let Y be non empty set ; 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 holds
(EqClass (u,((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M))) /\ (EqClass (z,A)) <> {}
let G be 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 holds
(EqClass (u,((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M))) /\ (EqClass (z,A)) <> {}
let A, B, C, D, E, F, J, M be 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 holds
(EqClass (u,((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M))) /\ (EqClass (z,A)) <> {}
let z, u be Element of Y; ( 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 implies (EqClass (u,((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M))) /\ (EqClass (z,A)) <> {} )
assume that
A1:
G is independent
and
A2:
G = {A,B,C,D,E,F,J,M}
and
A3:
( A <> B & A <> C & A <> D & A <> E & A <> F & A <> J )
and
A4:
A <> M
and
A5:
( 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 (u,((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M))) /\ (EqClass (z,A)) <> {}
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)));
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)))) . B = EqClass (u,B)
by A3, A5, Th62;
reconsider GG = EqClass (u,((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M)) as set ;
reconsider I = EqClass (z,A) as set ;
GG = (EqClass (u,(((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J))) /\ (EqClass (u,M))
by Th1;
then
GG = ((EqClass (u,((((B '/\' C) '/\' D) '/\' E) '/\' F))) /\ (EqClass (u,J))) /\ (EqClass (u,M))
by Th1;
then
GG = (((EqClass (u,(((B '/\' C) '/\' D) '/\' E))) /\ (EqClass (u,F))) /\ (EqClass (u,J))) /\ (EqClass (u,M))
by Th1;
then
GG = ((((EqClass (u,((B '/\' C) '/\' D))) /\ (EqClass (u,E))) /\ (EqClass (u,F))) /\ (EqClass (u,J))) /\ (EqClass (u,M))
by Th1;
then
GG = (((((EqClass (u,(B '/\' C))) /\ (EqClass (u,D))) /\ (EqClass (u,E))) /\ (EqClass (u,F))) /\ (EqClass (u,J))) /\ (EqClass (u,M))
by Th1;
then A7:
GG /\ I = (((((((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;
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)))) . A = EqClass (z,A)
by FUNCT_7:94;
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)))) . C = EqClass (u,C)
by A3, A5, Th62;
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)))) . M = EqClass (u,M)
by A4, Lm1;
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)))) . J = EqClass (u,J)
by A3, A5, Th62;
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)))) . F = EqClass (u,F)
by A3, A5, Th62;
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)))) . E = EqClass (u,E)
by A3, A5, Th62;
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)))) +* (M .--> (EqClass (u,M)))) +* (A .--> (EqClass (z,A)))) . D = EqClass (u,D)
by A3, A5, Th62;
A15:
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 Th64;
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
object ;
TARSKI:def 3 ( 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))))
;
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 A15, ENUMSET1:def 6;
hence
t in bool Y
by A8, A6, A9, A14, A13, A12, A11, A10;
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 ;
A16:
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, Th63;
then
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, 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)))) . 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 3;
then A18:
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 SETFAM_1:def 9;
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 ;
( 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
;
((((((((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 A8, A6, A9, A14, A13, A12, A11, A10;
verum
end;
then
Intersect FF <> {}
by A1, A16, BVFUNC_2:def 5;
then consider m being object such that
A19:
m in Intersect FF
by XBOOLE_0:def 1;
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, A16, ENUMSET1:def 6;
then
((((((((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 3;
then A20:
m in EqClass (u,C)
by A9, A18, A19, SETFAM_1:def 1;
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, A16, ENUMSET1:def 6;
then
((((((((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 3;
then
m in EqClass (u,B)
by A6, A18, A19, SETFAM_1:def 1;
then A21:
m in (EqClass (u,B)) /\ (EqClass (u,C))
by A20, XBOOLE_0:def 4;
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, A16, ENUMSET1:def 6;
then
((((((((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 3;
then
m in EqClass (u,D)
by A14, A18, A19, SETFAM_1:def 1;
then A22:
m in ((EqClass (u,B)) /\ (EqClass (u,C))) /\ (EqClass (u,D))
by A21, XBOOLE_0:def 4;
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, A16, ENUMSET1:def 6;
then
((((((((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 3;
then
m in EqClass (u,E)
by A13, A18, A19, SETFAM_1:def 1;
then A23:
m in (((EqClass (u,B)) /\ (EqClass (u,C))) /\ (EqClass (u,D))) /\ (EqClass (u,E))
by A22, XBOOLE_0:def 4;
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, A16, ENUMSET1:def 6;
then
((((((((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 3;
then
m in EqClass (u,F)
by A12, A18, A19, SETFAM_1:def 1;
then A24:
m in ((((EqClass (u,B)) /\ (EqClass (u,C))) /\ (EqClass (u,D))) /\ (EqClass (u,E))) /\ (EqClass (u,F))
by A23, XBOOLE_0:def 4;
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, A16, ENUMSET1:def 6;
then
((((((((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 3;
then
m in EqClass (u,J)
by A11, A18, A19, SETFAM_1:def 1;
then A25:
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;
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, A16, ENUMSET1:def 6;
then
((((((((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 3;
then
m in EqClass (u,M)
by A10, A18, A19, SETFAM_1:def 1;
then A26:
m in ((((((EqClass (u,B)) /\ (EqClass (u,C))) /\ (EqClass (u,D))) /\ (EqClass (u,E))) /\ (EqClass (u,F))) /\ (EqClass (u,J))) /\ (EqClass (u,M))
by A25, XBOOLE_0:def 4;
m in EqClass (z,A)
by A8, A17, A18, A19, SETFAM_1:def 1;
hence
(EqClass (u,((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M))) /\ (EqClass (z,A)) <> {}
by A7, A26, XBOOLE_0:def 4; verum