reserve Y for non empty set,
  G for Subset of PARTITIONS(Y),
  A,B,C,D,E,F for a_partition of Y;

theorem
  for G being Subset of PARTITIONS(Y), A,B,C,D,E,F being a_partition of
Y, z,u being Element of Y, 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)
proof
  let G be Subset of PARTITIONS(Y);
  let A,B,C,D,E,F be a_partition of Y;
  let z,u be Element of Y;
  let h be Function;
  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;
  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: h.A = EqClass(z,A) by A3,Th37;
  set GG=EqClass(u,B '/\' C '/\' D '/\' E '/\' F);
  GG = EqClass(u,B '/\' C '/\' D '/\' E) /\ EqClass(u,F) by Th1;
  then GG = EqClass(u,B '/\' C '/\' D) /\ EqClass(u,E) /\ EqClass(u,F) by Th1;
  then
  GG = EqClass(u,B '/\' C) /\ EqClass(u,D) /\ EqClass(u,E) /\ EqClass (u,
  F) by Th1;
  then
A5: GG /\ EqClass(z,A) = ((((EqClass(u,B) /\ EqClass(u,C)) /\ EqClass(u, D)
  ) /\ EqClass(u,E)) /\ EqClass(u,F)) /\ EqClass(z,A) by Th1;
A6: h.B = EqClass(u,B) by A3,Th37;
A7: h.D = EqClass(u,D) by A3,Th37;
A8: h.C = EqClass(u,C) by A3,Th37;
A9: h.F = EqClass(u,F) by A3,Th37;
A10: h.E = EqClass(u,E) by A3,Th37;
A11: rng h = {h.A,h.B,h.C,h.D,h.E,h.F} by Th39;
  rng h c= bool Y
  proof
    let t be object;
    assume
A12: t in rng h;
    now
      per cases by A11,A12,ENUMSET1:def 4;
      case
        t=h.A;
        hence thesis by A4;
      end;
      case
        t=h.B;
        hence thesis by A6;
      end;
      case
        t=h.C;
        hence thesis by A8;
      end;
      case
        t=h.D;
        hence thesis by A7;
      end;
      case
        t=h.E;
        hence thesis by A10;
      end;
      case
        t=h.F;
        hence thesis by A9;
      end;
    end;
    hence thesis;
  end;
  then reconsider FF=rng h as Subset-Family of Y;
A13: dom h = G by A2,Th38;
  then A in dom h by A2,ENUMSET1:def 4;
  then
A14: h.A in rng h by FUNCT_1:def 3;
  then
A15: 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
A16: d in G;
    now
      per cases by A2,A16,ENUMSET1:def 4;
      case
        d=A;
        hence thesis by A4;
      end;
      case
        d=B;
        hence thesis by A6;
      end;
      case
        d=C;
        hence thesis by A8;
      end;
      case
        d=D;
        hence thesis by A7;
      end;
      case
        d=E;
        hence thesis by A10;
      end;
      case
        d=F;
        hence thesis by A9;
      end;
    end;
    hence thesis;
  end;
  then Intersect FF <> {} by A1,A13,BVFUNC_2:def 5;
  then consider m being object such that
A17: m in Intersect FF by XBOOLE_0:def 1;
  C in dom h by A2,A13,ENUMSET1:def 4;
  then h.C in rng h by FUNCT_1:def 3;
  then
A18: m in EqClass(u,C) by A8,A15,A17,SETFAM_1:def 1;
  B in dom h by A2,A13,ENUMSET1:def 4;
  then h.B in rng h by FUNCT_1:def 3;
  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 h by A2,A13,ENUMSET1:def 4;
  then h.D in rng h by FUNCT_1:def 3;
  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 h by A2,A13,ENUMSET1:def 4;
  then h.E in rng h by FUNCT_1:def 3;
  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 h by A2,A13,ENUMSET1:def 4;
  then h.F in rng h by FUNCT_1:def 3;
  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 thesis by A5,XBOOLE_0:def 7;
end;
