Step |
Hyp |
Ref |
Expression |
1 |
|
df-3 |
|- 3 = ( 2 + 1 ) |
2 |
1
|
oveq2i |
|- ( ( A + B ) ^ 3 ) = ( ( A + B ) ^ ( 2 + 1 ) ) |
3 |
|
addcl |
|- ( ( A e. CC /\ B e. CC ) -> ( A + B ) e. CC ) |
4 |
|
2nn0 |
|- 2 e. NN0 |
5 |
|
expp1 |
|- ( ( ( A + B ) e. CC /\ 2 e. NN0 ) -> ( ( A + B ) ^ ( 2 + 1 ) ) = ( ( ( A + B ) ^ 2 ) x. ( A + B ) ) ) |
6 |
3 4 5
|
sylancl |
|- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) ^ ( 2 + 1 ) ) = ( ( ( A + B ) ^ 2 ) x. ( A + B ) ) ) |
7 |
2 6
|
eqtrid |
|- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) ^ 3 ) = ( ( ( A + B ) ^ 2 ) x. ( A + B ) ) ) |
8 |
|
sqcl |
|- ( ( A + B ) e. CC -> ( ( A + B ) ^ 2 ) e. CC ) |
9 |
3 8
|
syl |
|- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) ^ 2 ) e. CC ) |
10 |
|
simpl |
|- ( ( A e. CC /\ B e. CC ) -> A e. CC ) |
11 |
|
simpr |
|- ( ( A e. CC /\ B e. CC ) -> B e. CC ) |
12 |
9 10 11
|
adddid |
|- ( ( A e. CC /\ B e. CC ) -> ( ( ( A + B ) ^ 2 ) x. ( A + B ) ) = ( ( ( ( A + B ) ^ 2 ) x. A ) + ( ( ( A + B ) ^ 2 ) x. B ) ) ) |
13 |
|
binom2 |
|- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) ^ 2 ) = ( ( ( A ^ 2 ) + ( 2 x. ( A x. B ) ) ) + ( B ^ 2 ) ) ) |
14 |
13
|
oveq1d |
|- ( ( A e. CC /\ B e. CC ) -> ( ( ( A + B ) ^ 2 ) x. A ) = ( ( ( ( A ^ 2 ) + ( 2 x. ( A x. B ) ) ) + ( B ^ 2 ) ) x. A ) ) |
15 |
|
sqcl |
|- ( A e. CC -> ( A ^ 2 ) e. CC ) |
16 |
10 15
|
syl |
|- ( ( A e. CC /\ B e. CC ) -> ( A ^ 2 ) e. CC ) |
17 |
|
2cn |
|- 2 e. CC |
18 |
|
mulcl |
|- ( ( A e. CC /\ B e. CC ) -> ( A x. B ) e. CC ) |
19 |
|
mulcl |
|- ( ( 2 e. CC /\ ( A x. B ) e. CC ) -> ( 2 x. ( A x. B ) ) e. CC ) |
20 |
17 18 19
|
sylancr |
|- ( ( A e. CC /\ B e. CC ) -> ( 2 x. ( A x. B ) ) e. CC ) |
21 |
16 20
|
addcld |
|- ( ( A e. CC /\ B e. CC ) -> ( ( A ^ 2 ) + ( 2 x. ( A x. B ) ) ) e. CC ) |
22 |
|
sqcl |
|- ( B e. CC -> ( B ^ 2 ) e. CC ) |
23 |
11 22
|
syl |
|- ( ( A e. CC /\ B e. CC ) -> ( B ^ 2 ) e. CC ) |
24 |
21 23 10
|
adddird |
|- ( ( A e. CC /\ B e. CC ) -> ( ( ( ( A ^ 2 ) + ( 2 x. ( A x. B ) ) ) + ( B ^ 2 ) ) x. A ) = ( ( ( ( A ^ 2 ) + ( 2 x. ( A x. B ) ) ) x. A ) + ( ( B ^ 2 ) x. A ) ) ) |
25 |
16 20 10
|
adddird |
|- ( ( A e. CC /\ B e. CC ) -> ( ( ( A ^ 2 ) + ( 2 x. ( A x. B ) ) ) x. A ) = ( ( ( A ^ 2 ) x. A ) + ( ( 2 x. ( A x. B ) ) x. A ) ) ) |
26 |
1
|
oveq2i |
|- ( A ^ 3 ) = ( A ^ ( 2 + 1 ) ) |
27 |
|
expp1 |
|- ( ( A e. CC /\ 2 e. NN0 ) -> ( A ^ ( 2 + 1 ) ) = ( ( A ^ 2 ) x. A ) ) |
28 |
10 4 27
|
sylancl |
|- ( ( A e. CC /\ B e. CC ) -> ( A ^ ( 2 + 1 ) ) = ( ( A ^ 2 ) x. A ) ) |
29 |
26 28
|
eqtrid |
|- ( ( A e. CC /\ B e. CC ) -> ( A ^ 3 ) = ( ( A ^ 2 ) x. A ) ) |
30 |
|
sqval |
|- ( A e. CC -> ( A ^ 2 ) = ( A x. A ) ) |
31 |
10 30
|
syl |
|- ( ( A e. CC /\ B e. CC ) -> ( A ^ 2 ) = ( A x. A ) ) |
32 |
31
|
oveq1d |
|- ( ( A e. CC /\ B e. CC ) -> ( ( A ^ 2 ) x. B ) = ( ( A x. A ) x. B ) ) |
33 |
10 10 11
|
mul32d |
|- ( ( A e. CC /\ B e. CC ) -> ( ( A x. A ) x. B ) = ( ( A x. B ) x. A ) ) |
34 |
32 33
|
eqtrd |
|- ( ( A e. CC /\ B e. CC ) -> ( ( A ^ 2 ) x. B ) = ( ( A x. B ) x. A ) ) |
35 |
34
|
oveq2d |
|- ( ( A e. CC /\ B e. CC ) -> ( 2 x. ( ( A ^ 2 ) x. B ) ) = ( 2 x. ( ( A x. B ) x. A ) ) ) |
36 |
|
2cnd |
|- ( ( A e. CC /\ B e. CC ) -> 2 e. CC ) |
37 |
36 18 10
|
mulassd |
|- ( ( A e. CC /\ B e. CC ) -> ( ( 2 x. ( A x. B ) ) x. A ) = ( 2 x. ( ( A x. B ) x. A ) ) ) |
38 |
35 37
|
eqtr4d |
|- ( ( A e. CC /\ B e. CC ) -> ( 2 x. ( ( A ^ 2 ) x. B ) ) = ( ( 2 x. ( A x. B ) ) x. A ) ) |
39 |
29 38
|
oveq12d |
|- ( ( A e. CC /\ B e. CC ) -> ( ( A ^ 3 ) + ( 2 x. ( ( A ^ 2 ) x. B ) ) ) = ( ( ( A ^ 2 ) x. A ) + ( ( 2 x. ( A x. B ) ) x. A ) ) ) |
40 |
25 39
|
eqtr4d |
|- ( ( A e. CC /\ B e. CC ) -> ( ( ( A ^ 2 ) + ( 2 x. ( A x. B ) ) ) x. A ) = ( ( A ^ 3 ) + ( 2 x. ( ( A ^ 2 ) x. B ) ) ) ) |
41 |
23 10
|
mulcomd |
|- ( ( A e. CC /\ B e. CC ) -> ( ( B ^ 2 ) x. A ) = ( A x. ( B ^ 2 ) ) ) |
42 |
40 41
|
oveq12d |
|- ( ( A e. CC /\ B e. CC ) -> ( ( ( ( A ^ 2 ) + ( 2 x. ( A x. B ) ) ) x. A ) + ( ( B ^ 2 ) x. A ) ) = ( ( ( A ^ 3 ) + ( 2 x. ( ( A ^ 2 ) x. B ) ) ) + ( A x. ( B ^ 2 ) ) ) ) |
43 |
14 24 42
|
3eqtrd |
|- ( ( A e. CC /\ B e. CC ) -> ( ( ( A + B ) ^ 2 ) x. A ) = ( ( ( A ^ 3 ) + ( 2 x. ( ( A ^ 2 ) x. B ) ) ) + ( A x. ( B ^ 2 ) ) ) ) |
44 |
13
|
oveq1d |
|- ( ( A e. CC /\ B e. CC ) -> ( ( ( A + B ) ^ 2 ) x. B ) = ( ( ( ( A ^ 2 ) + ( 2 x. ( A x. B ) ) ) + ( B ^ 2 ) ) x. B ) ) |
45 |
21 23 11
|
adddird |
|- ( ( A e. CC /\ B e. CC ) -> ( ( ( ( A ^ 2 ) + ( 2 x. ( A x. B ) ) ) + ( B ^ 2 ) ) x. B ) = ( ( ( ( A ^ 2 ) + ( 2 x. ( A x. B ) ) ) x. B ) + ( ( B ^ 2 ) x. B ) ) ) |
46 |
|
sqval |
|- ( B e. CC -> ( B ^ 2 ) = ( B x. B ) ) |
47 |
11 46
|
syl |
|- ( ( A e. CC /\ B e. CC ) -> ( B ^ 2 ) = ( B x. B ) ) |
48 |
47
|
oveq2d |
|- ( ( A e. CC /\ B e. CC ) -> ( A x. ( B ^ 2 ) ) = ( A x. ( B x. B ) ) ) |
49 |
10 11 11
|
mulassd |
|- ( ( A e. CC /\ B e. CC ) -> ( ( A x. B ) x. B ) = ( A x. ( B x. B ) ) ) |
50 |
48 49
|
eqtr4d |
|- ( ( A e. CC /\ B e. CC ) -> ( A x. ( B ^ 2 ) ) = ( ( A x. B ) x. B ) ) |
51 |
50
|
oveq2d |
|- ( ( A e. CC /\ B e. CC ) -> ( 2 x. ( A x. ( B ^ 2 ) ) ) = ( 2 x. ( ( A x. B ) x. B ) ) ) |
52 |
36 18 11
|
mulassd |
|- ( ( A e. CC /\ B e. CC ) -> ( ( 2 x. ( A x. B ) ) x. B ) = ( 2 x. ( ( A x. B ) x. B ) ) ) |
53 |
51 52
|
eqtr4d |
|- ( ( A e. CC /\ B e. CC ) -> ( 2 x. ( A x. ( B ^ 2 ) ) ) = ( ( 2 x. ( A x. B ) ) x. B ) ) |
54 |
53
|
oveq2d |
|- ( ( A e. CC /\ B e. CC ) -> ( ( ( A ^ 2 ) x. B ) + ( 2 x. ( A x. ( B ^ 2 ) ) ) ) = ( ( ( A ^ 2 ) x. B ) + ( ( 2 x. ( A x. B ) ) x. B ) ) ) |
55 |
16 20 11
|
adddird |
|- ( ( A e. CC /\ B e. CC ) -> ( ( ( A ^ 2 ) + ( 2 x. ( A x. B ) ) ) x. B ) = ( ( ( A ^ 2 ) x. B ) + ( ( 2 x. ( A x. B ) ) x. B ) ) ) |
56 |
54 55
|
eqtr4d |
|- ( ( A e. CC /\ B e. CC ) -> ( ( ( A ^ 2 ) x. B ) + ( 2 x. ( A x. ( B ^ 2 ) ) ) ) = ( ( ( A ^ 2 ) + ( 2 x. ( A x. B ) ) ) x. B ) ) |
57 |
1
|
oveq2i |
|- ( B ^ 3 ) = ( B ^ ( 2 + 1 ) ) |
58 |
|
expp1 |
|- ( ( B e. CC /\ 2 e. NN0 ) -> ( B ^ ( 2 + 1 ) ) = ( ( B ^ 2 ) x. B ) ) |
59 |
11 4 58
|
sylancl |
|- ( ( A e. CC /\ B e. CC ) -> ( B ^ ( 2 + 1 ) ) = ( ( B ^ 2 ) x. B ) ) |
60 |
57 59
|
eqtrid |
|- ( ( A e. CC /\ B e. CC ) -> ( B ^ 3 ) = ( ( B ^ 2 ) x. B ) ) |
61 |
56 60
|
oveq12d |
|- ( ( A e. CC /\ B e. CC ) -> ( ( ( ( A ^ 2 ) x. B ) + ( 2 x. ( A x. ( B ^ 2 ) ) ) ) + ( B ^ 3 ) ) = ( ( ( ( A ^ 2 ) + ( 2 x. ( A x. B ) ) ) x. B ) + ( ( B ^ 2 ) x. B ) ) ) |
62 |
16 11
|
mulcld |
|- ( ( A e. CC /\ B e. CC ) -> ( ( A ^ 2 ) x. B ) e. CC ) |
63 |
10 23
|
mulcld |
|- ( ( A e. CC /\ B e. CC ) -> ( A x. ( B ^ 2 ) ) e. CC ) |
64 |
|
mulcl |
|- ( ( 2 e. CC /\ ( A x. ( B ^ 2 ) ) e. CC ) -> ( 2 x. ( A x. ( B ^ 2 ) ) ) e. CC ) |
65 |
17 63 64
|
sylancr |
|- ( ( A e. CC /\ B e. CC ) -> ( 2 x. ( A x. ( B ^ 2 ) ) ) e. CC ) |
66 |
|
3nn0 |
|- 3 e. NN0 |
67 |
|
expcl |
|- ( ( B e. CC /\ 3 e. NN0 ) -> ( B ^ 3 ) e. CC ) |
68 |
11 66 67
|
sylancl |
|- ( ( A e. CC /\ B e. CC ) -> ( B ^ 3 ) e. CC ) |
69 |
62 65 68
|
addassd |
|- ( ( A e. CC /\ B e. CC ) -> ( ( ( ( A ^ 2 ) x. B ) + ( 2 x. ( A x. ( B ^ 2 ) ) ) ) + ( B ^ 3 ) ) = ( ( ( A ^ 2 ) x. B ) + ( ( 2 x. ( A x. ( B ^ 2 ) ) ) + ( B ^ 3 ) ) ) ) |
70 |
61 69
|
eqtr3d |
|- ( ( A e. CC /\ B e. CC ) -> ( ( ( ( A ^ 2 ) + ( 2 x. ( A x. B ) ) ) x. B ) + ( ( B ^ 2 ) x. B ) ) = ( ( ( A ^ 2 ) x. B ) + ( ( 2 x. ( A x. ( B ^ 2 ) ) ) + ( B ^ 3 ) ) ) ) |
71 |
44 45 70
|
3eqtrd |
|- ( ( A e. CC /\ B e. CC ) -> ( ( ( A + B ) ^ 2 ) x. B ) = ( ( ( A ^ 2 ) x. B ) + ( ( 2 x. ( A x. ( B ^ 2 ) ) ) + ( B ^ 3 ) ) ) ) |
72 |
43 71
|
oveq12d |
|- ( ( A e. CC /\ B e. CC ) -> ( ( ( ( A + B ) ^ 2 ) x. A ) + ( ( ( A + B ) ^ 2 ) x. B ) ) = ( ( ( ( A ^ 3 ) + ( 2 x. ( ( A ^ 2 ) x. B ) ) ) + ( A x. ( B ^ 2 ) ) ) + ( ( ( A ^ 2 ) x. B ) + ( ( 2 x. ( A x. ( B ^ 2 ) ) ) + ( B ^ 3 ) ) ) ) ) |
73 |
|
expcl |
|- ( ( A e. CC /\ 3 e. NN0 ) -> ( A ^ 3 ) e. CC ) |
74 |
10 66 73
|
sylancl |
|- ( ( A e. CC /\ B e. CC ) -> ( A ^ 3 ) e. CC ) |
75 |
|
mulcl |
|- ( ( 2 e. CC /\ ( ( A ^ 2 ) x. B ) e. CC ) -> ( 2 x. ( ( A ^ 2 ) x. B ) ) e. CC ) |
76 |
17 62 75
|
sylancr |
|- ( ( A e. CC /\ B e. CC ) -> ( 2 x. ( ( A ^ 2 ) x. B ) ) e. CC ) |
77 |
74 76
|
addcld |
|- ( ( A e. CC /\ B e. CC ) -> ( ( A ^ 3 ) + ( 2 x. ( ( A ^ 2 ) x. B ) ) ) e. CC ) |
78 |
65 68
|
addcld |
|- ( ( A e. CC /\ B e. CC ) -> ( ( 2 x. ( A x. ( B ^ 2 ) ) ) + ( B ^ 3 ) ) e. CC ) |
79 |
77 63 62 78
|
add4d |
|- ( ( A e. CC /\ B e. CC ) -> ( ( ( ( A ^ 3 ) + ( 2 x. ( ( A ^ 2 ) x. B ) ) ) + ( A x. ( B ^ 2 ) ) ) + ( ( ( A ^ 2 ) x. B ) + ( ( 2 x. ( A x. ( B ^ 2 ) ) ) + ( B ^ 3 ) ) ) ) = ( ( ( ( A ^ 3 ) + ( 2 x. ( ( A ^ 2 ) x. B ) ) ) + ( ( A ^ 2 ) x. B ) ) + ( ( A x. ( B ^ 2 ) ) + ( ( 2 x. ( A x. ( B ^ 2 ) ) ) + ( B ^ 3 ) ) ) ) ) |
80 |
12 72 79
|
3eqtrd |
|- ( ( A e. CC /\ B e. CC ) -> ( ( ( A + B ) ^ 2 ) x. ( A + B ) ) = ( ( ( ( A ^ 3 ) + ( 2 x. ( ( A ^ 2 ) x. B ) ) ) + ( ( A ^ 2 ) x. B ) ) + ( ( A x. ( B ^ 2 ) ) + ( ( 2 x. ( A x. ( B ^ 2 ) ) ) + ( B ^ 3 ) ) ) ) ) |
81 |
74 76 62
|
addassd |
|- ( ( A e. CC /\ B e. CC ) -> ( ( ( A ^ 3 ) + ( 2 x. ( ( A ^ 2 ) x. B ) ) ) + ( ( A ^ 2 ) x. B ) ) = ( ( A ^ 3 ) + ( ( 2 x. ( ( A ^ 2 ) x. B ) ) + ( ( A ^ 2 ) x. B ) ) ) ) |
82 |
1
|
oveq1i |
|- ( 3 x. ( ( A ^ 2 ) x. B ) ) = ( ( 2 + 1 ) x. ( ( A ^ 2 ) x. B ) ) |
83 |
|
1cnd |
|- ( ( A e. CC /\ B e. CC ) -> 1 e. CC ) |
84 |
36 83 62
|
adddird |
|- ( ( A e. CC /\ B e. CC ) -> ( ( 2 + 1 ) x. ( ( A ^ 2 ) x. B ) ) = ( ( 2 x. ( ( A ^ 2 ) x. B ) ) + ( 1 x. ( ( A ^ 2 ) x. B ) ) ) ) |
85 |
82 84
|
eqtrid |
|- ( ( A e. CC /\ B e. CC ) -> ( 3 x. ( ( A ^ 2 ) x. B ) ) = ( ( 2 x. ( ( A ^ 2 ) x. B ) ) + ( 1 x. ( ( A ^ 2 ) x. B ) ) ) ) |
86 |
62
|
mulid2d |
|- ( ( A e. CC /\ B e. CC ) -> ( 1 x. ( ( A ^ 2 ) x. B ) ) = ( ( A ^ 2 ) x. B ) ) |
87 |
86
|
oveq2d |
|- ( ( A e. CC /\ B e. CC ) -> ( ( 2 x. ( ( A ^ 2 ) x. B ) ) + ( 1 x. ( ( A ^ 2 ) x. B ) ) ) = ( ( 2 x. ( ( A ^ 2 ) x. B ) ) + ( ( A ^ 2 ) x. B ) ) ) |
88 |
85 87
|
eqtrd |
|- ( ( A e. CC /\ B e. CC ) -> ( 3 x. ( ( A ^ 2 ) x. B ) ) = ( ( 2 x. ( ( A ^ 2 ) x. B ) ) + ( ( A ^ 2 ) x. B ) ) ) |
89 |
88
|
oveq2d |
|- ( ( A e. CC /\ B e. CC ) -> ( ( A ^ 3 ) + ( 3 x. ( ( A ^ 2 ) x. B ) ) ) = ( ( A ^ 3 ) + ( ( 2 x. ( ( A ^ 2 ) x. B ) ) + ( ( A ^ 2 ) x. B ) ) ) ) |
90 |
81 89
|
eqtr4d |
|- ( ( A e. CC /\ B e. CC ) -> ( ( ( A ^ 3 ) + ( 2 x. ( ( A ^ 2 ) x. B ) ) ) + ( ( A ^ 2 ) x. B ) ) = ( ( A ^ 3 ) + ( 3 x. ( ( A ^ 2 ) x. B ) ) ) ) |
91 |
|
1p2e3 |
|- ( 1 + 2 ) = 3 |
92 |
91
|
oveq1i |
|- ( ( 1 + 2 ) x. ( A x. ( B ^ 2 ) ) ) = ( 3 x. ( A x. ( B ^ 2 ) ) ) |
93 |
83 36 63
|
adddird |
|- ( ( A e. CC /\ B e. CC ) -> ( ( 1 + 2 ) x. ( A x. ( B ^ 2 ) ) ) = ( ( 1 x. ( A x. ( B ^ 2 ) ) ) + ( 2 x. ( A x. ( B ^ 2 ) ) ) ) ) |
94 |
92 93
|
eqtr3id |
|- ( ( A e. CC /\ B e. CC ) -> ( 3 x. ( A x. ( B ^ 2 ) ) ) = ( ( 1 x. ( A x. ( B ^ 2 ) ) ) + ( 2 x. ( A x. ( B ^ 2 ) ) ) ) ) |
95 |
63
|
mulid2d |
|- ( ( A e. CC /\ B e. CC ) -> ( 1 x. ( A x. ( B ^ 2 ) ) ) = ( A x. ( B ^ 2 ) ) ) |
96 |
95
|
oveq1d |
|- ( ( A e. CC /\ B e. CC ) -> ( ( 1 x. ( A x. ( B ^ 2 ) ) ) + ( 2 x. ( A x. ( B ^ 2 ) ) ) ) = ( ( A x. ( B ^ 2 ) ) + ( 2 x. ( A x. ( B ^ 2 ) ) ) ) ) |
97 |
94 96
|
eqtrd |
|- ( ( A e. CC /\ B e. CC ) -> ( 3 x. ( A x. ( B ^ 2 ) ) ) = ( ( A x. ( B ^ 2 ) ) + ( 2 x. ( A x. ( B ^ 2 ) ) ) ) ) |
98 |
97
|
oveq1d |
|- ( ( A e. CC /\ B e. CC ) -> ( ( 3 x. ( A x. ( B ^ 2 ) ) ) + ( B ^ 3 ) ) = ( ( ( A x. ( B ^ 2 ) ) + ( 2 x. ( A x. ( B ^ 2 ) ) ) ) + ( B ^ 3 ) ) ) |
99 |
63 65 68
|
addassd |
|- ( ( A e. CC /\ B e. CC ) -> ( ( ( A x. ( B ^ 2 ) ) + ( 2 x. ( A x. ( B ^ 2 ) ) ) ) + ( B ^ 3 ) ) = ( ( A x. ( B ^ 2 ) ) + ( ( 2 x. ( A x. ( B ^ 2 ) ) ) + ( B ^ 3 ) ) ) ) |
100 |
98 99
|
eqtr2d |
|- ( ( A e. CC /\ B e. CC ) -> ( ( A x. ( B ^ 2 ) ) + ( ( 2 x. ( A x. ( B ^ 2 ) ) ) + ( B ^ 3 ) ) ) = ( ( 3 x. ( A x. ( B ^ 2 ) ) ) + ( B ^ 3 ) ) ) |
101 |
90 100
|
oveq12d |
|- ( ( A e. CC /\ B e. CC ) -> ( ( ( ( A ^ 3 ) + ( 2 x. ( ( A ^ 2 ) x. B ) ) ) + ( ( A ^ 2 ) x. B ) ) + ( ( A x. ( B ^ 2 ) ) + ( ( 2 x. ( A x. ( B ^ 2 ) ) ) + ( B ^ 3 ) ) ) ) = ( ( ( A ^ 3 ) + ( 3 x. ( ( A ^ 2 ) x. B ) ) ) + ( ( 3 x. ( A x. ( B ^ 2 ) ) ) + ( B ^ 3 ) ) ) ) |
102 |
7 80 101
|
3eqtrd |
|- ( ( A e. CC /\ B e. CC ) -> ( ( A + B ) ^ 3 ) = ( ( ( A ^ 3 ) + ( 3 x. ( ( A ^ 2 ) x. B ) ) ) + ( ( 3 x. ( A x. ( B ^ 2 ) ) ) + ( B ^ 3 ) ) ) ) |