let Y be non empty set ; :: thesis: for G being Subset of ()
for A, B, C, D, E, F, J, M, N 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,N} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & A <> N & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & B <> N & C <> D & C <> E & C <> F & C <> J & C <> M & C <> N & D <> E & D <> F & D <> J & D <> M & D <> N & E <> F & E <> J & E <> M & E <> N & F <> J & F <> M & F <> N & J <> M & J <> N & M <> N holds
(EqClass (u,(((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N))) /\ (EqClass (z,A)) <> {}

let G be Subset of (); :: thesis: for A, B, C, D, E, F, J, M, N 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,N} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & A <> N & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & B <> N & C <> D & C <> E & C <> F & C <> J & C <> M & C <> N & D <> E & D <> F & D <> J & D <> M & D <> N & E <> F & E <> J & E <> M & E <> N & F <> J & F <> M & F <> N & J <> M & J <> N & M <> N holds
(EqClass (u,(((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N))) /\ (EqClass (z,A)) <> {}

let A, B, C, D, E, F, J, M, N 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,N} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & A <> N & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & B <> N & C <> D & C <> E & C <> F & C <> J & C <> M & C <> N & D <> E & D <> F & D <> J & D <> M & D <> N & E <> F & E <> J & E <> M & E <> N & F <> J & F <> M & F <> N & J <> M & J <> N & M <> N holds
(EqClass (u,(((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N))) /\ (EqClass (z,A)) <> {}

let z, u be Element of Y; :: thesis: ( G is independent & G = {A,B,C,D,E,F,J,M,N} & A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & A <> N & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & B <> N & C <> D & C <> E & C <> F & C <> J & C <> M & C <> N & D <> E & D <> F & D <> J & D <> M & D <> N & E <> F & E <> J & E <> M & E <> N & F <> J & F <> M & F <> N & J <> M & J <> N & M <> N implies (EqClass (u,(((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N))) /\ (EqClass (z,A)) <> {} )
assume that
A1: G is independent and
A2: G = {A,B,C,D,E,F,J,M,N} and
A3: ( A <> B & A <> C & A <> D & A <> E & A <> F & A <> J & A <> M & A <> N & B <> C & B <> D & B <> E & B <> F & B <> J & B <> M & B <> N & C <> D & C <> E & C <> F & C <> J & C <> M & C <> N & D <> E & D <> F & D <> J & D <> M & D <> N & E <> F & E <> J & E <> M & E <> N & F <> J & F <> M & F <> N & J <> M & J <> N & M <> N ) ; :: thesis: (EqClass (u,(((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N))) /\ (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)))) +* (N .--> (EqClass (u,N)))) +* (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)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . A = EqClass (z,A) by ;
set GG = EqClass (u,(((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N));
EqClass (u,(((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N)) = (EqClass (u,((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M))) /\ (EqClass (u,N)) by Th1;
then EqClass (u,(((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N)) = ((EqClass (u,(((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J))) /\ (EqClass (u,M))) /\ (EqClass (u,N)) by Th1;
then EqClass (u,(((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N)) = (((EqClass (u,((((B '/\' C) '/\' D) '/\' E) '/\' F))) /\ (EqClass (u,J))) /\ (EqClass (u,M))) /\ (EqClass (u,N)) by Th1;
then EqClass (u,(((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N)) = ((((EqClass (u,(((B '/\' C) '/\' D) '/\' E))) /\ (EqClass (u,F))) /\ (EqClass (u,J))) /\ (EqClass (u,M))) /\ (EqClass (u,N)) by Th1;
then EqClass (u,(((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N)) = (((((EqClass (u,((B '/\' C) '/\' D))) /\ (EqClass (u,E))) /\ (EqClass (u,F))) /\ (EqClass (u,J))) /\ (EqClass (u,M))) /\ (EqClass (u,N)) by Th1;
then EqClass (u,(((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N)) = ((((((EqClass (u,(B '/\' C))) /\ (EqClass (u,D))) /\ (EqClass (u,E))) /\ (EqClass (u,F))) /\ (EqClass (u,J))) /\ (EqClass (u,M))) /\ (EqClass (u,N)) by Th1;
then A5: (EqClass (u,(((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N))) /\ (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 (u,N))) /\ (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)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . B = EqClass (u,B) by ;
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)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . F = EqClass (u,F) by ;
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)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . E = EqClass (u,E) by ;
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)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . M = EqClass (u,M) by ;
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)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . J = EqClass (u,J) by ;
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)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . N = EqClass (u,N) by ;
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)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . D = EqClass (u,D) by ;
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)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . C = EqClass (u,C) by ;
A14: 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)))) +* (N .--> (EqClass (u,N)))) +* (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)))) +* (N .--> (EqClass (u,N)))) +* (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)))) +* (N .--> (EqClass (u,N)))) +* (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)))) +* (N .--> (EqClass (u,N)))) +* (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)))) +* (N .--> (EqClass (u,N)))) +* (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)))) +* (N .--> (EqClass (u,N)))) +* (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)))) +* (N .--> (EqClass (u,N)))) +* (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)))) +* (N .--> (EqClass (u,N)))) +* (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)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . M),((((((((((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)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . N)} by Th78;
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)))) +* (N .--> (EqClass (u,N)))) +* (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)))) +* (J .--> (EqClass (u,J)))) +* (M .--> (EqClass (u,M)))) +* (N .--> (EqClass (u,N)))) +* (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)))) +* (N .--> (EqClass (u,N)))) +* (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)))) +* (N .--> (EqClass (u,N)))) +* (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)))) +* (N .--> (EqClass (u,N)))) +* (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)))) +* (N .--> (EqClass (u,N)))) +* (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)))) +* (N .--> (EqClass (u,N)))) +* (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)))) +* (N .--> (EqClass (u,N)))) +* (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)))) +* (N .--> (EqClass (u,N)))) +* (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)))) +* (N .--> (EqClass (u,N)))) +* (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)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . M 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)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . N ) by ;
hence t in bool Y by A4, A6, A13, A12, A8, A7, A10, A9, A11; :: 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)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) as Subset-Family of Y ;
A15: 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)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) = G by ;
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)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) by ;
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)))) +* (N .--> (EqClass (u,N)))) +* (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)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def 3;
then A17: 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)))) +* (N .--> (EqClass (u,N)))) +* (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)))) +* (N .--> (EqClass (u,N)))) +* (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)))) +* (N .--> (EqClass (u,N)))) +* (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)))) +* (N .--> (EqClass (u,N)))) +* (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 or d = N ) by ;
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)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . d in d by A4, A6, A13, A12, A8, A7, A10, A9, A11; :: thesis: verum
end;
then Intersect FF <> {} by ;
then consider m being object such that
A18: 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)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) by ;
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)))) +* (N .--> (EqClass (u,N)))) +* (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)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def 3;
then A19: m in EqClass (u,C) by ;
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)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) by ;
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)))) +* (N .--> (EqClass (u,N)))) +* (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)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def 3;
then m in EqClass (u,B) by ;
then A20: m in (EqClass (u,B)) /\ (EqClass (u,C)) by ;
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)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) by ;
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)))) +* (N .--> (EqClass (u,N)))) +* (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)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def 3;
then m in EqClass (u,D) by ;
then A21: m in ((EqClass (u,B)) /\ (EqClass (u,C))) /\ (EqClass (u,D)) by ;
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)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) by ;
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)))) +* (N .--> (EqClass (u,N)))) +* (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)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def 3;
then m in EqClass (u,E) by ;
then A22: m in (((EqClass (u,B)) /\ (EqClass (u,C))) /\ (EqClass (u,D))) /\ (EqClass (u,E)) by ;
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)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) by ;
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)))) +* (N .--> (EqClass (u,N)))) +* (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)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def 3;
then m in EqClass (u,F) by ;
then A23: m in ((((EqClass (u,B)) /\ (EqClass (u,C))) /\ (EqClass (u,D))) /\ (EqClass (u,E))) /\ (EqClass (u,F)) by ;
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)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) by ;
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)))) +* (N .--> (EqClass (u,N)))) +* (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)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def 3;
then m in EqClass (u,J) by ;
then A24: m in (((((EqClass (u,B)) /\ (EqClass (u,C))) /\ (EqClass (u,D))) /\ (EqClass (u,E))) /\ (EqClass (u,F))) /\ (EqClass (u,J)) by ;
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)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) by ;
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)))) +* (N .--> (EqClass (u,N)))) +* (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)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def 3;
then m in EqClass (u,M) by ;
then A25: m in ((((((EqClass (u,B)) /\ (EqClass (u,C))) /\ (EqClass (u,D))) /\ (EqClass (u,E))) /\ (EqClass (u,F))) /\ (EqClass (u,J))) /\ (EqClass (u,M)) by ;
N 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)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) by ;
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)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) . N 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)))) +* (N .--> (EqClass (u,N)))) +* (A .--> (EqClass (z,A)))) by FUNCT_1:def 3;
then m in EqClass (u,N) by ;
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))) /\ (EqClass (u,N)) by ;
m in EqClass (z,A) by ;
hence (EqClass (u,(((((((B '/\' C) '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) '/\' N))) /\ (EqClass (z,A)) <> {} by ; :: thesis: verum