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 Th67:
  for G being Subset of PARTITIONS(Y), A,B,C,D,E,F,J,M,N being
a_partition of Y st 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 CompF(A,G) = B
  '/\' C '/\' D '/\' E '/\' F '/\' J '/\' M '/\' N
proof
  let G be Subset of PARTITIONS(Y);
  let A,B,C,D,E,F,J,M,N be a_partition of Y;
  assume that
A1: G={A,B,C,D,E,F,J,M,N} and
A2: A<>B and
A3: A<>C and
A4: A<>D & A<>E and
A5: A<>F & A<>J and
A6: A<>M & A<>N and
A7: 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 and
A8: M<>N;
A9: not B in {A} by A2,TARSKI:def 1;
  ( not D in {A})& not E in {A} by A4,TARSKI:def 1;
  then
A10: {D,E} \ {A} = {D,E} by ZFMISC_1:63;
A11: ( not F in {A})& not J in {A} by A5,TARSKI:def 1;
A12: not C in {A} by A3,TARSKI:def 1;
  G \ {A}={A} \/ {B,C,D,E,F,J,M,N} \ {A} by A1,ENUMSET1:77;
  then
A13: G \ {A} = ({A} \ {A}) \/ ({B,C,D,E,F,J,M,N} \ {A}) by XBOOLE_1:42;
A14: ( not M in {A})& not N in {A} by A6,TARSKI:def 1;
  {B,C,D,E,F,J,M,N} \ {A} = ({B} \/ {C,D,E,F,J,M,N}) \ {A} by ENUMSET1:22
    .= ({B} \ {A}) \/ ({C,D,E,F,J,M,N} \ {A}) by XBOOLE_1:42
    .= {B} \/ ({C,D,E,F,J,M,N} \ {A}) by A9,ZFMISC_1:59
    .= {B} \/ (({C} \/ {D,E,F,J,M,N}) \ {A}) by ENUMSET1:16
    .= {B} \/ (({C} \ {A}) \/ ({D,E,F,J,M,N} \ {A})) by XBOOLE_1:42
    .= {B} \/ (({C} \ {A}) \/ (({D,E} \/ {F,J,M,N}) \ {A})) by ENUMSET1:12
    .= {B} \/ (({C} \ {A}) \/ (({D,E} \ {A}) \/ ({F,J,M,N} \ {A}))) by
XBOOLE_1:42
    .= {B} \/ (({C} \ {A}) \/ ({D,E} \/ ({F,J} \/ {M,N} \ {A}))) by A10,
ENUMSET1:5
    .= {B} \/ (({C} \ {A}) \/ ({D,E} \/ (({F,J} \ {A}) \/ ({M,N} \ {A}))))
  by XBOOLE_1:42
    .= {B} \/ (({C} \ {A}) \/ ({D,E} \/ ({F,J} \/ ({M,N} \ {A})))) by A11,
ZFMISC_1:63
    .= {B} \/ ({C} \/ ({D,E} \/ ({F,J} \/ ({M,N} \ {A})))) by A12,ZFMISC_1:59
    .= {B} \/ ({C} \/ ({D,E} \/ ({F,J} \/ {M,N}))) by A14,ZFMISC_1:63
    .= {B} \/ ({C} \/ ({D,E} \/ {F,J,M,N})) by ENUMSET1:5
    .= {B} \/ ({C} \/ {D,E,F,J,M,N}) by ENUMSET1:12
    .= {B} \/ {C,D,E,F,J,M,N} by ENUMSET1:16
    .= {B,C,D,E,F,J,M,N} by ENUMSET1:22;
  then
A15: G \ {A} = {} \/ {B,C,D,E,F,J,M,N} by A13,XBOOLE_1:37;
A16: '/\' (G \ {A}) c= B '/\' C '/\' D '/\' E '/\' F '/\' J '/\' M '/\' N
  proof
    let x be object;
     reconsider xx=x as set by TARSKI:1;
    assume x in '/\' (G \ {A});
    then consider h being Function, FF being Subset-Family of Y such that
A17: dom h=(G \ {A}) and
A18: rng h = FF and
A19: for d being set st d in (G \ {A}) holds h.d in d and
A20: x=Intersect FF and
A21: x<>{} by BVFUNC_2:def 1;
A22: C in (G \ {A}) by A15,ENUMSET1:def 6;
    then
A23: h.C in C by A19;
    set mbcd=(h.B /\ h.C) /\ h.D;
A24: E in (G \ {A}) by A15,ENUMSET1:def 6;
    then
A25: h.E in rng h by A17,FUNCT_1:def 3;
A26: N in (G \ {A}) by A15,ENUMSET1:def 6;
    then
A27: h.N in rng h by A17,FUNCT_1:def 3;
    set mbc=h.B /\ h.C;
A28: B in (G \ {A}) by A15,ENUMSET1:def 6;
    then h.B in B by A19;
    then
A29: mbc in INTERSECTION(B,C) by A23,SETFAM_1:def 5;
A30: h.B in rng h by A17,A28,FUNCT_1:def 3;
    then
A31: Intersect FF = meet (rng h) by A18,SETFAM_1:def 9;
A32: h.C in rng h by A17,A22,FUNCT_1:def 3;
A33: F in (G \ {A}) by A15,ENUMSET1:def 6;
    then
A34: h.F in rng h by A17,FUNCT_1:def 3;
A35: M in (G \ {A}) by A15,ENUMSET1:def 6;
    then
A36: h.M in rng h by A17,FUNCT_1:def 3;
A37: J in (G \ {A}) by A15,ENUMSET1:def 6;
    then
A38: h.J in rng h by A17,FUNCT_1:def 3;
A39: D in (G \ {A}) by A15,ENUMSET1:def 6;
    then
A40: h.D in rng h by A17,FUNCT_1:def 3;
A41: xx c= ((((h.B /\ h.C) /\ h.D) /\ h.E) /\ h.F) /\ h.J /\ h.M /\ h.N
    proof
      let p be object;
      assume
A42:  p in xx;
      then p in h.B & p in h.C by A20,A30,A32,A31,SETFAM_1:def 1;
      then
A43:  p in h.B /\ h.C by XBOOLE_0:def 4;
      p in h.D by A20,A40,A31,A42,SETFAM_1:def 1;
      then
A44:  p in h.B /\ h.C /\ h.D by A43,XBOOLE_0:def 4;
      p in h.E by A20,A25,A31,A42,SETFAM_1:def 1;
      then
A45:  p in h.B /\ h.C /\ h.D /\ h.E by A44,XBOOLE_0:def 4;
      p in h.F by A20,A34,A31,A42,SETFAM_1:def 1;
      then
A46:  p in h.B /\ h.C /\ h.D /\ h.E /\ h.F by A45,XBOOLE_0:def 4;
      p in h.J by A20,A38,A31,A42,SETFAM_1:def 1;
      then
A47:  p in ((((h.B /\ h.C) /\ h.D) /\ h.E) /\ h.F) /\ h.J by A46,XBOOLE_0:def 4
;
      p in h.M by A20,A36,A31,A42,SETFAM_1:def 1;
      then
A48:  p in ((((h.B /\ h.C) /\ h.D) /\ h.E) /\ h.F) /\ h.J /\ h.M by A47,
XBOOLE_0:def 4;
      p in h.N by A20,A27,A31,A42,SETFAM_1:def 1;
      hence thesis by A48,XBOOLE_0:def 4;
    end;
    then mbcd<>{} by A21;
    then
A49: not mbcd in {{}} by TARSKI:def 1;
    mbc<>{} by A21,A41;
    then not mbc in {{}} by TARSKI:def 1;
    then mbc in INTERSECTION(B,C) \ {{}} by A29,XBOOLE_0:def 5;
    then
A50: mbc in B '/\' C by PARTIT1:def 4;
    h.D in D by A19,A39;
    then mbcd in INTERSECTION(B '/\' C,D) by A50,SETFAM_1:def 5;
    then mbcd in INTERSECTION(B '/\' C,D) \ {{}} by A49,XBOOLE_0:def 5;
    then
A51: mbcd in B '/\' C '/\' D by PARTIT1:def 4;
    set mbcdefjm=((h.B /\ h.C) /\ h.D) /\ h.E /\ h.F /\ h.J /\ h.M;
    set mbcdefj=((h.B /\ h.C) /\ h.D) /\ h.E /\ h.F /\ h.J;
A52: not x in {{}} by A21,TARSKI:def 1;
    set mbcdef=((h.B /\ h.C) /\ h.D) /\ h.E /\ h.F;
    set mbcde=(h.B /\ h.C) /\ h.D /\ h.E;
    mbcdef<>{} by A21,A41;
    then
A53: not mbcdef in {{}} by TARSKI:def 1;
    mbcde<>{} by A21,A41;
    then
A54: not mbcde in {{}} by TARSKI:def 1;
    ((((h.B /\ h.C) /\ h.D) /\ h.E) /\ h.F) /\ h.J /\ h.M /\ h.N c= xx
    proof
A55:  rng h c= {h.B,h.C,h.D,h.E,h.F,h.J,h.M,h.N}
      proof
        let u be object;
        assume u in rng h;
        then consider x1 being object such that
A56:    x1 in dom h and
A57:    u = h.x1 by FUNCT_1:def 3;
        x1=B or x1=C or x1=D or x1=E or x1=F or x1=J or x1=M or x1=N by A15,A17
,A56,ENUMSET1:def 6;
        hence thesis by A57,ENUMSET1:def 6;
      end;
      let p be object;
      assume
A58:  p in ((((h.B /\ h.C) /\ h.D) /\ h.E) /\ h.F) /\ h.J /\ h.M /\ h.N;
      then
A59:  p in h.N by XBOOLE_0:def 4;
A60:  p in ((((h.B /\ h.C) /\ h.D) /\ h.E) /\ h.F) /\ h.J /\ h.M by A58,
XBOOLE_0:def 4;
      then
A61:  p in ((((h.B /\ h.C) /\ h.D) /\ h.E) /\ h.F) /\ h.J by XBOOLE_0:def 4;
      then
A62:  p in h.J by XBOOLE_0:def 4;
A63:  p in h.M by A60,XBOOLE_0:def 4;
A64:  p in h.B /\ h.C /\ h.D /\ h.E /\ h.F by A61,XBOOLE_0:def 4;
      then
A65:  p in h.B /\ h.C /\ h.D /\ h.E by XBOOLE_0:def 4;
      then
A66:  p in h.E by XBOOLE_0:def 4;
A67:  p in h.B /\ h.C /\ h.D by A65,XBOOLE_0:def 4;
      then
A68:  p in h.D by XBOOLE_0:def 4;
      p in h.B /\ h.C by A67,XBOOLE_0:def 4;
      then
A69:  p in h.B & p in h.C by XBOOLE_0:def 4;
      p in h.F by A64,XBOOLE_0:def 4;
      then for y being set holds y in rng h implies p in y by A69,A68,A66,A62
,A63,A59,A55,ENUMSET1:def 6;
      hence thesis by A20,A30,A31,SETFAM_1:def 1;
    end;
    then
A70: ((((h.B /\ h.C) /\ h.D) /\ h.E) /\ h.F) /\ h.J /\ h.M /\ h.N = x by A41,
XBOOLE_0:def 10;
    h.E in E by A19,A24;
    then mbcde in INTERSECTION(B '/\' C '/\' D,E) by A51,SETFAM_1:def 5;
    then mbcde in INTERSECTION(B '/\' C '/\' D,E) \ {{}} by A54,XBOOLE_0:def 5;
    then
A71: mbcde in (B '/\' C '/\' D '/\' E) by PARTIT1:def 4;
    h.F in F by A19,A33;
    then mbcdef in INTERSECTION(B '/\' C '/\' D '/\' E,F) by A71,SETFAM_1:def 5
;
    then mbcdef in INTERSECTION(B '/\' C '/\' D '/\' E,F) \ {{}} by A53,
XBOOLE_0:def 5;
    then
A72: mbcdef in (B '/\' C '/\' D '/\' E '/\' F) by PARTIT1:def 4;
    mbcdefj<>{} by A21,A41;
    then
A73: not mbcdefj in {{}} by TARSKI:def 1;
    h.J in J by A19,A37;
    then mbcdefj in INTERSECTION(B '/\' C '/\' D '/\' E '/\' F,J) by A72,
SETFAM_1:def 5;
    then mbcdefj in INTERSECTION(B '/\' C '/\' D '/\' E '/\' F,J) \ {{}} by A73
,XBOOLE_0:def 5;
    then
A74: mbcdefj in (B '/\' C '/\' D '/\' E '/\' F '/\' J) by PARTIT1:def 4;
    mbcdefjm<>{} by A21,A41;
    then
A75: not mbcdefjm in {{}} by TARSKI:def 1;
    h.M in M by A19,A35;
    then mbcdefjm in INTERSECTION(B '/\' C '/\' D '/\' E '/\' F '/\' J,M) by
A74,SETFAM_1:def 5;
    then mbcdefjm in INTERSECTION(B '/\' C '/\' D '/\' E '/\' F '/\' J,M) \ {
    {}} by A75,XBOOLE_0:def 5;
    then
A76: mbcdefjm in (B '/\' C '/\' D '/\' E '/\' F '/\' J '/\' M) by PARTIT1:def 4
;
    h.N in N by A19,A26;
    then x in INTERSECTION(B '/\' C '/\' D '/\' E '/\' F '/\' J '/\' M,N) by
A70,A76,SETFAM_1:def 5;
    then x in INTERSECTION(B '/\' C '/\' D '/\' E '/\' F '/\' J '/\' M,N) \ {
    {}} by A52,XBOOLE_0:def 5;
    hence thesis by PARTIT1:def 4;
  end;
A77: B '/\' C '/\' D '/\' E '/\' F '/\' J '/\' M '/\' N c= '/\' (G \ {A})
  proof
    let x be object;
     reconsider xx=x as set by TARSKI:1;
    assume
A78: x in B '/\' C '/\' D '/\' E '/\' F '/\' J '/\' M '/\' N;
    then
A79: x<>{} by EQREL_1:def 4;
    x in INTERSECTION(B '/\' C '/\' D '/\' E '/\' F '/\' J '/\' M,N) \ {
    {}} by A78,PARTIT1:def 4;
    then consider bcdefjm,n being set such that
A80: bcdefjm in B '/\' C '/\' D '/\' E '/\' F '/\' J '/\' M and
A81: n in N and
A82: x = bcdefjm /\ n by SETFAM_1:def 5;
    bcdefjm in INTERSECTION(B '/\' C '/\' D '/\' E '/\' F '/\' J,M) \ {{}
    } by A80,PARTIT1:def 4;
    then consider bcdefj,m being set such that
A83: bcdefj in B '/\' C '/\' D '/\' E '/\' F '/\' J and
A84: m in M and
A85: bcdefjm = bcdefj /\ m by SETFAM_1:def 5;
    bcdefj in INTERSECTION(B '/\' C '/\' D '/\' E '/\' F,J) \ {{}} by A83,
PARTIT1:def 4;
    then consider bcdef,j being set such that
A86: bcdef in B '/\' C '/\' D '/\' E '/\' F and
A87: j in J and
A88: bcdefj = bcdef /\ j by SETFAM_1:def 5;
    bcdef in INTERSECTION(B '/\' C '/\' D '/\' E,F) \ {{}} by A86,PARTIT1:def 4
;
    then consider bcde,f being set such that
A89: bcde in B '/\' C '/\' D '/\' E and
A90: f in F and
A91: bcdef = bcde /\ f by SETFAM_1:def 5;
    bcde in INTERSECTION(B '/\' C '/\' D,E) \ {{}} by A89,PARTIT1:def 4;
    then consider bcd,e being set such that
A92: bcd in B '/\' C '/\' D and
A93: e in E and
A94: bcde = bcd /\ e by SETFAM_1:def 5;
    bcd in INTERSECTION(B '/\' C,D) \ {{}} by A92,PARTIT1:def 4;
    then consider bc,d being set such that
A95: bc in B '/\' C and
A96: d in D and
A97: bcd = bc /\ d by SETFAM_1:def 5;
    bc in INTERSECTION(B,C) \ {{}} by A95,PARTIT1:def 4;
    then consider b,c being set such that
A98: b in B and
A99: c in C and
A100: bc = b /\ c by SETFAM_1:def 5;
    set h = (B .--> b) +* (C .--> c) +* (D .--> d) +* (E .--> e) +* (F .--> f)
    +* (J .--> j) +* (M .--> m) +* (N .--> n);
A101: h.N = n by FUNCT_7:94;
A102: dom ((B .--> b) +* (C .--> c) +* (D .--> d) +* (E .--> e) +* (F .-->
    f) +* (J .--> j) +* (M .--> m) +* (N .--> n)) = {N,B,C,D,E,F,J,M} by Th63
      .= {N} \/ {B,C,D,E,F,J,M} by ENUMSET1:22
      .= {B,C,D,E,F,J,M,N} by ENUMSET1:28;
    then
A103: C in dom h by ENUMSET1:def 6;
A104: for p being set st p in (G \ {A}) holds h.p in p
    proof
      let p be set;
      assume p in (G \ {A});
      then p=B or p=C or p=D or p=E or p=F or p=J or p=M or p=N by A15,
ENUMSET1:def 6;
      hence thesis by A7,A8,A81,A84,A87,A90,A93,A96,A98,A99,Lm1,Th62,FUNCT_7:94
;
    end;
A105: D in dom h by A102,ENUMSET1:def 6;
    then
A106: h.D in rng h by FUNCT_1:def 3;
A107: N in dom h by A102,ENUMSET1:def 6;
A108: M in dom h by A102,ENUMSET1:def 6;
A109: J in dom h by A102,ENUMSET1:def 6;
A110: F in dom h by A102,ENUMSET1:def 6;
A111: h.B = b by A7,Th62;
A112: rng h c= {h.B,h.C,h.D,h.E,h.F,h.J,h.M,h.N}
    proof
      let t be object;
      assume t in rng h;
      then consider x1 being object such that
A113: x1 in dom h and
A114: t = h.x1 by FUNCT_1:def 3;
      now
        per cases by A102,A113,ENUMSET1:def 6;
        case
          x1=D;
          hence thesis by A114,ENUMSET1:def 6;
        end;
        case
          x1=B;
          hence thesis by A114,ENUMSET1:def 6;
        end;
        case
          x1=C;
          hence thesis by A114,ENUMSET1:def 6;
        end;
        case
          x1=E;
          hence thesis by A114,ENUMSET1:def 6;
        end;
        case
          x1=F;
          hence thesis by A114,ENUMSET1:def 6;
        end;
        case
          x1=J;
          hence thesis by A114,ENUMSET1:def 6;
        end;
        case
          x1=M;
          hence thesis by A114,ENUMSET1:def 6;
        end;
        case
          x1=N;
          hence thesis by A114,ENUMSET1:def 6;
        end;
      end;
      hence thesis;
    end;
A115: h.J = j by A7,Th62;
A116: h.F = f by A7,Th62;
A117: h.M = m by A8,Lm1;
A118: h.E = e by A7,Th62;
A119: h.C = c by A7,Th62;
A120: h.D = d by A7,Th62;
    rng h c= bool Y
    proof
      let t be object;
      assume
A121: t in rng h;
      now
        per cases by A112,A121,ENUMSET1:def 6;
        case
          t=h.D;
          hence thesis by A96,A120;
        end;
        case
          t=h.B;
          hence thesis by A98,A111;
        end;
        case
          t=h.C;
          hence thesis by A99,A119;
        end;
        case
          t=h.E;
          hence thesis by A93,A118;
        end;
        case
          t=h.F;
          hence thesis by A90,A116;
        end;
        case
          t=h.J;
          hence thesis by A87,A115;
        end;
        case
          t=h.M;
          hence thesis by A84,A117;
        end;
        case
          t=h.N;
          hence thesis by A81,A101;
        end;
      end;
      hence thesis;
    end;
    then reconsider FF=rng h as Subset-Family of Y;
A122: E in dom h by A102,ENUMSET1:def 6;
A123: B in dom h by A102,ENUMSET1:def 6;
    {h.B,h.C,h.D,h.E,h.F,h.J,h.M,h.N} c= rng h
    proof
      let t be object;
      assume
A124: t in {h.B,h.C,h.D,h.E,h.F,h.J,h.M,h.N};
      now
        per cases by A124,ENUMSET1:def 6;
        case
          t=h.D;
          hence thesis by A105,FUNCT_1:def 3;
        end;
        case
          t=h.B;
          hence thesis by A123,FUNCT_1:def 3;
        end;
        case
          t=h.C;
          hence thesis by A103,FUNCT_1:def 3;
        end;
        case
          t=h.E;
          hence thesis by A122,FUNCT_1:def 3;
        end;
        case
          t=h.F;
          hence thesis by A110,FUNCT_1:def 3;
        end;
        case
          t=h.J;
          hence thesis by A109,FUNCT_1:def 3;
        end;
        case
          t=h.M;
          hence thesis by A108,FUNCT_1:def 3;
        end;
        case
          t=h.N;
          hence thesis by A107,FUNCT_1:def 3;
        end;
      end;
      hence thesis;
    end;
    then
A125: rng h = {h.B,h.C,h.D,h.E,h.F,h.J,h.M,h.N} by A112,XBOOLE_0:def 10;
    reconsider h as Function;
A126: xx c= Intersect FF
    proof
      let u be object;
      assume
A127: u in xx;
      for y be set holds y in FF implies u in y
      proof
        let y be set;
        assume
A128:   y in FF;
        now
          per cases by A112,A128,ENUMSET1:def 6;
          case
A129:       y=h.D;
            u in (d /\ ((b /\ c) /\ e)) /\ f /\ j /\ m /\ n by A82,A85,A88,A91
,A94,A97,A100,A127,XBOOLE_1:16;
            then u in (d /\ ((b /\ c) /\ e /\ f)) /\ j /\ m /\ n by XBOOLE_1:16
;
            then u in d /\ (((b /\ c) /\ e /\ f) /\ j) /\ m /\ n by XBOOLE_1:16
;
            then u in d /\ ((((b /\ c) /\ e /\ f) /\ j) /\ m) /\ n by
XBOOLE_1:16;
            then u in d /\ ((((b /\ c) /\ e /\ f) /\ j) /\ m /\ n) by
XBOOLE_1:16;
            hence thesis by A120,A129,XBOOLE_0:def 4;
          end;
          case
A130:       y=h.B;
            u in (c /\ (d /\ b)) /\ e /\ f /\ j /\ m /\ n by A82,A85,A88,A91
,A94,A97,A100,A127,XBOOLE_1:16;
            then u in c /\ ((d /\ b) /\ e) /\ f /\ j /\ m /\ n by XBOOLE_1:16;
            then u in c /\ ((d /\ e) /\ b) /\ f /\ j /\ m /\ n by XBOOLE_1:16;
            then u in c /\ (((d /\ e) /\ b) /\ f) /\ j /\ m /\ n by XBOOLE_1:16
;
            then u in c /\ ((((d /\ e) /\ b) /\ f) /\ j) /\ m /\ n by
XBOOLE_1:16;
            then u in c /\ (((d /\ e) /\ (f /\ b)) /\ j) /\ m /\ n by
XBOOLE_1:16;
            then u in c /\ ((d /\ e) /\ ((f /\ b) /\ j)) /\ m /\ n by
XBOOLE_1:16;
            then u in c /\ ((d /\ e) /\ (f /\ (j /\ b))) /\ m /\ n by
XBOOLE_1:16;
            then u in (c /\ (d /\ e)) /\ (f /\ (j /\ b)) /\ m /\ n by
XBOOLE_1:16;
            then u in ((c /\ (d /\ e)) /\ f) /\ (j /\ b) /\ m /\ n by
XBOOLE_1:16;
            then u in (((c /\ (d /\ e)) /\ f) /\ j) /\ b /\ m /\ n by
XBOOLE_1:16;
            then u in (((c /\ (d /\ e)) /\ f) /\ j) /\ (m /\ b) /\ n by
XBOOLE_1:16;
            then u in (((c /\ (d /\ e)) /\ f) /\ j) /\ (m /\ b /\ n) by
XBOOLE_1:16;
            then u in (((c /\ (d /\ e)) /\ f) /\ j) /\ (m /\ (b /\ n)) by
XBOOLE_1:16;
            then u in (((c /\ (d /\ e)) /\ f) /\ j) /\ m /\ (n /\ b) by
XBOOLE_1:16;
            then u in (((c /\ (d /\ e)) /\ f) /\ j) /\ m /\ n /\ b by
XBOOLE_1:16;
            hence thesis by A111,A130,XBOOLE_0:def 4;
          end;
          case
A131:       y=h.C;
            u in (c /\ (d /\ b)) /\ e /\ f /\ j /\ m /\ n by A82,A85,A88,A91
,A94,A97,A100,A127,XBOOLE_1:16;
            then u in c /\ ((d /\ b) /\ e) /\ f /\ j /\ m /\ n by XBOOLE_1:16;
            then u in c /\ ((d /\ e) /\ b) /\ f /\ j /\ m /\ n by XBOOLE_1:16;
            then u in c /\ (((d /\ e) /\ b) /\ f) /\ j /\ m /\ n by XBOOLE_1:16
;
            then u in c /\ ((((d /\ e) /\ b) /\ f) /\ j) /\ m /\ n by
XBOOLE_1:16;
            then u in c /\ (((((d /\ e) /\ b) /\ f) /\ j) /\ m) /\ n by
XBOOLE_1:16;
            then u in c /\ ((((((d /\ e) /\ b) /\ f) /\ j) /\ m) /\ n) by
XBOOLE_1:16;
            hence thesis by A119,A131,XBOOLE_0:def 4;
          end;
          case
A132:       y=h.E;
            u in ((b /\ c) /\ d) /\ (f /\ e) /\ j /\ m /\ n by A82,A85,A88,A91
,A94,A97,A100,A127,XBOOLE_1:16;
            then u in ((b /\ c) /\ d) /\ ((f /\ e) /\ j) /\ m /\ n by
XBOOLE_1:16;
            then u in ((b /\ c) /\ d) /\ ((f /\ j) /\ e) /\ m /\ n by
XBOOLE_1:16;
            then u in (((b /\ c) /\ d) /\ (f /\ j)) /\ e /\ m /\ n by
XBOOLE_1:16;
            then u in (((b /\ c) /\ d) /\ (f /\ j)) /\ (e /\ m) /\ n by
XBOOLE_1:16;
            then u in (((b /\ c) /\ d) /\ (f /\ j)) /\ ((m /\ e) /\ n) by
XBOOLE_1:16;
            then u in (((b /\ c) /\ d) /\ (f /\ j)) /\ (m /\ (n /\ e)) by
XBOOLE_1:16;
            then u in (((b /\ c) /\ d) /\ (f /\ j)) /\ m /\ (n /\ e) by
XBOOLE_1:16;
            then u in (((b /\ c) /\ d) /\ (f /\ j)) /\ m /\ n /\ e by
XBOOLE_1:16;
            hence thesis by A118,A132,XBOOLE_0:def 4;
          end;
          case
A133:       y=h.F;
            u in (((b /\ c) /\ d) /\ e) /\ j /\ f /\ m /\ n by A82,A85,A88,A91
,A94,A97,A100,A127,XBOOLE_1:16;
            then u in (((b /\ c) /\ d) /\ e) /\ j /\ m /\ f /\ n by XBOOLE_1:16
;
            then u in (((b /\ c) /\ d) /\ e) /\ j /\ m /\ n /\ f by XBOOLE_1:16
;
            hence thesis by A116,A133,XBOOLE_0:def 4;
          end;
          case
A134:       y=h.J;
            u in (((b /\ c) /\ d) /\ e) /\ f /\ m /\ j /\ n by A82,A85,A88,A91
,A94,A97,A100,A127,XBOOLE_1:16;
            then u in (((b /\ c) /\ d) /\ e) /\ f /\ m /\ n /\ j by XBOOLE_1:16
;
            hence thesis by A115,A134,XBOOLE_0:def 4;
          end;
          case
A135:       y=h.M;
            u in (((b /\ c) /\ d) /\ e) /\ f /\ j /\ n /\ m by A82,A85,A88,A91
,A94,A97,A100,A127,XBOOLE_1:16;
            hence thesis by A117,A135,XBOOLE_0:def 4;
          end;
          case
            y=h.N;
            hence thesis by A82,A101,A127,XBOOLE_0:def 4;
          end;
        end;
        hence thesis;
      end;
      then u in meet FF by A125,SETFAM_1:def 1;
      hence thesis by A125,SETFAM_1:def 9;
    end;
A136: Intersect FF = meet (rng h) by A106,SETFAM_1:def 9;
    Intersect FF c= xx
    proof
      let t be object;
      assume
A137: t in Intersect FF;
      h.C in rng h by A125,ENUMSET1:def 6;
      then
A138: t in c by A119,A136,A137,SETFAM_1:def 1;
      h.B in rng h by A125,ENUMSET1:def 6;
      then t in b by A111,A136,A137,SETFAM_1:def 1;
      then
A139: t in b /\ c by A138,XBOOLE_0:def 4;
      h.D in rng h by A125,ENUMSET1:def 6;
      then t in d by A120,A136,A137,SETFAM_1:def 1;
      then
A140: t in (b /\ c) /\ d by A139,XBOOLE_0:def 4;
      h.E in rng h by A125,ENUMSET1:def 6;
      then t in e by A118,A136,A137,SETFAM_1:def 1;
      then
A141: t in (b /\ c) /\ d /\ e by A140,XBOOLE_0:def 4;
      h.F in rng h by A125,ENUMSET1:def 6;
      then t in f by A116,A136,A137,SETFAM_1:def 1;
      then
A142: t in (b /\ c) /\ d /\ e /\ f by A141,XBOOLE_0:def 4;
      h.J in rng h by A125,ENUMSET1:def 6;
      then t in j by A115,A136,A137,SETFAM_1:def 1;
      then
A143: t in (b /\ c) /\ d /\ e /\ f /\ j by A142,XBOOLE_0:def 4;
      h.M in rng h by A125,ENUMSET1:def 6;
      then t in m by A117,A136,A137,SETFAM_1:def 1;
      then
A144: t in (b /\ c) /\ d /\ e /\ f /\ j /\ m by A143,XBOOLE_0:def 4;
      h.N in rng h by A125,ENUMSET1:def 6;
      then t in n by A101,A136,A137,SETFAM_1:def 1;
      hence thesis by A82,A85,A88,A91,A94,A97,A100,A144,XBOOLE_0:def 4;
    end;
    then x = Intersect FF by A126,XBOOLE_0:def 10;
    hence thesis by A15,A102,A104,A79,BVFUNC_2:def 1;
  end;
  CompF(A,G)='/\' (G \ {A}) by BVFUNC_2:def 7;
  hence thesis by A77,A16,XBOOLE_0:def 10;
end;
