let Y be non empty set ; :: thesis: for G being Subset of (PARTITIONS Y)
for A, B, C, D, E, F, J being a_partition of Y
for z, u being Element of Y st G is independent & G = {A,B,C,D,E,F,J} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & B <> C & B <> D & B <> E & B <> F & B <> J & C <> D & C <> E & C <> F & C <> J & D <> E & D <> F & D <> J & E <> F & E <> J & F <> J & EqClass z,((((C '/\' D) '/\' E) '/\' F) '/\' J) = EqClass u,((((C '/\' D) '/\' E) '/\' F) '/\' J) holds
EqClass u,(CompF A,G) meets EqClass z,(CompF B,G)
let G be Subset of (PARTITIONS Y); :: thesis: for A, B, C, D, E, F, J being a_partition of Y
for z, u being Element of Y st G is independent & G = {A,B,C,D,E,F,J} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & B <> C & B <> D & B <> E & B <> F & B <> J & C <> D & C <> E & C <> F & C <> J & D <> E & D <> F & D <> J & E <> F & E <> J & F <> J & EqClass z,((((C '/\' D) '/\' E) '/\' F) '/\' J) = EqClass u,((((C '/\' D) '/\' E) '/\' F) '/\' J) holds
EqClass u,(CompF A,G) meets EqClass z,(CompF B,G)
let A, B, C, D, E, F, J be a_partition of Y; :: thesis: for z, u being Element of Y st G is independent & G = {A,B,C,D,E,F,J} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & B <> C & B <> D & B <> E & B <> F & B <> J & C <> D & C <> E & C <> F & C <> J & D <> E & D <> F & D <> J & E <> F & E <> J & F <> J & EqClass z,((((C '/\' D) '/\' E) '/\' F) '/\' J) = EqClass u,((((C '/\' D) '/\' E) '/\' F) '/\' J) holds
EqClass u,(CompF A,G) meets EqClass z,(CompF B,G)
let z, u be Element of Y; :: thesis: ( G is independent & G = {A,B,C,D,E,F,J} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & B <> C & B <> D & B <> E & B <> F & B <> J & C <> D & C <> E & C <> F & C <> J & D <> E & D <> F & D <> J & E <> F & E <> J & F <> J & EqClass z,((((C '/\' D) '/\' E) '/\' F) '/\' J) = EqClass u,((((C '/\' D) '/\' E) '/\' F) '/\' J) implies EqClass u,(CompF A,G) meets EqClass z,(CompF B,G) )
assume that
A1:
G is independent
and
A2:
( G = {A,B,C,D,E,F,J} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & B <> C & B <> D & B <> E & B <> F & B <> J & C <> D & C <> E & C <> F & C <> J & D <> E & D <> F & D <> J & E <> F & E <> J & F <> J & EqClass z,((((C '/\' D) '/\' E) '/\' F) '/\' J) = EqClass u,((((C '/\' D) '/\' E) '/\' F) '/\' J) )
; :: thesis: EqClass u,(CompF A,G) meets EqClass z,(CompF B,G)
A3:
CompF A,G = ((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J
by A2, TH1;
A4:
CompF B,G = ((((A '/\' C) '/\' D) '/\' E) '/\' F) '/\' J
by A2, Th2;
reconsider HH = EqClass z,(CompF B,G) as set ;
reconsider I = EqClass z,A as set ;
reconsider GG = EqClass u,(((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) as set ;
set h = ((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (J .--> (EqClass u,J))) +* (A .--> (EqClass z,A));
A5:
dom (((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (J .--> (EqClass u,J))) +* (A .--> (EqClass z,A))) = G
by A2, Th9;
A6:
(((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (J .--> (EqClass u,J))) +* (A .--> (EqClass z,A))) . A = EqClass z,A
by A2, Th8;
A7:
(((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (J .--> (EqClass u,J))) +* (A .--> (EqClass z,A))) . B = EqClass u,B
by A2, Th8;
A8:
(((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (J .--> (EqClass u,J))) +* (A .--> (EqClass z,A))) . C = EqClass u,C
by A2, Th8;
A9:
(((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (J .--> (EqClass u,J))) +* (A .--> (EqClass z,A))) . D = EqClass u,D
by A2, Th8;
A10:
(((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (J .--> (EqClass u,J))) +* (A .--> (EqClass z,A))) . E = EqClass u,E
by A2, Th8;
A11:
(((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (J .--> (EqClass u,J))) +* (A .--> (EqClass z,A))) . F = EqClass u,F
by A2, Th8;
A12:
(((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (J .--> (EqClass u,J))) +* (A .--> (EqClass z,A))) . J = EqClass u,J
by A2, Th8;
A13:
for d being set st d in G holds
(((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (J .--> (EqClass u,J))) +* (A .--> (EqClass z,A))) . d in d
proof
let d be
set ;
:: thesis: ( d in G implies (((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (J .--> (EqClass u,J))) +* (A .--> (EqClass z,A))) . d in d )
assume
d in G
;
:: thesis: (((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (J .--> (EqClass u,J))) +* (A .--> (EqClass z,A))) . d in d
then
(
d = A or
d = B or
d = C or
d = D or
d = E or
d = F or
d = J )
by A2, ENUMSET1:def 5;
hence
(((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (J .--> (EqClass u,J))) +* (A .--> (EqClass z,A))) . d in d
by A6, A7, A8, A9, A10, A11, A12;
:: thesis: verum
end;
A in dom (((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (J .--> (EqClass u,J))) +* (A .--> (EqClass z,A)))
by A2, A5, ENUMSET1:def 5;
then A14:
(((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (J .--> (EqClass u,J))) +* (A .--> (EqClass z,A))) . A in rng (((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (J .--> (EqClass u,J))) +* (A .--> (EqClass z,A)))
by FUNCT_1:def 5;
B in dom (((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (J .--> (EqClass u,J))) +* (A .--> (EqClass z,A)))
by A2, A5, ENUMSET1:def 5;
then A15:
(((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (J .--> (EqClass u,J))) +* (A .--> (EqClass z,A))) . B in rng (((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (J .--> (EqClass u,J))) +* (A .--> (EqClass z,A)))
by FUNCT_1:def 5;
C in dom (((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (J .--> (EqClass u,J))) +* (A .--> (EqClass z,A)))
by A2, A5, ENUMSET1:def 5;
then A16:
(((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (J .--> (EqClass u,J))) +* (A .--> (EqClass z,A))) . C in rng (((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (J .--> (EqClass u,J))) +* (A .--> (EqClass z,A)))
by FUNCT_1:def 5;
D in dom (((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (J .--> (EqClass u,J))) +* (A .--> (EqClass z,A)))
by A2, A5, ENUMSET1:def 5;
then A17:
(((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (J .--> (EqClass u,J))) +* (A .--> (EqClass z,A))) . D in rng (((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (J .--> (EqClass u,J))) +* (A .--> (EqClass z,A)))
by FUNCT_1:def 5;
E in dom (((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (J .--> (EqClass u,J))) +* (A .--> (EqClass z,A)))
by A2, A5, ENUMSET1:def 5;
then A18:
(((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (J .--> (EqClass u,J))) +* (A .--> (EqClass z,A))) . E in rng (((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (J .--> (EqClass u,J))) +* (A .--> (EqClass z,A)))
by FUNCT_1:def 5;
F in dom (((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (J .--> (EqClass u,J))) +* (A .--> (EqClass z,A)))
by A2, A5, ENUMSET1:def 5;
then A19:
(((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (J .--> (EqClass u,J))) +* (A .--> (EqClass z,A))) . F in rng (((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (J .--> (EqClass u,J))) +* (A .--> (EqClass z,A)))
by FUNCT_1:def 5;
J in dom (((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (J .--> (EqClass u,J))) +* (A .--> (EqClass z,A)))
by A2, A5, ENUMSET1:def 5;
then A20:
(((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (J .--> (EqClass u,J))) +* (A .--> (EqClass z,A))) . J in rng (((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (J .--> (EqClass u,J))) +* (A .--> (EqClass z,A)))
by FUNCT_1:def 5;
A21:
rng (((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (J .--> (EqClass u,J))) +* (A .--> (EqClass z,A))) = {((((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (J .--> (EqClass u,J))) +* (A .--> (EqClass z,A))) . A),((((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (J .--> (EqClass u,J))) +* (A .--> (EqClass z,A))) . B),((((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (J .--> (EqClass u,J))) +* (A .--> (EqClass z,A))) . C),((((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (J .--> (EqClass u,J))) +* (A .--> (EqClass z,A))) . D),((((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (J .--> (EqClass u,J))) +* (A .--> (EqClass z,A))) . E),((((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (J .--> (EqClass u,J))) +* (A .--> (EqClass z,A))) . F),((((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (J .--> (EqClass u,J))) +* (A .--> (EqClass z,A))) . J)}
by Th10;
rng (((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (J .--> (EqClass u,J))) +* (A .--> (EqClass z,A))) c= bool Y
proof
let t be
set ;
:: according to TARSKI:def 3 :: thesis: ( not t in rng (((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (J .--> (EqClass u,J))) +* (A .--> (EqClass z,A))) or t in bool Y )
assume
t in rng (((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (J .--> (EqClass u,J))) +* (A .--> (EqClass z,A)))
;
:: thesis: t in bool Y
then
(
t = (((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (J .--> (EqClass u,J))) +* (A .--> (EqClass z,A))) . A or
t = (((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (J .--> (EqClass u,J))) +* (A .--> (EqClass z,A))) . B or
t = (((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (J .--> (EqClass u,J))) +* (A .--> (EqClass z,A))) . C or
t = (((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (J .--> (EqClass u,J))) +* (A .--> (EqClass z,A))) . D or
t = (((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (J .--> (EqClass u,J))) +* (A .--> (EqClass z,A))) . E or
t = (((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (J .--> (EqClass u,J))) +* (A .--> (EqClass z,A))) . F or
t = (((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (J .--> (EqClass u,J))) +* (A .--> (EqClass z,A))) . J )
by A21, ENUMSET1:def 5;
hence
t in bool Y
by A6, A7, A8, A9, A10, A11, A12;
:: thesis: verum
end;
then reconsider FF = rng (((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (J .--> (EqClass u,J))) +* (A .--> (EqClass z,A))) as Subset-Family of Y ;
A22:
Intersect FF = meet (rng (((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (J .--> (EqClass u,J))) +* (A .--> (EqClass z,A))))
by A14, SETFAM_1:def 10;
Intersect FF <> {}
by A1, A5, A13, BVFUNC_2:def 5;
then consider m being set such that
A23:
m in Intersect FF
by XBOOLE_0:def 1;
A24:
( m in EqClass z,A & m in EqClass u,B & m in EqClass u,C & m in EqClass u,D & m in EqClass u,E & m in EqClass u,F & m in EqClass u,J )
by A6, A7, A8, A9, A10, A11, A12, A14, A15, A16, A17, A18, A19, A20, A22, A23, SETFAM_1:def 1;
GG = (EqClass u,((((B '/\' C) '/\' D) '/\' E) '/\' F)) /\ (EqClass u,J)
by Th1;
then
GG = ((EqClass u,(((B '/\' C) '/\' D) '/\' E)) /\ (EqClass u,F)) /\ (EqClass u,J)
by Th1;
then
GG = (((EqClass u,((B '/\' C) '/\' D)) /\ (EqClass u,E)) /\ (EqClass u,F)) /\ (EqClass u,J)
by Th1;
then
GG = ((((EqClass u,(B '/\' C)) /\ (EqClass u,D)) /\ (EqClass u,E)) /\ (EqClass u,F)) /\ (EqClass u,J)
by Th1;
then A25:
GG /\ I = ((((((EqClass u,B) /\ (EqClass u,C)) /\ (EqClass u,D)) /\ (EqClass u,E)) /\ (EqClass u,F)) /\ (EqClass u,J)) /\ (EqClass z,A)
by Th1;
m in (EqClass u,B) /\ (EqClass u,C)
by A24, XBOOLE_0:def 4;
then
m in ((EqClass u,B) /\ (EqClass u,C)) /\ (EqClass u,D)
by A24, XBOOLE_0:def 4;
then
m in (((EqClass u,B) /\ (EqClass u,C)) /\ (EqClass u,D)) /\ (EqClass u,E)
by A24, XBOOLE_0:def 4;
then
m in ((((EqClass u,B) /\ (EqClass u,C)) /\ (EqClass u,D)) /\ (EqClass u,E)) /\ (EqClass u,F)
by A24, XBOOLE_0:def 4;
then
m in (((((EqClass u,B) /\ (EqClass u,C)) /\ (EqClass u,D)) /\ (EqClass u,E)) /\ (EqClass u,F)) /\ (EqClass u,J)
by A24, XBOOLE_0:def 4;
then A26:
GG /\ I <> {}
by A24, A25, XBOOLE_0:def 4;
then consider p being set such that
A27:
p in GG /\ I
by XBOOLE_0:def 1;
( GG /\ I in INTERSECTION A,(((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) & not GG /\ I in {{} } )
by A26, SETFAM_1:def 5, TARSKI:def 1;
then
GG /\ I in (INTERSECTION A,(((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J)) \ {{} }
by XBOOLE_0:def 5;
then
GG /\ I in A '/\' (((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J)
by PARTIT1:def 4;
then reconsider p = p as Element of Y by A27;
reconsider K = EqClass p,((((C '/\' D) '/\' E) '/\' F) '/\' J) as set ;
A28:
( p in K & K in (((C '/\' D) '/\' E) '/\' F) '/\' J )
by EQREL_1:def 8;
( p in GG & p in I )
by A27, XBOOLE_0:def 4;
then A29:
p in I /\ K
by A28, XBOOLE_0:def 4;
then
( I /\ K in INTERSECTION A,((((C '/\' D) '/\' E) '/\' F) '/\' J) & not I /\ K in {{} } )
by SETFAM_1:def 5, TARSKI:def 1;
then
I /\ K in (INTERSECTION A,((((C '/\' D) '/\' E) '/\' F) '/\' J)) \ {{} }
by XBOOLE_0:def 5;
then A30:
I /\ K in A '/\' ((((C '/\' D) '/\' E) '/\' F) '/\' J)
by PARTIT1:def 4;
reconsider L = EqClass z,((((C '/\' D) '/\' E) '/\' F) '/\' J) as set ;
GG = EqClass u,((((B '/\' (C '/\' D)) '/\' E) '/\' F) '/\' J)
by PARTIT1:16;
then
GG = EqClass u,(((B '/\' ((C '/\' D) '/\' E)) '/\' F) '/\' J)
by PARTIT1:16;
then
GG = EqClass u,((B '/\' (((C '/\' D) '/\' E) '/\' F)) '/\' J)
by PARTIT1:16;
then
GG = EqClass u,(B '/\' ((((C '/\' D) '/\' E) '/\' F) '/\' J))
by PARTIT1:16;
then A31:
GG c= L
by A2, BVFUNC11:3;
A32:
p in GG
by A27, XBOOLE_0:def 4;
p in EqClass p,((((C '/\' D) '/\' E) '/\' F) '/\' J)
by EQREL_1:def 8;
then
K meets L
by A31, A32, XBOOLE_0:3;
then
K = L
by EQREL_1:50;
then A33:
z in K
by EQREL_1:def 8;
z in I
by EQREL_1:def 8;
then A34:
z in I /\ K
by A33, XBOOLE_0:def 4;
z in HH
by EQREL_1:def 8;
then A35:
I /\ K meets HH
by A34, XBOOLE_0:3;
A '/\' ((((C '/\' D) '/\' E) '/\' F) '/\' J) =
(A '/\' (((C '/\' D) '/\' E) '/\' F)) '/\' J
by PARTIT1:16
.=
((A '/\' ((C '/\' D) '/\' E)) '/\' F) '/\' J
by PARTIT1:16
.=
(((A '/\' (C '/\' D)) '/\' E) '/\' F) '/\' J
by PARTIT1:16
.=
((((A '/\' C) '/\' D) '/\' E) '/\' F) '/\' J
by PARTIT1:16
;
then
p in HH
by A4, A29, A30, A35, EQREL_1:def 6;
hence
EqClass u,(CompF A,G) meets EqClass z,(CompF B,G)
by A3, A32, XBOOLE_0:3; :: thesis: verum