let Y be non empty set ; :: thesis: for G being Subset of (PARTITIONS Y)
for A, B, C, D, E, F, J, M being a_partition of Y
for z, u being Element of Y st G is independent & G = {A,B,C,D,E,F,J,M} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & C <> D & C <> E & C <> F & C <> J & C <> M & D <> E & D <> F & D <> J & D <> M & E <> F & E <> J & E <> M & F <> J & F <> M & J <> M & EqClass z,(((((C '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) = EqClass u,(((((C '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) 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, M being a_partition of Y
for z, u being Element of Y st G is independent & G = {A,B,C,D,E,F,J,M} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & C <> D & C <> E & C <> F & C <> J & C <> M & D <> E & D <> F & D <> J & D <> M & E <> F & E <> J & E <> M & F <> J & F <> M & J <> M & EqClass z,(((((C '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) = EqClass u,(((((C '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) holds
EqClass u,(CompF A,G) meets EqClass z,(CompF B,G)
let A, B, C, D, E, F, J, M 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,M} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & C <> D & C <> E & C <> F & C <> J & C <> M & D <> E & D <> F & D <> J & D <> M & E <> F & E <> J & E <> M & F <> J & F <> M & J <> M & EqClass z,(((((C '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) = EqClass u,(((((C '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) 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,M} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & C <> D & C <> E & C <> F & C <> J & C <> M & D <> E & D <> F & D <> J & D <> M & E <> F & E <> J & E <> M & F <> J & F <> M & J <> M & EqClass z,(((((C '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) = EqClass u,(((((C '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) 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,M} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & C <> D & C <> E & C <> F & C <> J & C <> M & D <> E & D <> F & D <> J & D <> M & E <> F & E <> J & E <> M & F <> J & F <> M & J <> M & EqClass z,(((((C '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) = EqClass u,(((((C '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) )
; :: thesis: EqClass u,(CompF A,G) meets EqClass z,(CompF B,G)
A3:
CompF B,G = (((((A '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M
by A2, Th14;
set HH = EqClass z,(CompF B,G);
set I = EqClass z,A;
set GG = EqClass u,((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M);
A4:
EqClass u,((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) = EqClass u,(CompF A,G)
by A2, Th13;
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))) +* (M .--> (EqClass u,M))) +* (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))) +* (M .--> (EqClass u,M))) +* (A .--> (EqClass z,A))) = G
by A2, Th22;
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))) +* (M .--> (EqClass u,M))) +* (A .--> (EqClass z,A))) . A = EqClass z,A
by FUNCT_7:96;
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))) +* (M .--> (EqClass u,M))) +* (A .--> (EqClass z,A))) . B = EqClass u,B
by A2, Th21;
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))) +* (M .--> (EqClass u,M))) +* (A .--> (EqClass z,A))) . C = EqClass u,C
by A2, Th21;
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))) +* (M .--> (EqClass u,M))) +* (A .--> (EqClass z,A))) . D = EqClass u,D
by A2, Th21;
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))) +* (M .--> (EqClass u,M))) +* (A .--> (EqClass z,A))) . E = EqClass u,E
by A2, Th21;
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))) +* (M .--> (EqClass u,M))) +* (A .--> (EqClass z,A))) . F = EqClass u,F
by A2, Th21;
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))) +* (M .--> (EqClass u,M))) +* (A .--> (EqClass z,A))) . J = EqClass u,J
by A2, Th21;
A13:
((((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (J .--> (EqClass u,J))) +* (M .--> (EqClass u,M))) +* (A .--> (EqClass z,A))) . M = EqClass u,M
by A2, TH15;
A14:
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))) +* (M .--> (EqClass u,M))) +* (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))) +* (M .--> (EqClass u,M))) +* (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))) +* (M .--> (EqClass u,M))) +* (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 or
d = M )
by A2, ENUMSET1:def 6;
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))) +* (M .--> (EqClass u,M))) +* (A .--> (EqClass z,A))) . d in d
by A6, A7, A8, A9, A10, A11, A12, A13;
:: 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))) +* (M .--> (EqClass u,M))) +* (A .--> (EqClass z,A)))
by A2, A5, ENUMSET1:def 6;
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))) +* (M .--> (EqClass u,M))) +* (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))) +* (M .--> (EqClass u,M))) +* (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))) +* (M .--> (EqClass u,M))) +* (A .--> (EqClass z,A)))
by A2, A5, ENUMSET1:def 6;
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))) +* (M .--> (EqClass u,M))) +* (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))) +* (M .--> (EqClass u,M))) +* (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))) +* (M .--> (EqClass u,M))) +* (A .--> (EqClass z,A)))
by A2, A5, ENUMSET1:def 6;
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))) +* (M .--> (EqClass u,M))) +* (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))) +* (M .--> (EqClass u,M))) +* (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))) +* (M .--> (EqClass u,M))) +* (A .--> (EqClass z,A)))
by A2, A5, ENUMSET1:def 6;
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))) +* (M .--> (EqClass u,M))) +* (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))) +* (M .--> (EqClass u,M))) +* (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))) +* (M .--> (EqClass u,M))) +* (A .--> (EqClass z,A)))
by A2, A5, ENUMSET1:def 6;
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))) +* (M .--> (EqClass u,M))) +* (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))) +* (M .--> (EqClass u,M))) +* (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))) +* (M .--> (EqClass u,M))) +* (A .--> (EqClass z,A)))
by A2, A5, ENUMSET1:def 6;
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))) +* (M .--> (EqClass u,M))) +* (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))) +* (M .--> (EqClass u,M))) +* (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))) +* (M .--> (EqClass u,M))) +* (A .--> (EqClass z,A)))
by A2, A5, ENUMSET1:def 6;
then A21:
((((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (J .--> (EqClass u,J))) +* (M .--> (EqClass u,M))) +* (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))) +* (M .--> (EqClass u,M))) +* (A .--> (EqClass z,A)))
by FUNCT_1:def 5;
M 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))) +* (M .--> (EqClass u,M))) +* (A .--> (EqClass z,A)))
by A2, A5, ENUMSET1:def 6;
then A22:
((((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (J .--> (EqClass u,J))) +* (M .--> (EqClass u,M))) +* (A .--> (EqClass z,A))) . M 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))) +* (M .--> (EqClass u,M))) +* (A .--> (EqClass z,A)))
by FUNCT_1:def 5;
A23:
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))) +* (M .--> (EqClass u,M))) +* (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))) +* (M .--> (EqClass u,M))) +* (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))) +* (M .--> (EqClass u,M))) +* (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))) +* (M .--> (EqClass u,M))) +* (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))) +* (M .--> (EqClass u,M))) +* (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))) +* (M .--> (EqClass u,M))) +* (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))) +* (M .--> (EqClass u,M))) +* (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))) +* (M .--> (EqClass u,M))) +* (A .--> (EqClass z,A))) . J),(((((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (F .--> (EqClass u,F))) +* (J .--> (EqClass u,J))) +* (M .--> (EqClass u,M))) +* (A .--> (EqClass z,A))) . M)}
by Th23;
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))) +* (M .--> (EqClass u,M))) +* (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))) +* (M .--> (EqClass u,M))) +* (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))) +* (M .--> (EqClass u,M))) +* (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))) +* (M .--> (EqClass u,M))) +* (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))) +* (M .--> (EqClass u,M))) +* (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))) +* (M .--> (EqClass u,M))) +* (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))) +* (M .--> (EqClass u,M))) +* (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))) +* (M .--> (EqClass u,M))) +* (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))) +* (M .--> (EqClass u,M))) +* (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))) +* (M .--> (EqClass u,M))) +* (A .--> (EqClass z,A))) . J 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))) +* (M .--> (EqClass u,M))) +* (A .--> (EqClass z,A))) . M )
by A23, ENUMSET1:def 6;
hence
t in bool Y
by A6, A7, A8, A9, A10, A11, A12, A13;
:: 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))) +* (M .--> (EqClass u,M))) +* (A .--> (EqClass z,A))) as Subset-Family of Y ;
A24:
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))) +* (M .--> (EqClass u,M))) +* (A .--> (EqClass z,A))))
by A15, SETFAM_1:def 10;
Intersect FF <> {}
by A1, A5, A14, BVFUNC_2:def 5;
then consider m being set such that
A25:
m in Intersect FF
by XBOOLE_0:def 1;
A26:
( 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 & m in EqClass u,M )
by A6, A7, A8, A9, A10, A11, A12, A13, A15, A16, A17, A18, A19, A20, A21, A22, A24, A25, SETFAM_1:def 1;
EqClass u,((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) = (EqClass u,(((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J)) /\ (EqClass u,M)
by Th1;
then
EqClass u,((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) = ((EqClass u,((((B '/\' C) '/\' D) '/\' E) '/\' F)) /\ (EqClass u,J)) /\ (EqClass u,M)
by Th1;
then
EqClass u,((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) = (((EqClass u,(((B '/\' C) '/\' D) '/\' E)) /\ (EqClass u,F)) /\ (EqClass u,J)) /\ (EqClass u,M)
by Th1;
then
EqClass u,((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) = ((((EqClass u,((B '/\' C) '/\' D)) /\ (EqClass u,E)) /\ (EqClass u,F)) /\ (EqClass u,J)) /\ (EqClass u,M)
by Th1;
then
EqClass u,((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) = (((((EqClass u,(B '/\' C)) /\ (EqClass u,D)) /\ (EqClass u,E)) /\ (EqClass u,F)) /\ (EqClass u,J)) /\ (EqClass u,M)
by Th1;
then A27:
(EqClass u,((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M)) /\ (EqClass z,A) = (((((((EqClass u,B) /\ (EqClass u,C)) /\ (EqClass u,D)) /\ (EqClass u,E)) /\ (EqClass u,F)) /\ (EqClass u,J)) /\ (EqClass u,M)) /\ (EqClass z,A)
by Th1;
m in (EqClass u,B) /\ (EqClass u,C)
by A26, XBOOLE_0:def 4;
then
m in ((EqClass u,B) /\ (EqClass u,C)) /\ (EqClass u,D)
by A26, XBOOLE_0:def 4;
then
m in (((EqClass u,B) /\ (EqClass u,C)) /\ (EqClass u,D)) /\ (EqClass u,E)
by A26, XBOOLE_0:def 4;
then
m in ((((EqClass u,B) /\ (EqClass u,C)) /\ (EqClass u,D)) /\ (EqClass u,E)) /\ (EqClass u,F)
by A26, 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 A26, 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)) /\ (EqClass u,M)
by A26, XBOOLE_0:def 4;
then
(EqClass u,((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M)) /\ (EqClass z,A) <> {}
by A26, A27, XBOOLE_0:def 4;
then consider p being set such that
A28:
p in (EqClass u,((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M)) /\ (EqClass z,A)
by XBOOLE_0:def 1;
reconsider p = p as Element of Y by A28;
reconsider K = EqClass p,(((((C '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) as set ;
A29:
( p in K & K in ((((C '/\' D) '/\' E) '/\' F) '/\' J) '/\' M )
by EQREL_1:def 8;
( p in EqClass u,((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) & p in EqClass z,A )
by A28, XBOOLE_0:def 4;
then A30:
p in (EqClass z,A) /\ K
by A29, XBOOLE_0:def 4;
then
( (EqClass z,A) /\ K in INTERSECTION A,(((((C '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) & not (EqClass z,A) /\ K in {{} } )
by SETFAM_1:def 5, TARSKI:def 1;
then
(EqClass z,A) /\ K in (INTERSECTION A,(((((C '/\' D) '/\' E) '/\' F) '/\' J) '/\' M)) \ {{} }
by XBOOLE_0:def 5;
then A31:
(EqClass z,A) /\ K in A '/\' (((((C '/\' D) '/\' E) '/\' F) '/\' J) '/\' M)
by PARTIT1:def 4;
reconsider L = EqClass z,(((((C '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) as set ;
EqClass u,((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) = EqClass u,(((((B '/\' (C '/\' D)) '/\' E) '/\' F) '/\' J) '/\' M)
by PARTIT1:16;
then
EqClass u,((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) = EqClass u,((((B '/\' ((C '/\' D) '/\' E)) '/\' F) '/\' J) '/\' M)
by PARTIT1:16;
then
EqClass u,((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) = EqClass u,(((B '/\' (((C '/\' D) '/\' E) '/\' F)) '/\' J) '/\' M)
by PARTIT1:16;
then
EqClass u,((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) = EqClass u,((B '/\' ((((C '/\' D) '/\' E) '/\' F) '/\' J)) '/\' M)
by PARTIT1:16;
then
EqClass u,((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) = EqClass u,(B '/\' (((((C '/\' D) '/\' E) '/\' F) '/\' J) '/\' M))
by PARTIT1:16;
then A32:
EqClass u,((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) c= L
by A2, BVFUNC11:3;
A33:
p in EqClass u,((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M)
by A28, XBOOLE_0:def 4;
p in EqClass p,(((((C '/\' D) '/\' E) '/\' F) '/\' J) '/\' M)
by EQREL_1:def 8;
then
K meets L
by A32, A33, XBOOLE_0:3;
then
K = L
by EQREL_1:50;
then A34:
z in K
by EQREL_1:def 8;
z in EqClass z,A
by EQREL_1:def 8;
then A35:
z in (EqClass z,A) /\ K
by A34, XBOOLE_0:def 4;
z in EqClass z,(CompF B,G)
by EQREL_1:def 8;
then A36:
(EqClass z,A) /\ K meets EqClass z,(CompF B,G)
by A35, XBOOLE_0:3;
A '/\' (((((C '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) =
(A '/\' ((((C '/\' D) '/\' E) '/\' F) '/\' J)) '/\' M
by PARTIT1:16
.=
((A '/\' (((C '/\' D) '/\' E) '/\' F)) '/\' J) '/\' M
by PARTIT1:16
.=
(((A '/\' ((C '/\' D) '/\' E)) '/\' F) '/\' J) '/\' M
by PARTIT1:16
.=
((((A '/\' (C '/\' D)) '/\' E) '/\' F) '/\' J) '/\' M
by PARTIT1:16
.=
(((((A '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M
by PARTIT1:16
;
then
p in EqClass z,(CompF B,G)
by A3, A30, A31, A36, EQREL_1:def 6;
hence
EqClass u,(CompF A,G) meets EqClass z,(CompF B,G)
by A4, A33, XBOOLE_0:3; :: thesis: verum