reserve Y for non empty set,
  G for Subset of PARTITIONS(Y),
  A,B,C,D,E,F for a_partition of Y;
reserve Y for non empty set,
  G for Subset of PARTITIONS(Y),
  A, B, C, D, E, F, J, M for a_partition of Y,
  x,x1,x2,x3,x4,x5,x6,x7,x8,x9 for set;

theorem
  for G being Subset of PARTITIONS(Y), A,B,C,D,E,F,J,M,N being
a_partition of Y, 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 & EqClass(z,C '/\' D '/\' E '/\' F '/\' J '/\' M '/\' N)=
EqClass(u,C '/\' D '/\' E '/\' F '/\' J '/\' M '/\' N) holds EqClass(u,CompF(A,
  G)) meets EqClass(z,CompF(B,G))
proof
  let G be Subset of PARTITIONS(Y);
  let A,B,C,D,E,F,J,M,N be a_partition of Y;
  let z,u be Element of Y;
  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 and
A4: EqClass(z,C '/\' D '/\' E '/\' F '/\' J '/\' M '/\' N)= EqClass(u,C
  '/\' D '/\' E '/\' F '/\' J '/\' M '/\' N);
  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));
A5: h.A = EqClass(z,A) by A3,Th76;
  set L=EqClass(z,C '/\' D '/\' E '/\' F '/\' J '/\' M '/\' N);
  set GG=EqClass(u,(((B '/\' C) '/\' D) '/\' E '/\' F '/\' J '/\' M '/\' N));
  reconsider I=EqClass(z,A) as set;
  GG = EqClass(u,B '/\' C '/\' D '/\' E '/\' F '/\' J '/\' M) /\ EqClass(
  u,N) by Th1;
  then GG = EqClass(u,B '/\' C '/\' D '/\' E '/\' F '/\' J) /\ EqClass(u,M )
  /\ EqClass(u,N) by Th1;
  then GG = EqClass(u,B '/\' C '/\' D '/\' E '/\' F) /\ EqClass(u,J) /\
  EqClass(u,M) /\ EqClass(u,N) by Th1;
  then
  GG = EqClass(u,B '/\' C '/\' D '/\' E) /\ EqClass(u,F) /\ EqClass(u,J)
  /\ EqClass(u,M) /\ EqClass(u,N) by Th1;
  then GG = (((EqClass(u,B '/\' C '/\' D)) /\ EqClass(u,E)) /\ EqClass(u,F ))
  /\ EqClass(u,J) /\ EqClass(u,M) /\ EqClass(u,N) by Th1;
  then
  GG = ((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
A6: GG /\ I = ((((((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;
A7: CompF(A,G) = B '/\' C '/\' D '/\' E '/\' F '/\' J '/\' M '/\' N by A2,A3
,Th67;
  reconsider HH=EqClass(z,CompF(B,G)) as set;
A8: z in HH by EQREL_1:def 6;
A9: A '/\' (C '/\' D '/\' E '/\' F '/\' J '/\' M '/\' N) = A '/\' (C '/\'
  D '/\' E '/\' F '/\' J '/\' M) '/\' N by PARTIT1:14
    .= A '/\' (C '/\' D '/\' E '/\' F '/\' J) '/\' M '/\' N by PARTIT1:14
    .= A '/\' (C '/\' D '/\' E '/\' F) '/\' J '/\' M '/\' N by PARTIT1:14
    .= A '/\' (C '/\' D '/\' E) '/\' F '/\' J '/\' M '/\' N by PARTIT1:14
    .= A '/\' (C '/\' D) '/\' E '/\' F '/\' J '/\' M '/\' N by PARTIT1:14
    .= A '/\' C '/\' D '/\' E '/\' F '/\' J '/\' M '/\' N by PARTIT1:14;
A10: h.B = EqClass(u,B) by A3,Th76;
A11: h.N = EqClass(u,N) by A3,Th76;
A12: h.D = EqClass(u,D) by A3,Th76;
A13: h.C = EqClass(u,C) by A3,Th76;
A14: h.M = EqClass(u,M) by A3,Th76;
A15: h.J = EqClass(u,J) by A3,Th76;
A16: h.F = EqClass(u,F) by A3,Th76;
A17: h.E = EqClass(u,E) by A3,Th76;
A18: rng h = {h.A,h.B,h.C,h.D,h.E,h.F,h.J,h.M,h.N} by Th78;
  rng h c= bool Y
  proof
    let t be object;
    assume t in rng h;
    then t=h.A or t=h.B or t=h.C or t=h.D or t=h.E or t=h.F or t=h.J or t=h.M
    or t=h.N by A18,ENUMSET1:def 7;
    hence thesis by A5,A10,A13,A12,A17,A16,A15,A14,A11;
  end;
  then reconsider FF=rng h as Subset-Family of Y;
A19: dom h = G by A2,Th77;
  then A in dom h by A2,ENUMSET1:def 7;
  then
A20: h.A in rng h by FUNCT_1:def 3;
  then
A21: Intersect FF = meet (rng h) by SETFAM_1:def 9;
  for d being set st d in G holds h.d in d
  proof
    let d be set;
    assume d in G;
    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 A2,
ENUMSET1:def 7;
    hence thesis by A5,A10,A13,A12,A17,A16,A15,A14,A11;
  end;
  then (Intersect FF)<>{} by A1,A19,BVFUNC_2:def 5;
  then consider m being object such that
A22: m in Intersect FF by XBOOLE_0:def 1;
  C in dom h by A2,A19,ENUMSET1:def 7;
  then h.C in rng h by FUNCT_1:def 3;
  then
A23: m in EqClass(u,C) by A13,A21,A22,SETFAM_1:def 1;
  B in dom h by A2,A19,ENUMSET1:def 7;
  then h.B in rng h by FUNCT_1:def 3;
  then m in EqClass(u,B) by A10,A21,A22,SETFAM_1:def 1;
  then
A24: m in EqClass(u,B) /\ EqClass(u,C) by A23,XBOOLE_0:def 4;
  D in dom h by A2,A19,ENUMSET1:def 7;
  then h.D in rng h by FUNCT_1:def 3;
  then m in EqClass(u,D) by A12,A21,A22,SETFAM_1:def 1;
  then
A25: m in EqClass(u,B) /\ EqClass(u,C) /\ EqClass(u,D) by A24,XBOOLE_0:def 4;
  E in dom h by A2,A19,ENUMSET1:def 7;
  then h.E in rng h by FUNCT_1:def 3;
  then m in EqClass(u,E) by A17,A21,A22,SETFAM_1:def 1;
  then
A26: m in EqClass(u,B) /\ EqClass(u,C) /\ EqClass(u,D) /\ EqClass(u,E) by A25,
XBOOLE_0:def 4;
  F in dom h by A2,A19,ENUMSET1:def 7;
  then h.F in rng h by FUNCT_1:def 3;
  then m in EqClass(u,F) by A16,A21,A22,SETFAM_1:def 1;
  then
A27: m in EqClass(u,B) /\ EqClass(u,C) /\ EqClass(u,D) /\ EqClass(u,E) /\
  EqClass(u,F) by A26,XBOOLE_0:def 4;
  J in dom h by A2,A19,ENUMSET1:def 7;
  then h.J in rng h by FUNCT_1:def 3;
  then m in EqClass(u,J) by A15,A21,A22,SETFAM_1:def 1;
  then
A28: m in (((EqClass(u,B) /\ EqClass(u,C)) /\ EqClass(u,D)) /\ EqClass(u,E
  )) /\ EqClass(u,F) /\ EqClass(u,J) by A27,XBOOLE_0:def 4;
  M in dom h by A2,A19,ENUMSET1:def 7;
  then h.M in rng h by FUNCT_1:def 3;
  then m in EqClass(u,M) by A14,A21,A22,SETFAM_1:def 1;
  then
A29: m in ((((EqClass(u,B) /\ EqClass(u,C)) /\ EqClass(u,D)) /\ EqClass(u,E
  )) /\ EqClass(u,F) /\ EqClass(u,J)) /\ EqClass(u,M) by A28,XBOOLE_0:def 4;
  N in dom h by A2,A19,ENUMSET1:def 7;
  then h.N in rng h by FUNCT_1:def 3;
  then m in EqClass(u,N) by A11,A21,A22,SETFAM_1:def 1;
  then
A30: 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 A29,
XBOOLE_0:def 4;
  m in EqClass(z,A) by A5,A20,A21,A22,SETFAM_1:def 1;
  then GG /\ I <> {} by A6,A30,XBOOLE_0:def 4;
  then consider p being object such that
A31: p in GG /\ I by XBOOLE_0:def 1;
  reconsider p as Element of Y by A31;
  set K=EqClass(p,C '/\' D '/\' E '/\' F '/\' J '/\' M '/\' N);
A32: p in GG by A31,XBOOLE_0:def 4;
A33: p in EqClass(p,C '/\' D '/\' E '/\' F '/\' J '/\' M '/\' N) by
EQREL_1:def 6;
  GG=EqClass(u,(((B '/\' (C '/\' D)) '/\' E) '/\' F) '/\' J '/\' M '/\'
  N) by PARTIT1:14;
  then GG=EqClass(u,((B '/\' ((C '/\' D) '/\' E)) '/\' F) '/\' J '/\' M '/\'
  N) by PARTIT1:14;
  then GG=EqClass(u,(B '/\' (((C '/\' D) '/\' E) '/\' F)) '/\' J '/\' M '/\'
  N) by PARTIT1:14;
  then GG=EqClass(u,B '/\' ((((C '/\' D) '/\' E) '/\' F) '/\' J) '/\' M '/\'
  N) by PARTIT1:14;
  then GG= EqClass(u,B '/\' (((((C '/\' D) '/\' E) '/\' F) '/\' J) '/\' M)
  '/\' N) by PARTIT1:14;
  then GG=EqClass(u,B '/\' (C '/\' D '/\' E '/\' F '/\' J '/\' M '/\' N)) by
PARTIT1:14;
  then GG c= L by A4,BVFUNC11:3;
  then K meets L by A32,A33,XBOOLE_0:3;
  then K=L by EQREL_1:41;
  then
A34: z in K by EQREL_1:def 6;
  p in K & p in I by A31,EQREL_1:def 6,XBOOLE_0:def 4;
  then
A35: p in I /\ K by XBOOLE_0:def 4;
  then I /\ K in INTERSECTION(A,C '/\' D '/\' E '/\' F '/\' J '/\' M '/\' N)
  & not I /\ K in {{}} by SETFAM_1:def 5,TARSKI:def 1;
  then
A36: I /\ K in INTERSECTION(A,C '/\' D '/\' E '/\' F '/\' J '/\' M '/\' N)
  \ {{}} by XBOOLE_0:def 5;
  z in I by EQREL_1:def 6;
  then z in I /\ K by A34,XBOOLE_0:def 4;
  then
A37: I /\ K meets HH by A8,XBOOLE_0:3;
  CompF(B,G) = A '/\' C '/\' D '/\' E '/\' F '/\' J '/\' M '/\' N by A2,A3,Th68
;
  then I /\ K in CompF(B,G) by A36,A9,PARTIT1:def 4;
  then p in HH by A35,A37,EQREL_1:def 4;
  hence thesis by A7,A32,XBOOLE_0:3;
end;
