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, Th40;
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, Th40;
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, Th40;
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, Th40;
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, Th40;
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, Th40;
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 Th42;
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 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))) +* (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

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, Th41;
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 5;
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 10;
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

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 set 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 5;
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 5;
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 5;
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 5;
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 5;
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