Step |
Hyp |
Ref |
Expression |
1 |
|
ptunhmeo.x |
|- X = U. K |
2 |
|
ptunhmeo.y |
|- Y = U. L |
3 |
|
ptunhmeo.j |
|- J = ( Xt_ ` F ) |
4 |
|
ptunhmeo.k |
|- K = ( Xt_ ` ( F |` A ) ) |
5 |
|
ptunhmeo.l |
|- L = ( Xt_ ` ( F |` B ) ) |
6 |
|
ptunhmeo.g |
|- G = ( x e. X , y e. Y |-> ( x u. y ) ) |
7 |
|
ptunhmeo.c |
|- ( ph -> C e. V ) |
8 |
|
ptunhmeo.f |
|- ( ph -> F : C --> Top ) |
9 |
|
ptunhmeo.u |
|- ( ph -> C = ( A u. B ) ) |
10 |
|
ptunhmeo.i |
|- ( ph -> ( A i^i B ) = (/) ) |
11 |
|
vex |
|- x e. _V |
12 |
|
vex |
|- y e. _V |
13 |
11 12
|
op1std |
|- ( z = <. x , y >. -> ( 1st ` z ) = x ) |
14 |
11 12
|
op2ndd |
|- ( z = <. x , y >. -> ( 2nd ` z ) = y ) |
15 |
13 14
|
uneq12d |
|- ( z = <. x , y >. -> ( ( 1st ` z ) u. ( 2nd ` z ) ) = ( x u. y ) ) |
16 |
15
|
mpompt |
|- ( z e. ( X X. Y ) |-> ( ( 1st ` z ) u. ( 2nd ` z ) ) ) = ( x e. X , y e. Y |-> ( x u. y ) ) |
17 |
6 16
|
eqtr4i |
|- G = ( z e. ( X X. Y ) |-> ( ( 1st ` z ) u. ( 2nd ` z ) ) ) |
18 |
|
xp1st |
|- ( z e. ( X X. Y ) -> ( 1st ` z ) e. X ) |
19 |
18
|
adantl |
|- ( ( ph /\ z e. ( X X. Y ) ) -> ( 1st ` z ) e. X ) |
20 |
|
ixpeq2 |
|- ( A. n e. A U. ( ( F |` A ) ` n ) = U. ( F ` n ) -> X_ n e. A U. ( ( F |` A ) ` n ) = X_ n e. A U. ( F ` n ) ) |
21 |
|
fvres |
|- ( n e. A -> ( ( F |` A ) ` n ) = ( F ` n ) ) |
22 |
21
|
unieqd |
|- ( n e. A -> U. ( ( F |` A ) ` n ) = U. ( F ` n ) ) |
23 |
20 22
|
mprg |
|- X_ n e. A U. ( ( F |` A ) ` n ) = X_ n e. A U. ( F ` n ) |
24 |
|
ssun1 |
|- A C_ ( A u. B ) |
25 |
24 9
|
sseqtrrid |
|- ( ph -> A C_ C ) |
26 |
7 25
|
ssexd |
|- ( ph -> A e. _V ) |
27 |
8 25
|
fssresd |
|- ( ph -> ( F |` A ) : A --> Top ) |
28 |
4
|
ptuni |
|- ( ( A e. _V /\ ( F |` A ) : A --> Top ) -> X_ n e. A U. ( ( F |` A ) ` n ) = U. K ) |
29 |
26 27 28
|
syl2anc |
|- ( ph -> X_ n e. A U. ( ( F |` A ) ` n ) = U. K ) |
30 |
23 29
|
syl5eqr |
|- ( ph -> X_ n e. A U. ( F ` n ) = U. K ) |
31 |
30 1
|
eqtr4di |
|- ( ph -> X_ n e. A U. ( F ` n ) = X ) |
32 |
31
|
adantr |
|- ( ( ph /\ z e. ( X X. Y ) ) -> X_ n e. A U. ( F ` n ) = X ) |
33 |
19 32
|
eleqtrrd |
|- ( ( ph /\ z e. ( X X. Y ) ) -> ( 1st ` z ) e. X_ n e. A U. ( F ` n ) ) |
34 |
|
xp2nd |
|- ( z e. ( X X. Y ) -> ( 2nd ` z ) e. Y ) |
35 |
34
|
adantl |
|- ( ( ph /\ z e. ( X X. Y ) ) -> ( 2nd ` z ) e. Y ) |
36 |
9
|
eqcomd |
|- ( ph -> ( A u. B ) = C ) |
37 |
|
uneqdifeq |
|- ( ( A C_ C /\ ( A i^i B ) = (/) ) -> ( ( A u. B ) = C <-> ( C \ A ) = B ) ) |
38 |
25 10 37
|
syl2anc |
|- ( ph -> ( ( A u. B ) = C <-> ( C \ A ) = B ) ) |
39 |
36 38
|
mpbid |
|- ( ph -> ( C \ A ) = B ) |
40 |
39
|
ixpeq1d |
|- ( ph -> X_ n e. ( C \ A ) U. ( F ` n ) = X_ n e. B U. ( F ` n ) ) |
41 |
|
ixpeq2 |
|- ( A. n e. B U. ( ( F |` B ) ` n ) = U. ( F ` n ) -> X_ n e. B U. ( ( F |` B ) ` n ) = X_ n e. B U. ( F ` n ) ) |
42 |
|
fvres |
|- ( n e. B -> ( ( F |` B ) ` n ) = ( F ` n ) ) |
43 |
42
|
unieqd |
|- ( n e. B -> U. ( ( F |` B ) ` n ) = U. ( F ` n ) ) |
44 |
41 43
|
mprg |
|- X_ n e. B U. ( ( F |` B ) ` n ) = X_ n e. B U. ( F ` n ) |
45 |
|
ssun2 |
|- B C_ ( A u. B ) |
46 |
45 9
|
sseqtrrid |
|- ( ph -> B C_ C ) |
47 |
7 46
|
ssexd |
|- ( ph -> B e. _V ) |
48 |
8 46
|
fssresd |
|- ( ph -> ( F |` B ) : B --> Top ) |
49 |
5
|
ptuni |
|- ( ( B e. _V /\ ( F |` B ) : B --> Top ) -> X_ n e. B U. ( ( F |` B ) ` n ) = U. L ) |
50 |
47 48 49
|
syl2anc |
|- ( ph -> X_ n e. B U. ( ( F |` B ) ` n ) = U. L ) |
51 |
44 50
|
syl5eqr |
|- ( ph -> X_ n e. B U. ( F ` n ) = U. L ) |
52 |
51 2
|
eqtr4di |
|- ( ph -> X_ n e. B U. ( F ` n ) = Y ) |
53 |
40 52
|
eqtrd |
|- ( ph -> X_ n e. ( C \ A ) U. ( F ` n ) = Y ) |
54 |
53
|
adantr |
|- ( ( ph /\ z e. ( X X. Y ) ) -> X_ n e. ( C \ A ) U. ( F ` n ) = Y ) |
55 |
35 54
|
eleqtrrd |
|- ( ( ph /\ z e. ( X X. Y ) ) -> ( 2nd ` z ) e. X_ n e. ( C \ A ) U. ( F ` n ) ) |
56 |
25
|
adantr |
|- ( ( ph /\ z e. ( X X. Y ) ) -> A C_ C ) |
57 |
|
undifixp |
|- ( ( ( 1st ` z ) e. X_ n e. A U. ( F ` n ) /\ ( 2nd ` z ) e. X_ n e. ( C \ A ) U. ( F ` n ) /\ A C_ C ) -> ( ( 1st ` z ) u. ( 2nd ` z ) ) e. X_ n e. C U. ( F ` n ) ) |
58 |
33 55 56 57
|
syl3anc |
|- ( ( ph /\ z e. ( X X. Y ) ) -> ( ( 1st ` z ) u. ( 2nd ` z ) ) e. X_ n e. C U. ( F ` n ) ) |
59 |
|
ixpfn |
|- ( ( ( 1st ` z ) u. ( 2nd ` z ) ) e. X_ n e. C U. ( F ` n ) -> ( ( 1st ` z ) u. ( 2nd ` z ) ) Fn C ) |
60 |
58 59
|
syl |
|- ( ( ph /\ z e. ( X X. Y ) ) -> ( ( 1st ` z ) u. ( 2nd ` z ) ) Fn C ) |
61 |
|
dffn5 |
|- ( ( ( 1st ` z ) u. ( 2nd ` z ) ) Fn C <-> ( ( 1st ` z ) u. ( 2nd ` z ) ) = ( k e. C |-> ( ( ( 1st ` z ) u. ( 2nd ` z ) ) ` k ) ) ) |
62 |
60 61
|
sylib |
|- ( ( ph /\ z e. ( X X. Y ) ) -> ( ( 1st ` z ) u. ( 2nd ` z ) ) = ( k e. C |-> ( ( ( 1st ` z ) u. ( 2nd ` z ) ) ` k ) ) ) |
63 |
62
|
mpteq2dva |
|- ( ph -> ( z e. ( X X. Y ) |-> ( ( 1st ` z ) u. ( 2nd ` z ) ) ) = ( z e. ( X X. Y ) |-> ( k e. C |-> ( ( ( 1st ` z ) u. ( 2nd ` z ) ) ` k ) ) ) ) |
64 |
17 63
|
syl5eq |
|- ( ph -> G = ( z e. ( X X. Y ) |-> ( k e. C |-> ( ( ( 1st ` z ) u. ( 2nd ` z ) ) ` k ) ) ) ) |
65 |
|
pttop |
|- ( ( A e. _V /\ ( F |` A ) : A --> Top ) -> ( Xt_ ` ( F |` A ) ) e. Top ) |
66 |
26 27 65
|
syl2anc |
|- ( ph -> ( Xt_ ` ( F |` A ) ) e. Top ) |
67 |
4 66
|
eqeltrid |
|- ( ph -> K e. Top ) |
68 |
1
|
toptopon |
|- ( K e. Top <-> K e. ( TopOn ` X ) ) |
69 |
67 68
|
sylib |
|- ( ph -> K e. ( TopOn ` X ) ) |
70 |
|
pttop |
|- ( ( B e. _V /\ ( F |` B ) : B --> Top ) -> ( Xt_ ` ( F |` B ) ) e. Top ) |
71 |
47 48 70
|
syl2anc |
|- ( ph -> ( Xt_ ` ( F |` B ) ) e. Top ) |
72 |
5 71
|
eqeltrid |
|- ( ph -> L e. Top ) |
73 |
2
|
toptopon |
|- ( L e. Top <-> L e. ( TopOn ` Y ) ) |
74 |
72 73
|
sylib |
|- ( ph -> L e. ( TopOn ` Y ) ) |
75 |
|
txtopon |
|- ( ( K e. ( TopOn ` X ) /\ L e. ( TopOn ` Y ) ) -> ( K tX L ) e. ( TopOn ` ( X X. Y ) ) ) |
76 |
69 74 75
|
syl2anc |
|- ( ph -> ( K tX L ) e. ( TopOn ` ( X X. Y ) ) ) |
77 |
9
|
eleq2d |
|- ( ph -> ( k e. C <-> k e. ( A u. B ) ) ) |
78 |
77
|
biimpa |
|- ( ( ph /\ k e. C ) -> k e. ( A u. B ) ) |
79 |
|
elun |
|- ( k e. ( A u. B ) <-> ( k e. A \/ k e. B ) ) |
80 |
78 79
|
sylib |
|- ( ( ph /\ k e. C ) -> ( k e. A \/ k e. B ) ) |
81 |
|
ixpfn |
|- ( ( 1st ` z ) e. X_ n e. A U. ( F ` n ) -> ( 1st ` z ) Fn A ) |
82 |
33 81
|
syl |
|- ( ( ph /\ z e. ( X X. Y ) ) -> ( 1st ` z ) Fn A ) |
83 |
82
|
adantlr |
|- ( ( ( ph /\ k e. A ) /\ z e. ( X X. Y ) ) -> ( 1st ` z ) Fn A ) |
84 |
52
|
adantr |
|- ( ( ph /\ z e. ( X X. Y ) ) -> X_ n e. B U. ( F ` n ) = Y ) |
85 |
35 84
|
eleqtrrd |
|- ( ( ph /\ z e. ( X X. Y ) ) -> ( 2nd ` z ) e. X_ n e. B U. ( F ` n ) ) |
86 |
|
ixpfn |
|- ( ( 2nd ` z ) e. X_ n e. B U. ( F ` n ) -> ( 2nd ` z ) Fn B ) |
87 |
85 86
|
syl |
|- ( ( ph /\ z e. ( X X. Y ) ) -> ( 2nd ` z ) Fn B ) |
88 |
87
|
adantlr |
|- ( ( ( ph /\ k e. A ) /\ z e. ( X X. Y ) ) -> ( 2nd ` z ) Fn B ) |
89 |
10
|
ad2antrr |
|- ( ( ( ph /\ k e. A ) /\ z e. ( X X. Y ) ) -> ( A i^i B ) = (/) ) |
90 |
|
simplr |
|- ( ( ( ph /\ k e. A ) /\ z e. ( X X. Y ) ) -> k e. A ) |
91 |
|
fvun1 |
|- ( ( ( 1st ` z ) Fn A /\ ( 2nd ` z ) Fn B /\ ( ( A i^i B ) = (/) /\ k e. A ) ) -> ( ( ( 1st ` z ) u. ( 2nd ` z ) ) ` k ) = ( ( 1st ` z ) ` k ) ) |
92 |
83 88 89 90 91
|
syl112anc |
|- ( ( ( ph /\ k e. A ) /\ z e. ( X X. Y ) ) -> ( ( ( 1st ` z ) u. ( 2nd ` z ) ) ` k ) = ( ( 1st ` z ) ` k ) ) |
93 |
92
|
mpteq2dva |
|- ( ( ph /\ k e. A ) -> ( z e. ( X X. Y ) |-> ( ( ( 1st ` z ) u. ( 2nd ` z ) ) ` k ) ) = ( z e. ( X X. Y ) |-> ( ( 1st ` z ) ` k ) ) ) |
94 |
76
|
adantr |
|- ( ( ph /\ k e. A ) -> ( K tX L ) e. ( TopOn ` ( X X. Y ) ) ) |
95 |
13
|
mpompt |
|- ( z e. ( X X. Y ) |-> ( 1st ` z ) ) = ( x e. X , y e. Y |-> x ) |
96 |
69
|
adantr |
|- ( ( ph /\ k e. A ) -> K e. ( TopOn ` X ) ) |
97 |
74
|
adantr |
|- ( ( ph /\ k e. A ) -> L e. ( TopOn ` Y ) ) |
98 |
96 97
|
cnmpt1st |
|- ( ( ph /\ k e. A ) -> ( x e. X , y e. Y |-> x ) e. ( ( K tX L ) Cn K ) ) |
99 |
95 98
|
eqeltrid |
|- ( ( ph /\ k e. A ) -> ( z e. ( X X. Y ) |-> ( 1st ` z ) ) e. ( ( K tX L ) Cn K ) ) |
100 |
26
|
adantr |
|- ( ( ph /\ k e. A ) -> A e. _V ) |
101 |
27
|
adantr |
|- ( ( ph /\ k e. A ) -> ( F |` A ) : A --> Top ) |
102 |
|
simpr |
|- ( ( ph /\ k e. A ) -> k e. A ) |
103 |
1 4
|
ptpjcn |
|- ( ( A e. _V /\ ( F |` A ) : A --> Top /\ k e. A ) -> ( f e. X |-> ( f ` k ) ) e. ( K Cn ( ( F |` A ) ` k ) ) ) |
104 |
100 101 102 103
|
syl3anc |
|- ( ( ph /\ k e. A ) -> ( f e. X |-> ( f ` k ) ) e. ( K Cn ( ( F |` A ) ` k ) ) ) |
105 |
|
fvres |
|- ( k e. A -> ( ( F |` A ) ` k ) = ( F ` k ) ) |
106 |
105
|
adantl |
|- ( ( ph /\ k e. A ) -> ( ( F |` A ) ` k ) = ( F ` k ) ) |
107 |
106
|
oveq2d |
|- ( ( ph /\ k e. A ) -> ( K Cn ( ( F |` A ) ` k ) ) = ( K Cn ( F ` k ) ) ) |
108 |
104 107
|
eleqtrd |
|- ( ( ph /\ k e. A ) -> ( f e. X |-> ( f ` k ) ) e. ( K Cn ( F ` k ) ) ) |
109 |
|
fveq1 |
|- ( f = ( 1st ` z ) -> ( f ` k ) = ( ( 1st ` z ) ` k ) ) |
110 |
94 99 96 108 109
|
cnmpt11 |
|- ( ( ph /\ k e. A ) -> ( z e. ( X X. Y ) |-> ( ( 1st ` z ) ` k ) ) e. ( ( K tX L ) Cn ( F ` k ) ) ) |
111 |
93 110
|
eqeltrd |
|- ( ( ph /\ k e. A ) -> ( z e. ( X X. Y ) |-> ( ( ( 1st ` z ) u. ( 2nd ` z ) ) ` k ) ) e. ( ( K tX L ) Cn ( F ` k ) ) ) |
112 |
82
|
adantlr |
|- ( ( ( ph /\ k e. B ) /\ z e. ( X X. Y ) ) -> ( 1st ` z ) Fn A ) |
113 |
87
|
adantlr |
|- ( ( ( ph /\ k e. B ) /\ z e. ( X X. Y ) ) -> ( 2nd ` z ) Fn B ) |
114 |
10
|
ad2antrr |
|- ( ( ( ph /\ k e. B ) /\ z e. ( X X. Y ) ) -> ( A i^i B ) = (/) ) |
115 |
|
simplr |
|- ( ( ( ph /\ k e. B ) /\ z e. ( X X. Y ) ) -> k e. B ) |
116 |
|
fvun2 |
|- ( ( ( 1st ` z ) Fn A /\ ( 2nd ` z ) Fn B /\ ( ( A i^i B ) = (/) /\ k e. B ) ) -> ( ( ( 1st ` z ) u. ( 2nd ` z ) ) ` k ) = ( ( 2nd ` z ) ` k ) ) |
117 |
112 113 114 115 116
|
syl112anc |
|- ( ( ( ph /\ k e. B ) /\ z e. ( X X. Y ) ) -> ( ( ( 1st ` z ) u. ( 2nd ` z ) ) ` k ) = ( ( 2nd ` z ) ` k ) ) |
118 |
117
|
mpteq2dva |
|- ( ( ph /\ k e. B ) -> ( z e. ( X X. Y ) |-> ( ( ( 1st ` z ) u. ( 2nd ` z ) ) ` k ) ) = ( z e. ( X X. Y ) |-> ( ( 2nd ` z ) ` k ) ) ) |
119 |
76
|
adantr |
|- ( ( ph /\ k e. B ) -> ( K tX L ) e. ( TopOn ` ( X X. Y ) ) ) |
120 |
14
|
mpompt |
|- ( z e. ( X X. Y ) |-> ( 2nd ` z ) ) = ( x e. X , y e. Y |-> y ) |
121 |
69
|
adantr |
|- ( ( ph /\ k e. B ) -> K e. ( TopOn ` X ) ) |
122 |
74
|
adantr |
|- ( ( ph /\ k e. B ) -> L e. ( TopOn ` Y ) ) |
123 |
121 122
|
cnmpt2nd |
|- ( ( ph /\ k e. B ) -> ( x e. X , y e. Y |-> y ) e. ( ( K tX L ) Cn L ) ) |
124 |
120 123
|
eqeltrid |
|- ( ( ph /\ k e. B ) -> ( z e. ( X X. Y ) |-> ( 2nd ` z ) ) e. ( ( K tX L ) Cn L ) ) |
125 |
47
|
adantr |
|- ( ( ph /\ k e. B ) -> B e. _V ) |
126 |
48
|
adantr |
|- ( ( ph /\ k e. B ) -> ( F |` B ) : B --> Top ) |
127 |
|
simpr |
|- ( ( ph /\ k e. B ) -> k e. B ) |
128 |
2 5
|
ptpjcn |
|- ( ( B e. _V /\ ( F |` B ) : B --> Top /\ k e. B ) -> ( f e. Y |-> ( f ` k ) ) e. ( L Cn ( ( F |` B ) ` k ) ) ) |
129 |
125 126 127 128
|
syl3anc |
|- ( ( ph /\ k e. B ) -> ( f e. Y |-> ( f ` k ) ) e. ( L Cn ( ( F |` B ) ` k ) ) ) |
130 |
|
fvres |
|- ( k e. B -> ( ( F |` B ) ` k ) = ( F ` k ) ) |
131 |
130
|
adantl |
|- ( ( ph /\ k e. B ) -> ( ( F |` B ) ` k ) = ( F ` k ) ) |
132 |
131
|
oveq2d |
|- ( ( ph /\ k e. B ) -> ( L Cn ( ( F |` B ) ` k ) ) = ( L Cn ( F ` k ) ) ) |
133 |
129 132
|
eleqtrd |
|- ( ( ph /\ k e. B ) -> ( f e. Y |-> ( f ` k ) ) e. ( L Cn ( F ` k ) ) ) |
134 |
|
fveq1 |
|- ( f = ( 2nd ` z ) -> ( f ` k ) = ( ( 2nd ` z ) ` k ) ) |
135 |
119 124 122 133 134
|
cnmpt11 |
|- ( ( ph /\ k e. B ) -> ( z e. ( X X. Y ) |-> ( ( 2nd ` z ) ` k ) ) e. ( ( K tX L ) Cn ( F ` k ) ) ) |
136 |
118 135
|
eqeltrd |
|- ( ( ph /\ k e. B ) -> ( z e. ( X X. Y ) |-> ( ( ( 1st ` z ) u. ( 2nd ` z ) ) ` k ) ) e. ( ( K tX L ) Cn ( F ` k ) ) ) |
137 |
111 136
|
jaodan |
|- ( ( ph /\ ( k e. A \/ k e. B ) ) -> ( z e. ( X X. Y ) |-> ( ( ( 1st ` z ) u. ( 2nd ` z ) ) ` k ) ) e. ( ( K tX L ) Cn ( F ` k ) ) ) |
138 |
80 137
|
syldan |
|- ( ( ph /\ k e. C ) -> ( z e. ( X X. Y ) |-> ( ( ( 1st ` z ) u. ( 2nd ` z ) ) ` k ) ) e. ( ( K tX L ) Cn ( F ` k ) ) ) |
139 |
3 76 7 8 138
|
ptcn |
|- ( ph -> ( z e. ( X X. Y ) |-> ( k e. C |-> ( ( ( 1st ` z ) u. ( 2nd ` z ) ) ` k ) ) ) e. ( ( K tX L ) Cn J ) ) |
140 |
64 139
|
eqeltrd |
|- ( ph -> G e. ( ( K tX L ) Cn J ) ) |
141 |
1 2 3 4 5 6 7 8 9 10
|
ptuncnv |
|- ( ph -> `' G = ( z e. U. J |-> <. ( z |` A ) , ( z |` B ) >. ) ) |
142 |
|
pttop |
|- ( ( C e. V /\ F : C --> Top ) -> ( Xt_ ` F ) e. Top ) |
143 |
7 8 142
|
syl2anc |
|- ( ph -> ( Xt_ ` F ) e. Top ) |
144 |
3 143
|
eqeltrid |
|- ( ph -> J e. Top ) |
145 |
|
eqid |
|- U. J = U. J |
146 |
145
|
toptopon |
|- ( J e. Top <-> J e. ( TopOn ` U. J ) ) |
147 |
144 146
|
sylib |
|- ( ph -> J e. ( TopOn ` U. J ) ) |
148 |
145 3 4
|
ptrescn |
|- ( ( C e. V /\ F : C --> Top /\ A C_ C ) -> ( z e. U. J |-> ( z |` A ) ) e. ( J Cn K ) ) |
149 |
7 8 25 148
|
syl3anc |
|- ( ph -> ( z e. U. J |-> ( z |` A ) ) e. ( J Cn K ) ) |
150 |
145 3 5
|
ptrescn |
|- ( ( C e. V /\ F : C --> Top /\ B C_ C ) -> ( z e. U. J |-> ( z |` B ) ) e. ( J Cn L ) ) |
151 |
7 8 46 150
|
syl3anc |
|- ( ph -> ( z e. U. J |-> ( z |` B ) ) e. ( J Cn L ) ) |
152 |
147 149 151
|
cnmpt1t |
|- ( ph -> ( z e. U. J |-> <. ( z |` A ) , ( z |` B ) >. ) e. ( J Cn ( K tX L ) ) ) |
153 |
141 152
|
eqeltrd |
|- ( ph -> `' G e. ( J Cn ( K tX L ) ) ) |
154 |
|
ishmeo |
|- ( G e. ( ( K tX L ) Homeo J ) <-> ( G e. ( ( K tX L ) Cn J ) /\ `' G e. ( J Cn ( K tX L ) ) ) ) |
155 |
140 153 154
|
sylanbrc |
|- ( ph -> G e. ( ( K tX L ) Homeo J ) ) |