let Y be non empty set ; :: thesis: for G being Subset of (PARTITIONS Y)
for A, B, C, D, E, F being a_partition of Y
for z, u being Element of Y
for h being Function st G is independent & G = {A,B,C,D,E,F} & A <> B & A <> C & A <> D & A <> E & A <> F & B <> C & B <> D & B <> E & B <> F & C <> D & C <> E & C <> F & D <> E & D <> F & E <> F holds
EqClass (u,((((B '/\' C) '/\' D) '/\' E) '/\' F)) meets EqClass (z,A)
let G be Subset of (PARTITIONS Y); :: thesis: for A, B, C, D, E, F being a_partition of Y
for z, u being Element of Y
for h being Function st G is independent & G = {A,B,C,D,E,F} & A <> B & A <> C & A <> D & A <> E & A <> F & B <> C & B <> D & B <> E & B <> F & C <> D & C <> E & C <> F & D <> E & D <> F & E <> F holds
EqClass (u,((((B '/\' C) '/\' D) '/\' E) '/\' F)) meets EqClass (z,A)
let A, B, C, D, E, F be a_partition of Y; :: thesis: for z, u being Element of Y
for h being Function st G is independent & G = {A,B,C,D,E,F} & A <> B & A <> C & A <> D & A <> E & A <> F & B <> C & B <> D & B <> E & B <> F & C <> D & C <> E & C <> F & D <> E & D <> F & E <> F holds
EqClass (u,((((B '/\' C) '/\' D) '/\' E) '/\' F)) meets EqClass (z,A)
let z, u be Element of Y; :: thesis: for h being Function st G is independent & G = {A,B,C,D,E,F} & A <> B & A <> C & A <> D & A <> E & A <> F & B <> C & B <> D & B <> E & B <> F & C <> D & C <> E & C <> F & D <> E & D <> F & E <> F holds
EqClass (u,((((B '/\' C) '/\' D) '/\' E) '/\' F)) meets EqClass (z,A)
let h be Function; :: thesis: ( G is independent & G = {A,B,C,D,E,F} & A <> B & A <> C & A <> D & A <> E & A <> F & B <> C & B <> D & B <> E & B <> F & C <> D & C <> E & C <> F & D <> E & D <> F & E <> F implies EqClass (u,((((B '/\' C) '/\' D) '/\' E) '/\' F)) meets EqClass (z,A) )
assume that
A1: G is independent and
A2: G = {A,B,C,D,E,F} and
A3: ( A <> B & A <> C & A <> D & A <> E & A <> F & B <> C & B <> D & B <> E & B <> F & C <> D & C <> E & C <> F & D <> E & D <> F & E <> F ) ; :: thesis: EqClass (u,((((B '/\' C) '/\' D) '/\' E) '/\' F)) meets 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)))) +* (A .--> (EqClass (z,A)));
A4: ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . A = EqClass (z,A) by A3, Th37;
set GG = EqClass (u,((((B '/\' C) '/\' D) '/\' E) '/\' F));
EqClass (u,((((B '/\' C) '/\' D) '/\' E) '/\' F)) = (EqClass (u,(((B '/\' C) '/\' D) '/\' E))) /\ (EqClass (u,F)) by Th1;
then EqClass (u,((((B '/\' C) '/\' D) '/\' E) '/\' F)) = ((EqClass (u,((B '/\' C) '/\' D))) /\ (EqClass (u,E))) /\ (EqClass (u,F)) by Th1;
then EqClass (u,((((B '/\' C) '/\' D) '/\' E) '/\' F)) = (((EqClass (u,(B '/\' C))) /\ (EqClass (u,D))) /\ (EqClass (u,E))) /\ (EqClass (u,F)) by Th1;
then A5: (EqClass (u,((((B '/\' C) '/\' D) '/\' E) '/\' F))) /\ (EqClass (z,A)) = (((((EqClass (u,B)) /\ (EqClass (u,C))) /\ (EqClass (u,D))) /\ (EqClass (u,E))) /\ (EqClass (u,F))) /\ (EqClass (z,A)) by Th1;
A6: ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . B = EqClass (u,B) by A3, Th37;
A7: ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . D = EqClass (u,D) by A3, Th37;
A8: ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . C = EqClass (u,C) by A3, Th37;
A9: ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . F = EqClass (u,F) by A3, Th37;
A10: ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . E = EqClass (u,E) by A3, Th37;
A11: rng ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) = {(((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (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)))) +* (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)))) +* (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)))) +* (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)))) +* (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)))) +* (A .--> (EqClass (z,A)))) . F)} by Th39;
rng ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) c= bool Y

proof

let t be object ; :: 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)))) +* (A .--> (EqClass (z,A)))) or t in bool Y )
assume A12: t in rng ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) ; :: thesis: t in bool Y

now :: thesis: ( ( t = ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . A & t in bool Y ) or ( t = ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . B & t in bool Y ) or ( t = ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . C & t in bool Y ) or ( t = ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . D & t in bool Y ) or ( t = ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . E & t in bool Y ) or ( t = ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . F & t in bool Y ) )

per cases ( t = ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (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)))) +* (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)))) +* (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)))) +* (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)))) +* (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)))) +* (A .--> (EqClass (z,A)))) . F ) by A11, A12, ENUMSET1:def 4;

case t = ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . A ; :: thesis: t in bool Y

hence t in bool Y by A4; :: thesis: verum

end;

case t = ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . B ; :: thesis: t in bool Y

hence t in bool Y by A6; :: thesis: verum

end;

case t = ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . C ; :: thesis: t in bool Y

hence t in bool Y by A8; :: thesis: verum

end;

case t = ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . D ; :: thesis: t in bool Y

hence t in bool Y by A7; :: thesis: verum

end;

case t = ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . E ; :: thesis: t in bool Y

hence t in bool Y by A10; :: thesis: verum

end;

case t = ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . F ; :: thesis: t in bool Y

hence t in bool Y by A9; :: thesis: verum

end;

end;

end;

hence t in bool Y ; :: 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)))) +* (A .--> (EqClass (z,A)))) as Subset-Family of Y ;
A13: dom ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) = G by A2, Th38;
then A in dom ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) by A2, ENUMSET1:def 4;
then A14: ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (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)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def 3;
then A15: Intersect FF = meet (rng ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (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)))) +* (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)))) +* (A .--> (EqClass (z,A)))) . d in d )
assume A16: d in G ; :: thesis: ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . d in d

now :: thesis: ( ( d = A & ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . d in d ) or ( d = B & ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . d in d ) or ( d = C & ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . d in d ) or ( d = D & ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . d in d ) or ( d = E & ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . d in d ) or ( d = F & ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . d in d ) )

per cases ( d = A or d = B or d = C or d = D or d = E or d = F ) by A2, A16, ENUMSET1:def 4;

case d = A ; :: thesis: ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . d in d

hence ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . d in d by A4; :: thesis: verum

end;

case d = B ; :: thesis: ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . d in d

hence ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . d in d by A6; :: thesis: verum

end;

case d = C ; :: thesis: ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . d in d

hence ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . d in d by A8; :: thesis: verum

end;

case d = D ; :: thesis: ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . d in d

hence ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . d in d by A7; :: thesis: verum

end;

case d = E ; :: thesis: ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . d in d

hence ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . d in d by A10; :: thesis: verum

end;

case d = F ; :: thesis: ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . d in d

hence ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . d in d by A9; :: thesis: verum

end;

end;

end;

hence ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (A .--> (EqClass (z,A)))) . d in d ; :: thesis: verum

end;

then Intersect FF <> {} by A1, A13, BVFUNC_2:def 5;
then consider m being object such that
A17: 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)))) +* (A .--> (EqClass (z,A)))) by A2, A13, ENUMSET1:def 4;
then ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (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)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def 3;
then A18: m in EqClass (u,C) by A8, A15, A17, 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)))) +* (A .--> (EqClass (z,A)))) by A2, A13, ENUMSET1:def 4;
then ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (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)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def 3;
then m in EqClass (u,B) by A6, A15, A17, SETFAM_1:def 1;
then A19: m in (EqClass (u,B)) /\ (EqClass (u,C)) by A18, 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)))) +* (A .--> (EqClass (z,A)))) by A2, A13, ENUMSET1:def 4;
then ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (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)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def 3;
then m in EqClass (u,D) by A7, A15, A17, SETFAM_1:def 1;
then A20: m in ((EqClass (u,B)) /\ (EqClass (u,C))) /\ (EqClass (u,D)) by A19, 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)))) +* (A .--> (EqClass (z,A)))) by A2, A13, ENUMSET1:def 4;
then ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (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)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def 3;
then m in EqClass (u,E) by A10, A15, A17, SETFAM_1:def 1;
then A21: m in (((EqClass (u,B)) /\ (EqClass (u,C))) /\ (EqClass (u,D))) /\ (EqClass (u,E)) by A20, 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)))) +* (A .--> (EqClass (z,A)))) by A2, A13, ENUMSET1:def 4;
then ((((((B .--> (EqClass (u,B))) +* (C .--> (EqClass (u,C)))) +* (D .--> (EqClass (u,D)))) +* (E .--> (EqClass (u,E)))) +* (F .--> (EqClass (u,F)))) +* (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)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def 3;
then m in EqClass (u,F) by A9, A15, A17, SETFAM_1:def 1;
then A22: m in ((((EqClass (u,B)) /\ (EqClass (u,C))) /\ (EqClass (u,D))) /\ (EqClass (u,E))) /\ (EqClass (u,F)) by A21, XBOOLE_0:def 4;
m in EqClass (z,A) by A4, A14, A15, A17, SETFAM_1:def 1;
then m in (((((EqClass (u,B)) /\ (EqClass (u,C))) /\ (EqClass (u,D))) /\ (EqClass (u,E))) /\ (EqClass (u,F))) /\ (EqClass (z,A)) by A22, XBOOLE_0:def 4;
hence EqClass (u,((((B '/\' C) '/\' D) '/\' E) '/\' F)) meets EqClass (z,A) by A5, XBOOLE_0:def 7; :: thesis: verum