let Y be non empty set ; :: thesis: for G being Subset of (PARTITIONS Y)
for A, B, C, D, E 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} & A <> B & A <> C & A <> D & A <> E & B <> C & B <> D & B <> E & C <> D & C <> E & D <> E holds
EqClass u,(((B '/\' C) '/\' D) '/\' E) meets EqClass z,A
let G be Subset of (PARTITIONS Y); :: thesis: for A, B, C, D, E 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} & A <> B & A <> C & A <> D & A <> E & B <> C & B <> D & B <> E & C <> D & C <> E & D <> E holds
EqClass u,(((B '/\' C) '/\' D) '/\' E) meets EqClass z,A
let A, B, C, D, E 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} & A <> B & A <> C & A <> D & A <> E & B <> C & B <> D & B <> E & C <> D & C <> E & D <> E holds
EqClass u,(((B '/\' C) '/\' D) '/\' E) 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} & A <> B & A <> C & A <> D & A <> E & B <> C & B <> D & B <> E & C <> D & C <> E & D <> E holds
EqClass u,(((B '/\' C) '/\' D) '/\' E) meets EqClass z,A
let h be Function; :: thesis: ( G is independent & G = {A,B,C,D,E} & A <> B & A <> C & A <> D & A <> E & B <> C & B <> D & B <> E & C <> D & C <> E & D <> E implies EqClass u,(((B '/\' C) '/\' D) '/\' E) meets EqClass z,A )
assume that
A1: G is independent and
A2: G = {A,B,C,D,E} and
A3: ( A <> B & A <> C & A <> D & A <> E & B <> C & B <> D & B <> E & C <> D & C <> E & D <> E ) ; :: thesis: EqClass u,(((B '/\' C) '/\' D) '/\' E) meets EqClass z,A
set h = ((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (A .--> (EqClass z,A));
A4: (((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (A .--> (EqClass z,A))) . B = EqClass u,B by A3, Th29;
A5: (((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (A .--> (EqClass z,A))) . D = EqClass u,D by A3, Th29;
A6: (((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (A .--> (EqClass z,A))) . C = EqClass u,C by A3, Th29;
A7: (((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (A .--> (EqClass z,A))) . E = EqClass u,E by A3, Th29;
A8: rng (((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (A .--> (EqClass z,A))) = {((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (A .--> (EqClass z,A))) . A),((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (A .--> (EqClass z,A))) . B),((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (A .--> (EqClass z,A))) . C),((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (A .--> (EqClass z,A))) . D),((((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (A .--> (EqClass z,A))) . E)} by Th31;
rng (((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (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))) +* (A .--> (EqClass z,A))) or t in bool Y )
assume A9: t in rng (((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (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))) +* (A .--> (EqClass z,A))) . A or t = (((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (A .--> (EqClass z,A))) . B or t = (((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (A .--> (EqClass z,A))) . C or t = (((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (A .--> (EqClass z,A))) . D or t = (((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (A .--> (EqClass z,A))) . E ) by A8, A9, ENUMSET1:def 3;

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

then t = EqClass z,A by A3, Th29;
hence t in bool Y ; :: thesis: verum

end;

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

hence t in bool Y by A5; :: thesis: verum

end;

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

hence t in bool Y by A7; :: 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))) +* (A .--> (EqClass z,A))) as Subset-Family of Y ;
A10: dom (((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (A .--> (EqClass z,A))) = G by A2, Th30;
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))) +* (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))) +* (A .--> (EqClass z,A))) . d in d )
assume A11: d in G ; :: thesis: (((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (A .--> (EqClass z,A))) . d in d

now

per cases ( d = A or d = B or d = C or d = D or d = E ) by A2, A11, ENUMSET1:def 3;

case A12: d = A ; :: thesis: (((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (A .--> (EqClass z,A))) . d in d

(((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (A .--> (EqClass z,A))) . A = EqClass z,A by A3, Th29;
hence (((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (A .--> (EqClass z,A))) . d in d by A12; :: thesis: verum

end;

case A13: d = B ; :: thesis: (((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (A .--> (EqClass z,A))) . d in d

(((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (A .--> (EqClass z,A))) . B = EqClass u,B by A3, Th29;
hence (((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (A .--> (EqClass z,A))) . d in d by A13; :: thesis: verum

end;

case A14: d = C ; :: thesis: (((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (A .--> (EqClass z,A))) . d in d

(((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (A .--> (EqClass z,A))) . C = EqClass u,C by A3, Th29;
hence (((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (A .--> (EqClass z,A))) . d in d by A14; :: thesis: verum

end;

case A15: d = D ; :: thesis: (((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (A .--> (EqClass z,A))) . d in d

(((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (A .--> (EqClass z,A))) . D = EqClass u,D by A3, Th29;
hence (((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (A .--> (EqClass z,A))) . d in d by A15; :: thesis: verum

end;

case A16: d = E ; :: thesis: (((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (A .--> (EqClass z,A))) . d in d

(((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (A .--> (EqClass z,A))) . E = EqClass u,E by A3, Th29;
hence (((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (A .--> (EqClass z,A))) . d in d by A16; :: thesis: verum

end;

end;

end;

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

end;

then Intersect FF <> {} by A1, A10, BVFUNC_2:def 5;
then consider m being set such that
A17: m in Intersect FF by XBOOLE_0:def 1;
A in dom (((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (A .--> (EqClass z,A))) by A2, A10, ENUMSET1:def 3;
then A18: (((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (A .--> (EqClass z,A))) . A in rng (((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (A .--> (EqClass z,A))) by FUNCT_1:def 5;
then A19: m in meet FF by A17, SETFAM_1:def 10;
then A20: m in (((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (A .--> (EqClass z,A))) . A by A18, SETFAM_1:def 1;
D in dom (((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (A .--> (EqClass z,A))) by A2, A10, ENUMSET1:def 3;
then (((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (A .--> (EqClass z,A))) . D in rng (((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (A .--> (EqClass z,A))) by FUNCT_1:def 5;
then A21: m in (((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (A .--> (EqClass z,A))) . D by A19, SETFAM_1:def 1;
C in dom (((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (A .--> (EqClass z,A))) by A2, A10, ENUMSET1:def 3;
then (((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (A .--> (EqClass z,A))) . C in rng (((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (A .--> (EqClass z,A))) by FUNCT_1:def 5;
then A22: m in (((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (A .--> (EqClass z,A))) . C by A19, SETFAM_1:def 1;
B in dom (((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (A .--> (EqClass z,A))) by A2, A10, ENUMSET1:def 3;
then (((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (A .--> (EqClass z,A))) . B in rng (((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (A .--> (EqClass z,A))) by FUNCT_1:def 5;
then m in (((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (A .--> (EqClass z,A))) . B by A19, SETFAM_1:def 1;
then m in (EqClass u,B) /\ (EqClass u,C) by A4, A6, A22, XBOOLE_0:def 4;
then A23: m in ((EqClass u,B) /\ (EqClass u,C)) /\ (EqClass u,D) by A5, A21, XBOOLE_0:def 4;
E in dom (((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (A .--> (EqClass z,A))) by A2, A10, ENUMSET1:def 3;
then (((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (A .--> (EqClass z,A))) . E in rng (((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (A .--> (EqClass z,A))) by FUNCT_1:def 5;
then m in (((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (A .--> (EqClass z,A))) . E by A19, SETFAM_1:def 1;
then A24: m in (((EqClass u,B) /\ (EqClass u,C)) /\ (EqClass u,D)) /\ (EqClass u,E) by A7, A23, XBOOLE_0:def 4;
set GG = EqClass u,(((B '/\' C) '/\' D) '/\' E);
EqClass u,(((B '/\' C) '/\' D) '/\' E) = (EqClass u,((B '/\' C) '/\' D)) /\ (EqClass u,E) by Th1;
then A25: EqClass u,(((B '/\' C) '/\' D) '/\' E) = ((EqClass u,(B '/\' C)) /\ (EqClass u,D)) /\ (EqClass u,E) by Th1;
(((((B .--> (EqClass u,B)) +* (C .--> (EqClass u,C))) +* (D .--> (EqClass u,D))) +* (E .--> (EqClass u,E))) +* (A .--> (EqClass z,A))) . A = EqClass z,A by A3, Th29;
then m in ((((EqClass u,B) /\ (EqClass u,C)) /\ (EqClass u,D)) /\ (EqClass u,E)) /\ (EqClass z,A) by A20, A24, XBOOLE_0:def 4;
then (((EqClass u,B) /\ (EqClass u,C)) /\ (EqClass u,D)) /\ (EqClass u,E) meets EqClass z,A by XBOOLE_0:4;
hence EqClass u,(((B '/\' C) '/\' D) '/\' E) meets EqClass z,A by A25, Th1; :: thesis: verum