝ ⟼ if (if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) = 0, 0, if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{*} (f^{-1} [ran f ∖ \{0\}])}), 0) + y)) \leq_{f} F$
416		fveq2	$⊢ (z \in ℝ ⟼ if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0)) = g \to \int^{1} ((z \in ℝ ⟼ if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0))) = \int^{1} (g)$
417	416	eqcoms	$⊢ g = (z \in ℝ ⟼ if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0)) \to \int^{1} ((z \in ℝ ⟼ if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0))) = \int^{1} (g)$
418	417	biantrud	$⊢ g = (z \in ℝ ⟼ if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0)) \to (\exists y \in ℝ^{+} (z \in ℝ ⟼ if (g (z) = 0, 0, g (z) + y)) \leq_{f} F \leftrightarrow (\exists y \in ℝ^{+} (z \in ℝ ⟼ if (g (z) = 0, 0, g (z) + y)) \leq_{f} F \land \int^{1} ((z \in ℝ ⟼ if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0))) = \int^{1} (g)))$
419		nfmpt1	$⊢ \underline{Ⅎ} z (z \in ℝ ⟼ if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0))$
420	419	nfeq2	$⊢ Ⅎ z g = (z \in ℝ ⟼ if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0))$
421		fveq1	$⊢ g = (z \in ℝ ⟼ if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0)) \to g (z) = (z \in ℝ ⟼ if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0)) (z)$
422	310 105	ifex	$⊢ if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) \in V$
423		eqid	$⊢ (z \in ℝ ⟼ if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0)) = (z \in ℝ ⟼ if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0))$
424	423	fvmpt2	$⊢ (z \in ℝ \land if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) \in V) \to (z \in ℝ ⟼ if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0)) (z) = if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0)$
425	422 424	mpan2	$⊢ z \in ℝ \to (z \in ℝ ⟼ if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0)) (z) = if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0)$
426	421 425	sylan9eq	$⊢ (g = (z \in ℝ ⟼ if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0)) \land z \in ℝ) \to g (z) = if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0)$
427	426	eqeq1d	$⊢ (g = (z \in ℝ ⟼ if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0)) \land z \in ℝ) \to (g (z) = 0 \leftrightarrow if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) = 0)$
428	426	oveq1d	$⊢ (g = (z \in ℝ ⟼ if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0)) \land z \in ℝ) \to g (z) + y = if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) + y$
429	427 428	ifbieq2d	$⊢ (g = (z \in ℝ ⟼ if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0)) \land z \in ℝ) \to if (g (z) = 0, 0, g (z) + y) = if (if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) = 0, 0, if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) + y)$
430	420 429	mpteq2da	$⊢ g = (z \in ℝ ⟼ if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0)) \to (z \in ℝ ⟼ if (g (z) = 0, 0, g (z) + y)) = (z \in ℝ ⟼ if (if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) = 0, 0, if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) + y))$
431	430	breq1d	$⊢ g = (z \in ℝ ⟼ if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0)) \to ((z \in ℝ ⟼ if (g (z) = 0, 0, g (z) + y)) \leq_{f} F \leftrightarrow (z \in ℝ ⟼ if (if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) = 0, 0, if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) + y)) \leq_{f} F)$
432	431	rexbidv	$⊢ g = (z \in ℝ ⟼ if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0)) \to (\exists y \in ℝ^{+} (z \in ℝ ⟼ if (g (z) = 0, 0, g (z) + y)) \leq_{f} F \leftrightarrow \exists y \in ℝ^{+} (z \in ℝ ⟼ if (if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) = 0, 0, if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) + y)) \leq_{f} F)$
433	418 432	bitr3d	$⊢ g = (z \in ℝ ⟼ if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0)) \to ((\exists y \in ℝ^{+} (z \in ℝ ⟼ if (g (z) = 0, 0, g (z) + y)) \leq_{f} F \land \int^{1} ((z \in ℝ ⟼ if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0))) = \int^{1} (g)) \leftrightarrow \exists y \in ℝ^{+} (z \in ℝ ⟼ if (if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) = 0, 0, if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) + y)) \leq_{f} F)$
434	433	rspcev	$⊢ ((z \in ℝ ⟼ if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0)) \in dom \int^{1} \land \exists y \in ℝ^{+} (z \in ℝ ⟼ if (if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) = 0, 0, if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) + y)) \leq_{f} F) \to \exists g \in dom \int^{1} (\exists y \in ℝ^{+} (z \in ℝ ⟼ if (g (z) = 0, 0, g (z) + y)) \leq_{f} F \land \int^{1} ((z \in ℝ ⟼ if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0))) = \int^{1} (g))$
435	319 415 434	syl2anc	$⊢ (((F : ℝ ⟶ [0, +∞] \land (f \in dom \int^{1} \land (f \leq_{f} F \land s = \int^{1} (f)))) \land (b < \int^{1} (f) \land b \in ℝ)) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to \exists g \in dom \int^{1} (\exists y \in ℝ^{+} (z \in ℝ ⟼ if (g (z) = 0, 0, g (z) + y)) \leq_{f} F \land \int^{1} ((z \in ℝ ⟼ if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{*} (f^{-1} [ran f ∖ \{0\}])}), 0))) = \int^{1} (g))$
436		simplrr	$⊢ ((f \in dom \int^{1} \land (b < \int^{1} (f) \land b \in ℝ)) \land {vol}^{*} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to b \in ℝ$
437	199	a1i	$⊢ (((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \land z \in ℝ) \to \frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])} \in V$
438	235 105	ifex	$⊢ if (z \in f^{-1} [ran f ∖ \{0\}], 1, 0) \in V$
439	438	a1i	$⊢ (((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{*} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \land z \in ℝ) \to if (z \in f^{-1} [ran f ∖ \{0\}], 1, 0) \in V$
440		fconstmpt	$⊢ ℝ \times \{\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}\} = (z \in ℝ ⟼ \frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])})$
441	440	a1i	$⊢ ((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to ℝ \times \{\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}\} = (z \in ℝ ⟼ \frac{\int^{1} (f) - b}{2 {vol}^{*} (f^{-1} [ran f ∖ \{0\}])})$
442		eqidd	$⊢ ((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{*} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to (z \in ℝ ⟼ if (z \in f^{-1} [ran f ∖ \{0\}], 1, 0)) = (z \in ℝ ⟼ if (z \in f^{-1} [ran f ∖ \{0\}], 1, 0))$
443	196 437 439 441 442	offval2	$⊢ ((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to (ℝ \times \{\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}\}) \times_{f} (z \in ℝ ⟼ if (z \in f^{-1} [ran f ∖ \{0\}], 1, 0)) = (z \in ℝ ⟼ (\frac{\int^{1} (f) - b}{2 {vol}^{*} (f^{-1} [ran f ∖ \{0\}])}) if (z \in f^{-1} [ran f ∖ \{0\}], 1, 0))$
444		ovif2	$⊢ (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) if (z \in f^{-1} [ran f ∖ \{0\}], 1, 0) = if (z \in f^{-1} [ran f ∖ \{0\}], (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \cdot 1, (\frac{\int^{1} (f) - b}{2 {vol}^{*} (f^{-1} [ran f ∖ \{0\}])}) \cdot 0)$
445	266 267	ifeq12d	$⊢ ((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to if (z \in f^{-1} [ran f ∖ \{0\}], (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \cdot 1, (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \cdot 0) = if (z \in f^{-1} [ran f ∖ \{0\}], \frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}, 0)$
446	444 445	eqtrid	$⊢ ((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) if (z \in f^{-1} [ran f ∖ \{0\}], 1, 0) = if (z \in f^{-1} [ran f ∖ \{0\}], \frac{\int^{1} (f) - b}{2 {vol}^{*} (f^{-1} [ran f ∖ \{0\}])}, 0)$
447	446	mpteq2dv	$⊢ ((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to (z \in ℝ ⟼ (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) if (z \in f^{-1} [ran f ∖ \{0\}], 1, 0)) = (z \in ℝ ⟼ if (z \in f^{-1} [ran f ∖ \{0\}], \frac{\int^{1} (f) - b}{2 {vol}^{*} (f^{-1} [ran f ∖ \{0\}])}, 0))$
448	443 447	eqtrd	$⊢ ((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to (ℝ \times \{\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}\}) \times_{f} (z \in ℝ ⟼ if (z \in f^{-1} [ran f ∖ \{0\}], 1, 0)) = (z \in ℝ ⟼ if (z \in f^{-1} [ran f ∖ \{0\}], \frac{\int^{1} (f) - b}{2 {vol}^{*} (f^{-1} [ran f ∖ \{0\}])}, 0))$
449		eqid	$⊢ (z \in ℝ ⟼ if (z \in f^{-1} [ran f ∖ \{0\}], 1, 0)) = (z \in ℝ ⟼ if (z \in f^{-1} [ran f ∖ \{0\}], 1, 0))$
450	449	i1f1	$⊢ (f^{-1} [ran f ∖ \{0\}] \in dom vol \land vol (f^{-1} [ran f ∖ \{0\}]) \in ℝ) \to (z \in ℝ ⟼ if (z \in f^{-1} [ran f ∖ \{0\}], 1, 0)) \in dom \int^{1}$
451	247 252 450	syl2anc	$⊢ f \in dom \int^{1} \to (z \in ℝ ⟼ if (z \in f^{-1} [ran f ∖ \{0\}], 1, 0)) \in dom \int^{1}$
452	451	ad2antrr	$⊢ ((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{*} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to (z \in ℝ ⟼ if (z \in f^{-1} [ran f ∖ \{0\}], 1, 0)) \in dom \int^{1}$
453	452 264	i1fmulc	$⊢ ((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to (ℝ \times \{\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}\}) \times_{f} (z \in ℝ ⟼ if (z \in f^{-1} [ran f ∖ \{0\}], 1, 0)) \in dom \int^{1}$
454	448 453	eqeltrrd	$⊢ ((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to (z \in ℝ ⟼ if (z \in f^{-1} [ran f ∖ \{0\}], \frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}, 0)) \in dom \int^{1}$
455		i1fsub	$⊢ (f \in dom \int^{1} \land (z \in ℝ ⟼ if (z \in f^{-1} [ran f ∖ \{0\}], \frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}, 0)) \in dom \int^{1}) \to f -_{f} (z \in ℝ ⟼ if (z \in f^{-1} [ran f ∖ \{0\}], \frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}, 0)) \in dom \int^{1}$
456	233 454 455	syl2anc	$⊢ ((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to f -_{f} (z \in ℝ ⟼ if (z \in f^{-1} [ran f ∖ \{0\}], \frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}, 0)) \in dom \int^{1}$
457		itg1cl	$⊢ f -_{f} (z \in ℝ ⟼ if (z \in f^{-1} [ran f ∖ \{0\}], \frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}, 0)) \in dom \int^{1} \to \int^{1} (f -_{f} (z \in ℝ ⟼ if (z \in f^{-1} [ran f ∖ \{0\}], \frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}, 0))) \in ℝ$
458	456 457	syl	$⊢ ((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to \int^{1} (f -_{f} (z \in ℝ ⟼ if (z \in f^{-1} [ran f ∖ \{0\}], \frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}, 0))) \in ℝ$
459	458	adantlrl	$⊢ ((f \in dom \int^{1} \land (b < \int^{1} (f) \land b \in ℝ)) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to \int^{1} (f -_{f} (z \in ℝ ⟼ if (z \in f^{-1} [ran f ∖ \{0\}], \frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}, 0))) \in ℝ$
460	318	adantlrl	$⊢ ((f \in dom \int^{1} \land (b < \int^{1} (f) \land b \in ℝ)) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to (z \in ℝ ⟼ if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{*} (f^{-1} [ran f ∖ \{0\}])}), 0)) \in dom \int^{1}$
461		itg1cl	$⊢ (z \in ℝ ⟼ if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0)) \in dom \int^{1} \to \int^{1} ((z \in ℝ ⟼ if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0))) \in ℝ$
462	460 461	syl	$⊢ ((f \in dom \int^{1} \land (b < \int^{1} (f) \land b \in ℝ)) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to \int^{1} ((z \in ℝ ⟼ if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{*} (f^{-1} [ran f ∖ \{0\}])}), 0))) \in ℝ$
463		simplrl	$⊢ ((f \in dom \int^{1} \land (b < \int^{1} (f) \land b \in ℝ)) \land {vol}^{*} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to b < \int^{1} (f)$
464		simpr	$⊢ (f \in dom \int^{1} \land b \in ℝ) \to b \in ℝ$
465	8	adantr	$⊢ (f \in dom \int^{1} \land b \in ℝ) \to \int^{1} (f) \in ℝ$
466	97	a1i	$⊢ (f \in dom \int^{1} \land b \in ℝ) \to 2 \in ℝ^{+}$
467	464 465 466	ltdiv1d	$⊢ (f \in dom \int^{1} \land b \in ℝ) \to (b < \int^{1} (f) \leftrightarrow \frac{b}{2} < \frac{\int^{1} (f)}{2})$
468		recn	$⊢ b \in ℝ \to b \in ℂ$
469	468	2halvesd	$⊢ b \in ℝ \to (\frac{b}{2}) + (\frac{b}{2}) = b$
470	469	oveq1d	$⊢ b \in ℝ \to (\frac{b}{2}) + (\frac{b}{2}) - (\frac{b}{2}) = b - (\frac{b}{2})$
471	468	halfcld	$⊢ b \in ℝ \to \frac{b}{2} \in ℂ$
472	471 471	pncand	$⊢ b \in ℝ \to (\frac{b}{2}) + (\frac{b}{2}) - (\frac{b}{2}) = \frac{b}{2}$
473	470 472	eqtr3d	$⊢ b \in ℝ \to b - (\frac{b}{2}) = \frac{b}{2}$
474	473	breq1d	$⊢ b \in ℝ \to (b - (\frac{b}{2}) < \frac{\int^{1} (f)}{2} \leftrightarrow \frac{b}{2} < \frac{\int^{1} (f)}{2})$
475	474	adantl	$⊢ (f \in dom \int^{1} \land b \in ℝ) \to (b - (\frac{b}{2}) < \frac{\int^{1} (f)}{2} \leftrightarrow \frac{b}{2} < \frac{\int^{1} (f)}{2})$
476		rehalfcl	$⊢ b \in ℝ \to \frac{b}{2} \in ℝ$
477	476	adantl	$⊢ (f \in dom \int^{1} \land b \in ℝ) \to \frac{b}{2} \in ℝ$
478	8	rehalfcld	$⊢ f \in dom \int^{1} \to \frac{\int^{1} (f)}{2} \in ℝ$
479	478	adantr	$⊢ (f \in dom \int^{1} \land b \in ℝ) \to \frac{\int^{1} (f)}{2} \in ℝ$
480	464 477 479	ltsubaddd	$⊢ (f \in dom \int^{1} \land b \in ℝ) \to (b - (\frac{b}{2}) < \frac{\int^{1} (f)}{2} \leftrightarrow b < (\frac{\int^{1} (f)}{2}) + (\frac{b}{2}))$
481	467 475 480	3bitr2d	$⊢ (f \in dom \int^{1} \land b \in ℝ) \to (b < \int^{1} (f) \leftrightarrow b < (\frac{\int^{1} (f)}{2}) + (\frac{b}{2}))$
482	481	adantr	$⊢ ((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{*} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to (b < \int^{1} (f) \leftrightarrow b < (\frac{\int^{1} (f)}{2}) + (\frac{b}{2}))$
483	482	adantlrl	$⊢ ((f \in dom \int^{1} \land (b < \int^{1} (f) \land b \in ℝ)) \land {vol}^{*} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to (b < \int^{1} (f) \leftrightarrow b < (\frac{\int^{1} (f)}{2}) + (\frac{b}{2}))$
484	463 483	mpbid	$⊢ ((f \in dom \int^{1} \land (b < \int^{1} (f) \land b \in ℝ)) \land {vol}^{*} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to b < (\frac{\int^{1} (f)}{2}) + (\frac{b}{2})$
485	452 264	itg1mulc	$⊢ ((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to \int^{1} ((ℝ \times \{\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}\}) \times_{f} (z \in ℝ ⟼ if (z \in f^{-1} [ran f ∖ \{0\}], 1, 0))) = (\frac{\int^{1} (f) - b}{2 {vol}^{*} (f^{-1} [ran f ∖ \{0\}])}) \int^{1} ((z \in ℝ ⟼ if (z \in f^{-1} [ran f ∖ \{0\}], 1, 0)))$
486	448	fveq2d	$⊢ ((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to \int^{1} ((ℝ \times \{\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}\}) \times_{f} (z \in ℝ ⟼ if (z \in f^{-1} [ran f ∖ \{0\}], 1, 0))) = \int^{1} ((z \in ℝ ⟼ if (z \in f^{-1} [ran f ∖ \{0\}], \frac{\int^{1} (f) - b}{2 {vol}^{*} (f^{-1} [ran f ∖ \{0\}])}, 0)))$
487	449	itg11	$⊢ (f^{-1} [ran f ∖ \{0\}] \in dom vol \land vol (f^{-1} [ran f ∖ \{0\}]) \in ℝ) \to \int^{1} ((z \in ℝ ⟼ if (z \in f^{-1} [ran f ∖ \{0\}], 1, 0))) = vol (f^{-1} [ran f ∖ \{0\}])$
488	247 252 487	syl2anc	$⊢ f \in dom \int^{1} \to \int^{1} ((z \in ℝ ⟼ if (z \in f^{-1} [ran f ∖ \{0\}], 1, 0))) = vol (f^{-1} [ran f ∖ \{0\}])$
489	488	oveq2d	$⊢ f \in dom \int^{1} \to (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \int^{1} ((z \in ℝ ⟼ if (z \in f^{-1} [ran f ∖ \{0\}], 1, 0))) = (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) vol (f^{-1} [ran f ∖ \{0\}])$
490	489	ad2antrr	$⊢ ((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \int^{1} ((z \in ℝ ⟼ if (z \in f^{-1} [ran f ∖ \{0\}], 1, 0))) = (\frac{\int^{1} (f) - b}{2 {vol}^{*} (f^{-1} [ran f ∖ \{0\}])}) vol (f^{-1} [ran f ∖ \{0\}])$
491	252	recnd	$⊢ f \in dom \int^{1} \to vol (f^{-1} [ran f ∖ \{0\}]) \in ℂ$
492	491	ad2antrr	$⊢ ((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{*} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to vol (f^{-1} [ran f ∖ \{0\}]) \in ℂ$
493	265 492	mulcomd	$⊢ ((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) vol (f^{-1} [ran f ∖ \{0\}]) = vol (f^{-1} [ran f ∖ \{0\}]) (\frac{\int^{1} (f) - b}{2 {vol}^{*} (f^{-1} [ran f ∖ \{0\}])})$
494	249	ad2antrr	$⊢ ((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to vol (f^{-1} [ran f ∖ \{0\}]) = {vol}^{} (f^{-1} [ran f ∖ \{0\}])$
495	494	oveq1d	$⊢ ((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to vol (f^{-1} [ran f ∖ \{0\}]) (\int^{1} (f) - b) = {vol}^{} (f^{-1} [ran f ∖ \{0\}]) (\int^{1} (f) - b)$
496	259 382	mulcomd	$⊢ ((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to {vol}^{} (f^{-1} [ran f ∖ \{0\}]) (\int^{1} (f) - b) = (\int^{1} (f) - b) {vol}^{*} (f^{-1} [ran f ∖ \{0\}])$
497	495 496	eqtrd	$⊢ ((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to vol (f^{-1} [ran f ∖ \{0\}]) (\int^{1} (f) - b) = (\int^{1} (f) - b) {vol}^{} (f^{-1} [ran f ∖ \{0\}])$
498	497	oveq1d	$⊢ ((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to \frac{vol (f^{-1} [ran f ∖ \{0\}]) (\int^{1} (f) - b)}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])} = \frac{(\int^{1} (f) - b) {vol}^{} (f^{-1} [ran f ∖ \{0\}])}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}$
499	492 382 384 263	divassd	$⊢ ((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to \frac{vol (f^{-1} [ran f ∖ \{0\}]) (\int^{1} (f) - b)}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])} = vol (f^{-1} [ran f ∖ \{0\}]) (\frac{\int^{1} (f) - b}{2 {vol}^{*} (f^{-1} [ran f ∖ \{0\}])})$
500	382 257 259 261 262	divcan5rd	$⊢ ((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to \frac{(\int^{1} (f) - b) {vol}^{} (f^{-1} [ran f ∖ \{0\}])}{2 {vol}^{*} (f^{-1} [ran f ∖ \{0\}])} = \frac{\int^{1} (f) - b}{2}$
501	498 499 500	3eqtr3d	$⊢ ((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to vol (f^{-1} [ran f ∖ \{0\}]) (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) = \frac{\int^{1} (f) - b}{2}$
502	490 493 501	3eqtrd	$⊢ ((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \int^{1} ((z \in ℝ ⟼ if (z \in f^{-1} [ran f ∖ \{0\}], 1, 0))) = \frac{\int^{1} (f) - b}{2}$
503	485 486 502	3eqtr3d	$⊢ ((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to \int^{1} ((z \in ℝ ⟼ if (z \in f^{-1} [ran f ∖ \{0\}], \frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}, 0))) = \frac{\int^{1} (f) - b}{2}$
504	503	oveq2d	$⊢ ((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to \int^{1} (f) - \int^{1} ((z \in ℝ ⟼ if (z \in f^{-1} [ran f ∖ \{0\}], \frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}, 0))) = \int^{1} (f) - (\frac{\int^{1} (f) - b}{2})$
505		itg1sub	$⊢ (f \in dom \int^{1} \land (z \in ℝ ⟼ if (z \in f^{-1} [ran f ∖ \{0\}], \frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}, 0)) \in dom \int^{1}) \to \int^{1} (f -_{f} (z \in ℝ ⟼ if (z \in f^{-1} [ran f ∖ \{0\}], \frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}, 0))) = \int^{1} (f) - \int^{1} ((z \in ℝ ⟼ if (z \in f^{-1} [ran f ∖ \{0\}], \frac{\int^{1} (f) - b}{2 {vol}^{*} (f^{-1} [ran f ∖ \{0\}])}, 0)))$
506	233 454 505	syl2anc	$⊢ ((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to \int^{1} (f -_{f} (z \in ℝ ⟼ if (z \in f^{-1} [ran f ∖ \{0\}], \frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}, 0))) = \int^{1} (f) - \int^{1} ((z \in ℝ ⟼ if (z \in f^{-1} [ran f ∖ \{0\}], \frac{\int^{1} (f) - b}{2 {vol}^{*} (f^{-1} [ran f ∖ \{0\}])}, 0)))$
507	8	recnd	$⊢ f \in dom \int^{1} \to \int^{1} (f) \in ℂ$
508	507	ad2antrr	$⊢ ((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{*} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to \int^{1} (f) \in ℂ$
509	468	ad2antlr	$⊢ ((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{*} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to b \in ℂ$
510	508 509 257 261	divsubdird	$⊢ ((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{*} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to \frac{\int^{1} (f) - b}{2} = (\frac{\int^{1} (f)}{2}) - (\frac{b}{2})$
511	510	oveq2d	$⊢ ((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{*} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to \int^{1} (f) - (\frac{\int^{1} (f) - b}{2}) = \int^{1} (f) - ((\frac{\int^{1} (f)}{2}) - (\frac{b}{2}))$
512	507	adantr	$⊢ (f \in dom \int^{1} \land b \in ℝ) \to \int^{1} (f) \in ℂ$
513	512	halfcld	$⊢ (f \in dom \int^{1} \land b \in ℝ) \to \frac{\int^{1} (f)}{2} \in ℂ$
514	471	adantl	$⊢ (f \in dom \int^{1} \land b \in ℝ) \to \frac{b}{2} \in ℂ$
515	512 513 514	subsubd	$⊢ (f \in dom \int^{1} \land b \in ℝ) \to \int^{1} (f) - ((\frac{\int^{1} (f)}{2}) - (\frac{b}{2})) = \int^{1} (f) - (\frac{\int^{1} (f)}{2}) + (\frac{b}{2})$
516	515	adantr	$⊢ ((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{*} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to \int^{1} (f) - ((\frac{\int^{1} (f)}{2}) - (\frac{b}{2})) = \int^{1} (f) - (\frac{\int^{1} (f)}{2}) + (\frac{b}{2})$
517	507	2halvesd	$⊢ f \in dom \int^{1} \to (\frac{\int^{1} (f)}{2}) + (\frac{\int^{1} (f)}{2}) = \int^{1} (f)$
518	517	oveq1d	$⊢ f \in dom \int^{1} \to (\frac{\int^{1} (f)}{2}) + (\frac{\int^{1} (f)}{2}) - (\frac{\int^{1} (f)}{2}) = \int^{1} (f) - (\frac{\int^{1} (f)}{2})$
519	507	halfcld	$⊢ f \in dom \int^{1} \to \frac{\int^{1} (f)}{2} \in ℂ$
520	519 519	pncand	$⊢ f \in dom \int^{1} \to (\frac{\int^{1} (f)}{2}) + (\frac{\int^{1} (f)}{2}) - (\frac{\int^{1} (f)}{2}) = \frac{\int^{1} (f)}{2}$
521	518 520	eqtr3d	$⊢ f \in dom \int^{1} \to \int^{1} (f) - (\frac{\int^{1} (f)}{2}) = \frac{\int^{1} (f)}{2}$
522	521	oveq1d	$⊢ f \in dom \int^{1} \to \int^{1} (f) - (\frac{\int^{1} (f)}{2}) + (\frac{b}{2}) = (\frac{\int^{1} (f)}{2}) + (\frac{b}{2})$
523	522	ad2antrr	$⊢ ((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{*} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to \int^{1} (f) - (\frac{\int^{1} (f)}{2}) + (\frac{b}{2}) = (\frac{\int^{1} (f)}{2}) + (\frac{b}{2})$
524	511 516 523	3eqtrrd	$⊢ ((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{*} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to (\frac{\int^{1} (f)}{2}) + (\frac{b}{2}) = \int^{1} (f) - (\frac{\int^{1} (f) - b}{2})$
525	504 506 524	3eqtr4d	$⊢ ((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to \int^{1} (f -_{f} (z \in ℝ ⟼ if (z \in f^{-1} [ran f ∖ \{0\}], \frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}, 0))) = (\frac{\int^{1} (f)}{2}) + (\frac{b}{2})$
526	525	adantlrl	$⊢ ((f \in dom \int^{1} \land (b < \int^{1} (f) \land b \in ℝ)) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to \int^{1} (f -_{f} (z \in ℝ ⟼ if (z \in f^{-1} [ran f ∖ \{0\}], \frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}, 0))) = (\frac{\int^{1} (f)}{2}) + (\frac{b}{2})$
527	484 526	breqtrrd	$⊢ ((f \in dom \int^{1} \land (b < \int^{1} (f) \land b \in ℝ)) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to b < \int^{1} (f -_{f} (z \in ℝ ⟼ if (z \in f^{-1} [ran f ∖ \{0\}], \frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}, 0)))$
528	456	adantlrl	$⊢ ((f \in dom \int^{1} \land (b < \int^{1} (f) \land b \in ℝ)) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to f -_{f} (z \in ℝ ⟼ if (z \in f^{-1} [ran f ∖ \{0\}], \frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}, 0)) \in dom \int^{1}$
529		id	$⊢ ((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to ((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0)$
530	529	adantlrl	$⊢ ((f \in dom \int^{1} \land (b < \int^{1} (f) \land b \in ℝ)) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to ((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0)$
531	233 36	sylan	$⊢ (((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{*} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \land z \in ℝ) \to f (z) \in ℝ$
532	264	adantr	$⊢ (((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \land z \in ℝ) \to \frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])} \in ℝ$
533	531 532	resubcld	$⊢ (((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \land z \in ℝ) \to f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \in ℝ$
534	533	leidd	$⊢ (((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \land z \in ℝ) \to f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{*} (f^{-1} [ran f ∖ \{0\}])})$
535	534	adantr	$⊢ ((((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \land z \in ℝ) \land 0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])})) \to f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])})$
536	285	breq2d	$⊢ 0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \to (f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \leq if (0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) \leftrightarrow f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}))$
537	536	adantl	$⊢ ((((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \land z \in ℝ) \land 0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])})) \to (f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \leq if (0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) \leftrightarrow f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{*} (f^{-1} [ran f ∖ \{0\}])}))$
538	535 537	mpbird	$⊢ ((((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \land z \in ℝ) \land 0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])})) \to f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \leq if (0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{*} (f^{-1} [ran f ∖ \{0\}])}), 0)$
539	533	adantr	$⊢ ((((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \land z \in ℝ) \land \neg 0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])})) \to f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{*} (f^{-1} [ran f ∖ \{0\}])}) \in ℝ$
540	48	a1i	$⊢ ((((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \land z \in ℝ) \land \neg 0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])})) \to 0 \in ℝ$
541	48	a1i	$⊢ (((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{*} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \land z \in ℝ) \to 0 \in ℝ$
542	533 541	ltnled	$⊢ (((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \land z \in ℝ) \to (f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) < 0 \leftrightarrow \neg 0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{*} (f^{-1} [ran f ∖ \{0\}])}))$
543	542	biimpar	$⊢ ((((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \land z \in ℝ) \land \neg 0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])})) \to f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{*} (f^{-1} [ran f ∖ \{0\}])}) < 0$
544	539 540 543	ltled	$⊢ ((((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \land z \in ℝ) \land \neg 0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])})) \to f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{*} (f^{-1} [ran f ∖ \{0\}])}) \leq 0$
545		iffalse	$⊢ \neg 0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \to if (0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{*} (f^{-1} [ran f ∖ \{0\}])}), 0) = 0$
546	545	breq2d	$⊢ \neg 0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \to (f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \leq if (0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) \leftrightarrow f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{*} (f^{-1} [ran f ∖ \{0\}])}) \leq 0)$
547	546	adantl	$⊢ ((((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \land z \in ℝ) \land \neg 0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])})) \to (f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \leq if (0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) \leftrightarrow f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \leq 0)$
548	544 547	mpbird	$⊢ ((((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \land z \in ℝ) \land \neg 0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])})) \to f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \leq if (0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{*} (f^{-1} [ran f ∖ \{0\}])}), 0)$
549	538 548	pm2.61dan	$⊢ (((f \in dom \int^{1} \land b \in ℝ) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \land z \in ℝ) \to f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \leq if (0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0)$
550	530 549	sylan	$⊢ (((f \in dom \int^{1} \land (b < \int^{1} (f) \land b \in ℝ)) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \land z \in ℝ) \to f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \leq if (0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0)$
551	550	adantr	$⊢ ((((f \in dom \int^{1} \land (b < \int^{1} (f) \land b \in ℝ)) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \land z \in ℝ) \land z \in f^{-1} [ran f ∖ \{0\}]) \to f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \leq if (0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0)$
552		iftrue	$⊢ z \in f^{-1} [ran f ∖ \{0\}] \to if (z \in f^{-1} [ran f ∖ \{0\}], \frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}, 0) = \frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}$
553	552	oveq2d	$⊢ z \in f^{-1} [ran f ∖ \{0\}] \to f (z) - if (z \in f^{-1} [ran f ∖ \{0\}], \frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}, 0) = f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])})$
554		iba	$⊢ z \in f^{-1} [ran f ∖ \{0\}] \to (0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \leftrightarrow (0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]))$
555	554	bicomd	$⊢ z \in f^{-1} [ran f ∖ \{0\}] \to ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]) \leftrightarrow 0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}))$
556	555	ifbid	$⊢ z \in f^{-1} [ran f ∖ \{0\}] \to if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) = if (0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0)$
557	553 556	breq12d	$⊢ z \in f^{-1} [ran f ∖ \{0\}] \to (f (z) - if (z \in f^{-1} [ran f ∖ \{0\}], \frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}, 0) \leq if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) \leftrightarrow f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \leq if (0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0))$
558	557	adantl	$⊢ ((((f \in dom \int^{1} \land (b < \int^{1} (f) \land b \in ℝ)) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \land z \in ℝ) \land z \in f^{-1} [ran f ∖ \{0\}]) \to (f (z) - if (z \in f^{-1} [ran f ∖ \{0\}], \frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}, 0) \leq if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) \leftrightarrow f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \leq if (0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{*} (f^{-1} [ran f ∖ \{0\}])}), 0))$
559	551 558	mpbird	$⊢ ((((f \in dom \int^{1} \land (b < \int^{1} (f) \land b \in ℝ)) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \land z \in ℝ) \land z \in f^{-1} [ran f ∖ \{0\}]) \to f (z) - if (z \in f^{-1} [ran f ∖ \{0\}], \frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}, 0) \leq if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0)$
560	35	ad2antrr	$⊢ ((f \in dom \int^{1} \land (b < \int^{1} (f) \land b \in ℝ)) \land {vol}^{*} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to f : ℝ ⟶ ℝ$
561	170	eleq2d	$⊢ f : ℝ ⟶ ℝ \to (z \in f^{-1} [ran f ∖ \{0\}] \leftrightarrow z \in (f^{-1} [ran f] ∖ f^{-1} [\{0\}]))$
562		eldif	$⊢ z \in (f^{-1} [ran f] ∖ f^{-1} [\{0\}]) \leftrightarrow (z \in f^{-1} [ran f] \land \neg z \in f^{-1} [\{0\}])$
563	561 562	bitrdi	$⊢ f : ℝ ⟶ ℝ \to (z \in f^{-1} [ran f ∖ \{0\}] \leftrightarrow (z \in f^{-1} [ran f] \land \neg z \in f^{-1} [\{0\}]))$
564	563	notbid	$⊢ f : ℝ ⟶ ℝ \to (\neg z \in f^{-1} [ran f ∖ \{0\}] \leftrightarrow \neg (z \in f^{-1} [ran f] \land \neg z \in f^{-1} [\{0\}]))$
565	564	adantr	$⊢ (f : ℝ ⟶ ℝ \land z \in ℝ) \to (\neg z \in f^{-1} [ran f ∖ \{0\}] \leftrightarrow \neg (z \in f^{-1} [ran f] \land \neg z \in f^{-1} [\{0\}]))$
566		pm4.53	$⊢ \neg (z \in f^{-1} [ran f] \land \neg z \in f^{-1} [\{0\}]) \leftrightarrow (\neg z \in f^{-1} [ran f] \lor z \in f^{-1} [\{0\}])$
567	207	eleq2d	$⊢ f : ℝ ⟶ ℝ \to (z \in f^{-1} [ran f] \leftrightarrow z \in ℝ)$
568	567	biimpar	$⊢ (f : ℝ ⟶ ℝ \land z \in ℝ) \to z \in f^{-1} [ran f]$
569	568	pm2.24d	$⊢ (f : ℝ ⟶ ℝ \land z \in ℝ) \to (\neg z \in f^{-1} [ran f] \to f (z) = 0)$
570	181	simplbda	$⊢ (f Fn ℝ \land z \in f^{-1} [\{0\}]) \to f (z) = 0$
571	570	ex	$⊢ f Fn ℝ \to (z \in f^{-1} [\{0\}] \to f (z) = 0)$
572	180 571	syl	$⊢ f : ℝ ⟶ ℝ \to (z \in f^{-1} [\{0\}] \to f (z) = 0)$
573	572	adantr	$⊢ (f : ℝ ⟶ ℝ \land z \in ℝ) \to (z \in f^{-1} [\{0\}] \to f (z) = 0)$
574	569 573	jaod	$⊢ (f : ℝ ⟶ ℝ \land z \in ℝ) \to ((\neg z \in f^{-1} [ran f] \lor z \in f^{-1} [\{0\}]) \to f (z) = 0)$
575	566 574	biimtrid	$⊢ (f : ℝ ⟶ ℝ \land z \in ℝ) \to (\neg (z \in f^{-1} [ran f] \land \neg z \in f^{-1} [\{0\}]) \to f (z) = 0)$
576	565 575	sylbid	$⊢ (f : ℝ ⟶ ℝ \land z \in ℝ) \to (\neg z \in f^{-1} [ran f ∖ \{0\}] \to f (z) = 0)$
577	576	imp	$⊢ ((f : ℝ ⟶ ℝ \land z \in ℝ) \land \neg z \in f^{-1} [ran f ∖ \{0\}]) \to f (z) = 0$
578	560 577	sylanl1	$⊢ ((((f \in dom \int^{1} \land (b < \int^{1} (f) \land b \in ℝ)) \land {vol}^{*} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \land z \in ℝ) \land \neg z \in f^{-1} [ran f ∖ \{0\}]) \to f (z) = 0$
579	578	oveq1d	$⊢ ((((f \in dom \int^{1} \land (b < \int^{1} (f) \land b \in ℝ)) \land {vol}^{*} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \land z \in ℝ) \land \neg z \in f^{-1} [ran f ∖ \{0\}]) \to f (z) - 0 = 0 - 0$
580	579 224	eqtrdi	$⊢ ((((f \in dom \int^{1} \land (b < \int^{1} (f) \land b \in ℝ)) \land {vol}^{*} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \land z \in ℝ) \land \neg z \in f^{-1} [ran f ∖ \{0\}]) \to f (z) - 0 = 0$
581	580 30	eqbrtrdi	$⊢ ((((f \in dom \int^{1} \land (b < \int^{1} (f) \land b \in ℝ)) \land {vol}^{*} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \land z \in ℝ) \land \neg z \in f^{-1} [ran f ∖ \{0\}]) \to f (z) - 0 \leq 0$
582		iffalse	$⊢ \neg z \in f^{-1} [ran f ∖ \{0\}] \to if (z \in f^{-1} [ran f ∖ \{0\}], \frac{\int^{1} (f) - b}{2 {vol}^{*} (f^{-1} [ran f ∖ \{0\}])}, 0) = 0$
583	582	oveq2d	$⊢ \neg z \in f^{-1} [ran f ∖ \{0\}] \to f (z) - if (z \in f^{-1} [ran f ∖ \{0\}], \frac{\int^{1} (f) - b}{2 {vol}^{*} (f^{-1} [ran f ∖ \{0\}])}, 0) = f (z) - 0$
584	289 288	sylbir	$⊢ (\neg 0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \lor \neg z \in f^{-1} [ran f ∖ \{0\}]) \to if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{*} (f^{-1} [ran f ∖ \{0\}])}), 0) = 0$
585	584	olcs	$⊢ \neg z \in f^{-1} [ran f ∖ \{0\}] \to if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) = 0$
586	583 585	breq12d	$⊢ \neg z \in f^{-1} [ran f ∖ \{0\}] \to (f (z) - if (z \in f^{-1} [ran f ∖ \{0\}], \frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}, 0) \leq if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{*} (f^{-1} [ran f ∖ \{0\}])}), 0) \leftrightarrow f (z) - 0 \leq 0)$
587	586	adantl	$⊢ ((((f \in dom \int^{1} \land (b < \int^{1} (f) \land b \in ℝ)) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \land z \in ℝ) \land \neg z \in f^{-1} [ran f ∖ \{0\}]) \to (f (z) - if (z \in f^{-1} [ran f ∖ \{0\}], \frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}, 0) \leq if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0) \leftrightarrow f (z) - 0 \leq 0)$
588	581 587	mpbird	$⊢ ((((f \in dom \int^{1} \land (b < \int^{1} (f) \land b \in ℝ)) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \land z \in ℝ) \land \neg z \in f^{-1} [ran f ∖ \{0\}]) \to f (z) - if (z \in f^{-1} [ran f ∖ \{0\}], \frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}, 0) \leq if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0)$
589	559 588	pm2.61dan	$⊢ (((f \in dom \int^{1} \land (b < \int^{1} (f) \land b \in ℝ)) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \land z \in ℝ) \to f (z) - if (z \in f^{-1} [ran f ∖ \{0\}], \frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}, 0) \leq if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0)$
590	589	ralrimiva	$⊢ ((f \in dom \int^{1} \land (b < \int^{1} (f) \land b \in ℝ)) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to \forall z \in ℝ f (z) - if (z \in f^{-1} [ran f ∖ \{0\}], \frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}, 0) \leq if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0)$
591	63	a1i	$⊢ ((f \in dom \int^{1} \land (b < \int^{1} (f) \land b \in ℝ)) \land {vol}^{*} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to ℝ \in V$
592		ovex	$⊢ f (z) - if (z \in f^{-1} [ran f ∖ \{0\}], \frac{\int^{1} (f) - b}{2 {vol}^{*} (f^{-1} [ran f ∖ \{0\}])}, 0) \in V$
593	592	a1i	$⊢ (((f \in dom \int^{1} \land (b < \int^{1} (f) \land b \in ℝ)) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \land z \in ℝ) \to f (z) - if (z \in f^{-1} [ran f ∖ \{0\}], \frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}, 0) \in V$
594	422	a1i	$⊢ (((f \in dom \int^{1} \land (b < \int^{1} (f) \land b \in ℝ)) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \land z \in ℝ) \to if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{*} (f^{-1} [ran f ∖ \{0\}])}), 0) \in V$
595		fvex	$⊢ f (z) \in V$
596	595	a1i	$⊢ (((f \in dom \int^{1} \land (b < \int^{1} (f) \land b \in ℝ)) \land {vol}^{*} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \land z \in ℝ) \to f (z) \in V$
597	199 105	ifex	$⊢ if (z \in f^{-1} [ran f ∖ \{0\}], \frac{\int^{1} (f) - b}{2 {vol}^{*} (f^{-1} [ran f ∖ \{0\}])}, 0) \in V$
598	597	a1i	$⊢ (((f \in dom \int^{1} \land (b < \int^{1} (f) \land b \in ℝ)) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \land z \in ℝ) \to if (z \in f^{-1} [ran f ∖ \{0\}], \frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}, 0) \in V$
599	70	ad2antrr	$⊢ ((f \in dom \int^{1} \land (b < \int^{1} (f) \land b \in ℝ)) \land {vol}^{*} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to f = (z \in ℝ ⟼ f (z))$
600		eqidd	$⊢ ((f \in dom \int^{1} \land (b < \int^{1} (f) \land b \in ℝ)) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to (z \in ℝ ⟼ if (z \in f^{-1} [ran f ∖ \{0\}], \frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}, 0)) = (z \in ℝ ⟼ if (z \in f^{-1} [ran f ∖ \{0\}], \frac{\int^{1} (f) - b}{2 {vol}^{*} (f^{-1} [ran f ∖ \{0\}])}, 0))$
601	591 596 598 599 600	offval2	$⊢ ((f \in dom \int^{1} \land (b < \int^{1} (f) \land b \in ℝ)) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to f -_{f} (z \in ℝ ⟼ if (z \in f^{-1} [ran f ∖ \{0\}], \frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}, 0)) = (z \in ℝ ⟼ f (z) - if (z \in f^{-1} [ran f ∖ \{0\}], \frac{\int^{1} (f) - b}{2 {vol}^{*} (f^{-1} [ran f ∖ \{0\}])}, 0))$
602		eqidd	$⊢ ((f \in dom \int^{1} \land (b < \int^{1} (f) \land b \in ℝ)) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to (z \in ℝ ⟼ if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0)) = (z \in ℝ ⟼ if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{*} (f^{-1} [ran f ∖ \{0\}])}), 0))$
603	591 593 594 601 602	ofrfval2	$⊢ ((f \in dom \int^{1} \land (b < \int^{1} (f) \land b \in ℝ)) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to ((f -_{f} (z \in ℝ ⟼ if (z \in f^{-1} [ran f ∖ \{0\}], \frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}, 0))) \leq_{f} (z \in ℝ ⟼ if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0)) \leftrightarrow \forall z \in ℝ f (z) - if (z \in f^{-1} [ran f ∖ \{0\}], \frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}, 0) \leq if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{*} (f^{-1} [ran f ∖ \{0\}])}), 0))$
604	590 603	mpbird	$⊢ ((f \in dom \int^{1} \land (b < \int^{1} (f) \land b \in ℝ)) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to (f -_{f} (z \in ℝ ⟼ if (z \in f^{-1} [ran f ∖ \{0\}], \frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}, 0))) \leq_{f} (z \in ℝ ⟼ if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0))$
605		itg1le	$⊢ (f -_{f} (z \in ℝ ⟼ if (z \in f^{-1} [ran f ∖ \{0\}], \frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}, 0)) \in dom \int^{1} \land (z \in ℝ ⟼ if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0)) \in dom \int^{1} \land (f -_{f} (z \in ℝ ⟼ if (z \in f^{-1} [ran f ∖ \{0\}], \frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}, 0))) \leq_{f} (z \in ℝ ⟼ if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0))) \to \int^{1} (f -_{f} (z \in ℝ ⟼ if (z \in f^{-1} [ran f ∖ \{0\}], \frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}, 0))) \leq \int^{1} ((z \in ℝ ⟼ if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{*} (f^{-1} [ran f ∖ \{0\}])}), 0)))$
606	528 460 604 605	syl3anc	$⊢ ((f \in dom \int^{1} \land (b < \int^{1} (f) \land b \in ℝ)) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to \int^{1} (f -_{f} (z \in ℝ ⟼ if (z \in f^{-1} [ran f ∖ \{0\}], \frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}, 0))) \leq \int^{1} ((z \in ℝ ⟼ if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0)))$
607	436 459 462 527 606	ltletrd	$⊢ ((f \in dom \int^{1} \land (b < \int^{1} (f) \land b \in ℝ)) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to b < \int^{1} ((z \in ℝ ⟼ if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{*} (f^{-1} [ran f ∖ \{0\}])}), 0)))$
608	607	adantllr	$⊢ (((f \in dom \int^{1} \land (f \leq_{f} F \land s = \int^{1} (f))) \land (b < \int^{1} (f) \land b \in ℝ)) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to b < \int^{1} ((z \in ℝ ⟼ if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{*} (f^{-1} [ran f ∖ \{0\}])}), 0)))$
609	608	adantlll	$⊢ (((F : ℝ ⟶ [0, +∞] \land (f \in dom \int^{1} \land (f \leq_{f} F \land s = \int^{1} (f)))) \land (b < \int^{1} (f) \land b \in ℝ)) \land {vol}^{} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to b < \int^{1} ((z \in ℝ ⟼ if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{*} (f^{-1} [ran f ∖ \{0\}])}), 0)))$
610		fvex	$⊢ \int^{1} ((z \in ℝ ⟼ if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0))) \in V$
611		eqeq1	$⊢ a = \int^{1} ((z \in ℝ ⟼ if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0))) \to (a = \int^{1} (g) \leftrightarrow \int^{1} ((z \in ℝ ⟼ if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0))) = \int^{1} (g))$
612	611	anbi2d	$⊢ a = \int^{1} ((z \in ℝ ⟼ if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0))) \to ((\exists y \in ℝ^{+} (z \in ℝ ⟼ if (g (z) = 0, 0, g (z) + y)) \leq_{f} F \land a = \int^{1} (g)) \leftrightarrow (\exists y \in ℝ^{+} (z \in ℝ ⟼ if (g (z) = 0, 0, g (z) + y)) \leq_{f} F \land \int^{1} ((z \in ℝ ⟼ if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0))) = \int^{1} (g)))$
613	612	rexbidv	$⊢ a = \int^{1} ((z \in ℝ ⟼ if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0))) \to (\exists g \in dom \int^{1} (\exists y \in ℝ^{+} (z \in ℝ ⟼ if (g (z) = 0, 0, g (z) + y)) \leq_{f} F \land a = \int^{1} (g)) \leftrightarrow \exists g \in dom \int^{1} (\exists y \in ℝ^{+} (z \in ℝ ⟼ if (g (z) = 0, 0, g (z) + y)) \leq_{f} F \land \int^{1} ((z \in ℝ ⟼ if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0))) = \int^{1} (g)))$
614		breq2	$⊢ a = \int^{1} ((z \in ℝ ⟼ if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0))) \to (b < a \leftrightarrow b < \int^{1} ((z \in ℝ ⟼ if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0))))$
615	613 614	anbi12d	$⊢ a = \int^{1} ((z \in ℝ ⟼ if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0))) \to ((\exists g \in dom \int^{1} (\exists y \in ℝ^{+} (z \in ℝ ⟼ if (g (z) = 0, 0, g (z) + y)) \leq_{f} F \land a = \int^{1} (g)) \land b < a) \leftrightarrow (\exists g \in dom \int^{1} (\exists y \in ℝ^{+} (z \in ℝ ⟼ if (g (z) = 0, 0, g (z) + y)) \leq_{f} F \land \int^{1} ((z \in ℝ ⟼ if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0))) = \int^{1} (g)) \land b < \int^{1} ((z \in ℝ ⟼ if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0)))))$
616	610 615	spcev	$⊢ (\exists g \in dom \int^{1} (\exists y \in ℝ^{+} (z \in ℝ ⟼ if (g (z) = 0, 0, g (z) + y)) \leq_{f} F \land \int^{1} ((z \in ℝ ⟼ if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0))) = \int^{1} (g)) \land b < \int^{1} ((z \in ℝ ⟼ if ((0 \leq f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}) \land z \in f^{-1} [ran f ∖ \{0\}]), f (z) - (\frac{\int^{1} (f) - b}{2 {vol}^{} (f^{-1} [ran f ∖ \{0\}])}), 0)))) \to \exists a (\exists g \in dom \int^{1} (\exists y \in ℝ^{+} (z \in ℝ ⟼ if (g (z) = 0, 0, g (z) + y)) \leq_{f} F \land a = \int^{1} (g)) \land b < a)$
617	435 609 616	syl2anc	$⊢ (((F : ℝ ⟶ [0, +∞] \land (f \in dom \int^{1} \land (f \leq_{f} F \land s = \int^{1} (f)))) \land (b < \int^{1} (f) \land b \in ℝ)) \land {vol}^{*} (f^{-1} [ran f ∖ \{0\}]) \neq 0) \to \exists a (\exists g \in dom \int^{1} (\exists y \in ℝ^{+} (z \in ℝ ⟼ if (g (z) = 0, 0, g (z) + y)) \leq_{f} F \land a = \int^{1} (g)) \land b < a)$
618	192 617	pm2.61dane	$⊢ ((F : ℝ ⟶ [0, +∞] \land (f \in dom \int^{1} \land (f \leq_{f} F \land s = \int^{1} (f)))) \land (b < \int^{1} (f) \land b \in ℝ)) \to \exists a (\exists g \in dom \int^{1} (\exists y \in ℝ^{+} (z \in ℝ ⟼ if (g (z) = 0, 0, g (z) + y)) \leq_{f} F \land a = \int^{1} (g)) \land b < a)$
619	618	expr	$⊢ ((F : ℝ ⟶ [0, +∞] \land (f \in dom \int^{1} \land (f \leq_{f} F \land s = \int^{1} (f)))) \land b < \int^{1} (f)) \to (b \in ℝ \to \exists a (\exists g \in dom \int^{1} (\exists y \in ℝ^{+} (z \in ℝ ⟼ if (g (z) = 0, 0, g (z) + y)) \leq_{f} F \land a = \int^{1} (g)) \land b < a))$
620	619	adantllr	$⊢ (((F : ℝ ⟶ [0, +∞] \land b \in ℝ^{*}) \land (f \in dom \int^{1} \land (f \leq_{f} F \land s = \int^{1} (f)))) \land b < \int^{1} (f)) \to (b \in ℝ \to \exists a (\exists g \in dom \int^{1} (\exists y \in ℝ^{+} (z \in ℝ ⟼ if (g (z) = 0, 0, g (z) + y)) \leq_{f} F \land a = \int^{1} (g)) \land b < a))$
621	620	adantr	$⊢ ((((F : ℝ ⟶ [0, +∞] \land b \in ℝ^{*}) \land (f \in dom \int^{1} \land (f \leq_{f} F \land s = \int^{1} (f)))) \land b < \int^{1} (f)) \land b \neq −∞) \to (b \in ℝ \to \exists a (\exists g \in dom \int^{1} (\exists y \in ℝ^{+} (z \in ℝ ⟼ if (g (z) = 0, 0, g (z) + y)) \leq_{f} F \land a = \int^{1} (g)) \land b < a))$
622	157 621	mpd	$⊢ ((((F : ℝ ⟶ [0, +∞] \land b \in ℝ^{*}) \land (f \in dom \int^{1} \land (f \leq_{f} F \land s = \int^{1} (f)))) \land b < \int^{1} (f)) \land b \neq −∞) \to \exists a (\exists g \in dom \int^{1} (\exists y \in ℝ^{+} (z \in ℝ ⟼ if (g (z) = 0, 0, g (z) + y)) \leq_{f} F \land a = \int^{1} (g)) \land b < a)$
623	139 622	pm2.61dane	$⊢ (((F : ℝ ⟶ [0, +∞] \land b \in ℝ^{*}) \land (f \in dom \int^{1} \land (f \leq_{f} F \land s = \int^{1} (f)))) \land b < \int^{1} (f)) \to \exists a (\exists g \in dom \int^{1} (\exists y \in ℝ^{+} (z \in ℝ ⟼ if (g (z) = 0, 0, g (z) + y)) \leq_{f} F \land a = \int^{1} (g)) \land b < a)$
624	623	ex	$⊢ ((F : ℝ ⟶ [0, +∞] \land b \in ℝ^{*}) \land (f \in dom \int^{1} \land (f \leq_{f} F \land s = \int^{1} (f)))) \to (b < \int^{1} (f) \to \exists a (\exists g \in dom \int^{1} (\exists y \in ℝ^{+} (z \in ℝ ⟼ if (g (z) = 0, 0, g (z) + y)) \leq_{f} F \land a = \int^{1} (g)) \land b < a))$
625	94 624	sylbid	$⊢ ((F : ℝ ⟶ [0, +∞] \land b \in ℝ^{*}) \land (f \in dom \int^{1} \land (f \leq_{f} F \land s = \int^{1} (f)))) \to (b < s \to \exists a (\exists g \in dom \int^{1} (\exists y \in ℝ^{+} (z \in ℝ ⟼ if (g (z) = 0, 0, g (z) + y)) \leq_{f} F \land a = \int^{1} (g)) \land b < a))$
626	625	imp	$⊢ (((F : ℝ ⟶ [0, +∞] \land b \in ℝ^{*}) \land (f \in dom \int^{1} \land (f \leq_{f} F \land s = \int^{1} (f)))) \land b < s) \to \exists a (\exists g \in dom \int^{1} (\exists y \in ℝ^{+} (z \in ℝ ⟼ if (g (z) = 0, 0, g (z) + y)) \leq_{f} F \land a = \int^{1} (g)) \land b < a)$
627	626	an32s	$⊢ (((F : ℝ ⟶ [0, +∞] \land b \in ℝ^{*}) \land b < s) \land (f \in dom \int^{1} \land (f \leq_{f} F \land s = \int^{1} (f)))) \to \exists a (\exists g \in dom \int^{1} (\exists y \in ℝ^{+} (z \in ℝ ⟼ if (g (z) = 0, 0, g (z) + y)) \leq_{f} F \land a = \int^{1} (g)) \land b < a)$
628	627	rexlimdvaa	$⊢ ((F : ℝ ⟶ [0, +∞] \land b \in ℝ^{*}) \land b < s) \to (\exists f \in dom \int^{1} (f \leq_{f} F \land s = \int^{1} (f)) \to \exists a (\exists g \in dom \int^{1} (\exists y \in ℝ^{+} (z \in ℝ ⟼ if (g (z) = 0, 0, g (z) + y)) \leq_{f} F \land a = \int^{1} (g)) \land b < a))$
629	628	expimpd	$⊢ (F : ℝ ⟶ [0, +∞] \land b \in ℝ^{*}) \to ((b < s \land \exists f \in dom \int^{1} (f \leq_{f} F \land s = \int^{1} (f))) \to \exists a (\exists g \in dom \int^{1} (\exists y \in ℝ^{+} (z \in ℝ ⟼ if (g (z) = 0, 0, g (z) + y)) \leq_{f} F \land a = \int^{1} (g)) \land b < a))$
630	629	ancomsd	$⊢ (F : ℝ ⟶ [0, +∞] \land b \in ℝ^{*}) \to ((\exists f \in dom \int^{1} (f \leq_{f} F \land s = \int^{1} (f)) \land b < s) \to \exists a (\exists g \in dom \int^{1} (\exists y \in ℝ^{+} (z \in ℝ ⟼ if (g (z) = 0, 0, g (z) + y)) \leq_{f} F \land a = \int^{1} (g)) \land b < a))$
631	630	exlimdv	$⊢ (F : ℝ ⟶ [0, +∞] \land b \in ℝ^{*}) \to (\exists s (\exists f \in dom \int^{1} (f \leq_{f} F \land s = \int^{1} (f)) \land b < s) \to \exists a (\exists g \in dom \int^{1} (\exists y \in ℝ^{+} (z \in ℝ ⟼ if (g (z) = 0, 0, g (z) + y)) \leq_{f} F \land a = \int^{1} (g)) \land b < a))$
632		eqeq1	$⊢ x = s \to (x = \int^{1} (f) \leftrightarrow s = \int^{1} (f))$
633	632	anbi2d	$⊢ x = s \to ((f \leq_{f} F \land x = \int^{1} (f)) \leftrightarrow (f \leq_{f} F \land s = \int^{1} (f)))$
634	633	rexbidv	$⊢ x = s \to (\exists f \in dom \int^{1} (f \leq_{f} F \land x = \int^{1} (f)) \leftrightarrow \exists f \in dom \int^{1} (f \leq_{f} F \land s = \int^{1} (f)))$
635	634	rexab	$⊢ \exists s \in \{x \| \exists f \in dom \int^{1} (f \leq_{f} F \land x = \int^{1} (f))\} b < s \leftrightarrow \exists s (\exists f \in dom \int^{1} (f \leq_{f} F \land s = \int^{1} (f)) \land b < s)$
636		eqeq1	$⊢ x = a \to (x = \int^{1} (g) \leftrightarrow a = \int^{1} (g))$
637	636	anbi2d	$⊢ x = a \to ((\exists y \in ℝ^{+} (z \in ℝ ⟼ if (g (z) = 0, 0, g (z) + y)) \leq_{f} F \land x = \int^{1} (g)) \leftrightarrow (\exists y \in ℝ^{+} (z \in ℝ ⟼ if (g (z) = 0, 0, g (z) + y)) \leq_{f} F \land a = \int^{1} (g)))$
638	637	rexbidv	$⊢ x = a \to (\exists g \in dom \int^{1} (\exists y \in ℝ^{+} (z \in ℝ ⟼ if (g (z) = 0, 0, g (z) + y)) \leq_{f} F \land x = \int^{1} (g)) \leftrightarrow \exists g \in dom \int^{1} (\exists y \in ℝ^{+} (z \in ℝ ⟼ if (g (z) = 0, 0, g (z) + y)) \leq_{f} F \land a = \int^{1} (g)))$
639	638	rexab	$⊢ \exists a \in \{x \| \exists g \in dom \int^{1} (\exists y \in ℝ^{+} (z \in ℝ ⟼ if (g (z) = 0, 0, g (z) + y)) \leq_{f} F \land x = \int^{1} (g))\} b < a \leftrightarrow \exists a (\exists g \in dom \int^{1} (\exists y \in ℝ^{+} (z \in ℝ ⟼ if (g (z) = 0, 0, g (z) + y)) \leq_{f} F \land a = \int^{1} (g)) \land b < a)$
640	631 635 639	3imtr4g	$⊢ (F : ℝ ⟶ [0, +∞] \land b \in ℝ^{*}) \to (\exists s \in \{x \| \exists f \in dom \int^{1} (f \leq_{f} F \land x = \int^{1} (f))\} b < s \to \exists a \in \{x \| \exists g \in dom \int^{1} (\exists y \in ℝ^{+} (z \in ℝ ⟼ if (g (z) = 0, 0, g (z) + y)) \leq_{f} F \land x = \int^{1} (g))\} b < a)$
641	92 640	sylbid	$⊢ (F : ℝ ⟶ [0, +∞] \land b \in ℝ^{}) \to (b < sup (\{x \| \exists f \in dom \int^{1} (f \leq_{f} F \land x = \int^{1} (f))\} ℝ^{} <) \to \exists a \in \{x \| \exists g \in dom \int^{1} (\exists y \in ℝ^{+} (z \in ℝ ⟼ if (g (z) = 0, 0, g (z) + y)) \leq_{f} F \land x = \int^{1} (g))\} b < a)$
642	641	impr	$⊢ (F : ℝ ⟶ [0, +∞] \land (b \in ℝ^{} \land b < sup (\{x \| \exists f \in dom \int^{1} (f \leq_{f} F \land x = \int^{1} (f))\} ℝ^{} <))) \to \exists a \in \{x \| \exists g \in dom \int^{1} (\exists y \in ℝ^{+} (z \in ℝ ⟼ if (g (z) = 0, 0, g (z) + y)) \leq_{f} F \land x = \int^{1} (g))\} b < a$
643	6 15 89 642	eqsupd	$⊢ F : ℝ ⟶ [0, +∞] \to sup (\{x \| \exists g \in dom \int^{1} (\exists y \in ℝ^{+} (z \in ℝ ⟼ if (g (z) = 0, 0, g (z) + y)) \leq_{f} F \land x = \int^{1} (g))\} ℝ^{} <) = sup (\{x \| \exists f \in dom \int^{1} (f \leq_{f} F \land x = \int^{1} (f))\} ℝ^{} <)$
644	4 643	eqtrid	$⊢ F : ℝ ⟶ [0, +∞] \to sup (L ℝ^{} <) = sup (\{x \| \exists f \in dom \int^{1} (f \leq_{f} F \land x = \int^{1} (f))\} ℝ^{} <)$
645	3 644	eqtr4d	$⊢ F : ℝ ⟶ [0, +∞] \to \int^{2} (F) = sup (L ℝ^{*} <)$

Metamath Proof Explorer

Theorem itg2addnclem

Proof