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 being a_partition
of Y, z,u being Element of Y st G is independent & G={A,B,C,D,E,F,J,M} & A<>B &
A<>C & A<>D & A<>E & A<>F & A<>J & A<>M & B<>C & B<>D & B<>E & B<>F & B<>J & B
<>M & C<>D & C<>E & C<>F & C<>J & C<>M & D<>E & D<>F & D<>J & D<>M & E<>F & E<>
J & E<>M & F<>J & F<>M & J<>M & EqClass(z,C '/\' D '/\' E '/\' F '/\' J '/\' M)
  = EqClass(u,C '/\' D '/\' E '/\' F '/\' J '/\' M) 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 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} and
A3: A<>B & A<>C & A<>D & A<>E & A<>F & A<>J and
A4: A<>M and
A5: B<>C & B<>D & B<>E & B<>F & B<>J & B<>M & C<>D & C<>E & C<>F & C<>J
  & C<> M & D<>E & D<>F & D<>J & D<>M & E<>F & E<>J & E<>M & F<>J & F<>M & J<>M
  and
A6: EqClass(z,C '/\' D '/\' E '/\' F '/\' J '/\' M)= EqClass(u,C '/\' D
  '/\' E '/\' F '/\' J '/\' M);
  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)) +* (A .--> EqClass(z,A));
A7: h.B = EqClass(u,B) by A3,A5,Th62;
  set HH=EqClass(z,CompF(B,G)), I=EqClass(z,A), GG=EqClass(u,(((B '/\' C) '/\'
  D) '/\' E '/\' F '/\' J '/\' M));
A8: GG=EqClass(u,CompF(A,G)) by A2,A3,A4,A5,Th54;
  GG = EqClass(u,B '/\' C '/\' D '/\' E '/\' F '/\' J) /\ EqClass(u,M) by Th1;
  then GG = EqClass(u,B '/\' C '/\' D '/\' E '/\' F) /\ EqClass(u,J) /\
  EqClass(u,M) by Th1;
  then
  GG = EqClass(u,B '/\' C '/\' D '/\' E) /\ EqClass(u,F) /\ EqClass(u,J)
  /\ EqClass(u,M) by Th1;
  then GG = EqClass(u,B '/\' C '/\' D) /\ EqClass(u,E) /\ EqClass(u,F) /\
  EqClass(u,J) /\ EqClass(u,M) by Th1;
  then
  GG = ((EqClass(u,B '/\' C) /\ EqClass(u,D)) /\ EqClass(u,E)) /\ EqClass
  (u,F) /\ EqClass(u,J) /\ EqClass(u,M) by Th1;
  then
A9: GG /\ I = ((((EqClass(u,B) /\ EqClass(u,C)) /\ EqClass(u,D)) /\ EqClass
  (u,E)) /\ EqClass(u,F) /\ EqClass(u,J) /\ EqClass(u,M)) /\ EqClass(z,A) by
Th1;
A10: h.A = EqClass(z,A) by FUNCT_7:94;
A11: h.C = EqClass(u,C) by A3,A5,Th62;
A12: h.M = EqClass(u,M) by A4,Lm1;
A13: h.J = EqClass(u,J) by A3,A5,Th62;
A14: h.F = EqClass(u,F) by A3,A5,Th62;
A15: h.E = EqClass(u,E) by A3,A5,Th62;
A16: h.D = EqClass(u,D) by A3,A5,Th62;
A17: rng h = {h.A,h.B,h.C,h.D,h.E,h.F,h.J,h.M} by Th64;
  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
    by A17,ENUMSET1:def 6;
    hence thesis by A10,A7,A11,A16,A15,A14,A13,A12;
  end;
  then reconsider FF=rng h as Subset-Family of Y;
A18: dom h = G by A2,Th63;
  then A in dom h by A2,ENUMSET1:def 6;
  then
A19: h.A in rng h by FUNCT_1:def 3;
  then
A20: 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 by A2,
ENUMSET1:def 6;
    hence thesis by A10,A7,A11,A16,A15,A14,A13,A12;
  end;
  then (Intersect FF)<>{} by A1,A18,BVFUNC_2:def 5;
  then consider m being object such that
A21: m in Intersect FF by XBOOLE_0:def 1;
  C in dom h by A2,A18,ENUMSET1:def 6;
  then h.C in rng h by FUNCT_1:def 3;
  then
A22: m in EqClass(u,C) by A11,A20,A21,SETFAM_1:def 1;
  B in dom h by A2,A18,ENUMSET1:def 6;
  then h.B in rng h by FUNCT_1:def 3;
  then m in EqClass(u,B) by A7,A20,A21,SETFAM_1:def 1;
  then
A23: m in EqClass(u,B) /\ EqClass(u,C) by A22,XBOOLE_0:def 4;
  D in dom h by A2,A18,ENUMSET1:def 6;
  then h.D in rng h by FUNCT_1:def 3;
  then m in EqClass(u,D) by A16,A20,A21,SETFAM_1:def 1;
  then
A24: m in EqClass(u,B) /\ EqClass(u,C) /\ EqClass(u,D) by A23,XBOOLE_0:def 4;
  E in dom h by A2,A18,ENUMSET1:def 6;
  then h.E in rng h by FUNCT_1:def 3;
  then m in EqClass(u,E) by A15,A20,A21,SETFAM_1:def 1;
  then
A25: m in EqClass(u,B) /\ EqClass(u,C) /\ EqClass(u,D) /\ EqClass(u,E) by A24,
XBOOLE_0:def 4;
  F in dom h by A2,A18,ENUMSET1:def 6;
  then h.F in rng h by FUNCT_1:def 3;
  then m in EqClass(u,F) by A14,A20,A21,SETFAM_1:def 1;
  then
A26: m in EqClass(u,B) /\ EqClass(u,C) /\ EqClass(u,D) /\ EqClass(u,E) /\
  EqClass(u,F) by A25,XBOOLE_0:def 4;
  J in dom h by A2,A18,ENUMSET1:def 6;
  then h.J in rng h by FUNCT_1:def 3;
  then m in EqClass(u,J) by A13,A20,A21,SETFAM_1:def 1;
  then
A27: m in (((EqClass(u,B) /\ EqClass(u,C)) /\ EqClass(u,D)) /\ EqClass(u,E)
  ) /\ EqClass(u,F) /\ EqClass(u,J) by A26,XBOOLE_0:def 4;
  M in dom h by A2,A18,ENUMSET1:def 6;
  then h.M in rng h by FUNCT_1:def 3;
  then m in EqClass(u,M) by A12,A20,A21,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)) /\ EqClass(u,M) by A27,XBOOLE_0:def 4;
  m in EqClass(z,A) by A10,A19,A20,A21,SETFAM_1:def 1;
  then GG /\ I <> {} by A9,A28,XBOOLE_0:def 4;
  then consider p being object such that
A29: p in GG /\ I by XBOOLE_0:def 1;
  reconsider p as Element of Y by A29;
  reconsider K=EqClass(p,C '/\' D '/\' E '/\' F '/\' J '/\' M) as set;
A30: p in GG by A29,XBOOLE_0:def 4;
  reconsider L=EqClass(z,C '/\' D '/\' E '/\' F '/\' J '/\' M) as set;
A31: p in EqClass(p,C '/\' D '/\' E '/\' F '/\' J '/\' M) by EQREL_1:def 6;
  GG = EqClass(u,(((B '/\' (C '/\' D)) '/\' E) '/\' F) '/\' J '/\' M) by
PARTIT1:14;
  then GG = EqClass(u,((B '/\' ((C '/\' D) '/\' E)) '/\' F) '/\' J '/\' M) by
PARTIT1:14;
  then GG = EqClass(u,(B '/\' (((C '/\' D) '/\' E) '/\' F)) '/\' J '/\' M) by
PARTIT1:14;
  then GG = EqClass(u,B '/\' ((((C '/\' D) '/\' E) '/\' F) '/\' J) '/\' M) by
PARTIT1:14;
  then GG = EqClass(u,B '/\' (C '/\' D '/\' E '/\' F '/\' J '/\' M)) by
PARTIT1:14;
  then GG c= L by A6,BVFUNC11:3;
  then K meets L by A30,A31,XBOOLE_0:3;
  then K=L by EQREL_1:41;
  then
A32: z in K by EQREL_1:def 6;
A33: z in HH by EQREL_1:def 6;
  z in I by EQREL_1:def 6;
  then z in I /\ K by A32,XBOOLE_0:def 4;
  then
A34: I /\ K meets HH by A33,XBOOLE_0:3;
A35: A '/\' (C '/\' D '/\' E '/\' F '/\' J '/\' M) = A '/\' (C '/\' D '/\'
  E '/\' F '/\' J) '/\' M by PARTIT1:14
    .= A '/\' (C '/\' D '/\' E '/\' F) '/\' J '/\' M by PARTIT1:14
    .= A '/\' (C '/\' D '/\' E) '/\' F '/\' J '/\' M by PARTIT1:14
    .= A '/\' (C '/\' D) '/\' E '/\' F '/\' J '/\' M by PARTIT1:14
    .= A '/\' C '/\' D '/\' E '/\' F '/\' J '/\' M by PARTIT1:14;
  p in K & p in I by A29,EQREL_1:def 6,XBOOLE_0:def 4;
  then
A36: p in I /\ K by XBOOLE_0:def 4;
  then I /\ K in INTERSECTION(A,C '/\' D '/\' E '/\' F '/\' J '/\' M) & not I
  /\ K in {{}} by SETFAM_1:def 5,TARSKI:def 1;
  then I /\ K in INTERSECTION(A,C '/\' D '/\' E '/\' F '/\' J '/\' M) \ { {}}
  by XBOOLE_0:def 5;
  then
A37: I /\ K in A '/\' (C '/\' D '/\' E '/\' F '/\' J '/\' M) by PARTIT1:def 4;
  CompF(B,G) = A '/\' C '/\' D '/\' E '/\' F '/\' J '/\' M by A2,A3,A4,A5,Th55;
  then p in HH by A36,A37,A34,A35,EQREL_1:def 4;
  hence thesis by A8,A30,XBOOLE_0:3;
end;
