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

theorem Th31:
  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 implies CompF(A,G) = B
  '/\' C '/\' D '/\' E '/\' F
proof
  assume that
A1: G={A,B,C,D,E,F} and
A2: A<>B and
A3: A<>C and
A4: A<>D & A<>E and
A5: A<>F and
A6: B<>C & B<>D & B<>E & B<>F & C<>D & C<>E & C<>F & D<>E & D<>F & E<>F;
A7: G \ {A}={A} \/ {B,C,D,E,F} \ {A} by A1,ENUMSET1:11
    .= ({A} \ {A}) \/ ({B,C,D,E,F} \ {A}) by XBOOLE_1:42;
A8: not F in {A} by A5,TARSKI:def 1;
A9: ( not D in {A})& not E in {A} by A4,TARSKI:def 1;
A10: not C in {A} by A3,TARSKI:def 1;
A11: not B in {A} by A2,TARSKI:def 1;
  A in {A} by TARSKI:def 1;
  then
A12: {A} \ {A}={} by ZFMISC_1:60;
A13: {B,C,D,E,F} \ {A} = ({B} \/ {C,D,E,F}) \ {A} by ENUMSET1:7
    .= ({B} \ {A}) \/ ({C,D,E,F} \ {A}) by XBOOLE_1:42
    .= {B} \/ ({C,D,E,F} \ {A}) by A11,ZFMISC_1:59
    .= {B} \/ (({C} \/ {D,E,F}) \ {A}) by ENUMSET1:4
    .= {B} \/ (({C} \ {A}) \/ ({D,E,F} \ {A})) by XBOOLE_1:42
    .= {B} \/ (({C} \ {A}) \/ (({D,E} \/ {F}) \ {A})) by ENUMSET1:3
    .= {B} \/ (({C} \ {A}) \/ (({D,E} \ {A}) \/ ({F} \ {A}))) by XBOOLE_1:42
    .= {B} \/ (({C} \ {A}) \/ ({D,E} \/ ({F} \ {A}))) by A9,ZFMISC_1:63
    .= {B} \/ (({C} \ {A}) \/ ({D,E} \/ {F})) by A8,ZFMISC_1:59
    .= {B} \/ ({C} \/ ({D,E} \/ {F})) by A10,ZFMISC_1:59
    .= {B} \/ ({C} \/ {D,E,F}) by ENUMSET1:3
    .= {B} \/ {C,D,E,F} by ENUMSET1:4
    .= {B,C,D,E,F} by ENUMSET1:7;
A14: B '/\' C '/\' D '/\' E '/\' F c= '/\' (G \ {A})
  proof
    let x be object;
     reconsider xx=x as set by TARSKI:1;
    assume
A15: x in B '/\' C '/\' D '/\' E '/\' F;
    then
A16: x<>{} by EQREL_1:def 4;
    x in INTERSECTION(B '/\' C '/\' D '/\' E,F) \ {{}} by A15,PARTIT1:def 4;
    then consider bcde,f being set such that
A17: bcde in B '/\' C '/\' D '/\' E and
A18: f in F and
A19: x = bcde /\ f by SETFAM_1:def 5;
    bcde in INTERSECTION(B '/\' C '/\' D,E) \ {{}} by A17,PARTIT1:def 4;
    then consider bcd,e being set such that
A20: bcd in B '/\' C '/\' D and
A21: e in E and
A22: bcde = bcd /\ e by SETFAM_1:def 5;
    bcd in INTERSECTION(B '/\' C,D) \ {{}} by A20,PARTIT1:def 4;
    then consider bc,d being set such that
A23: bc in B '/\' C and
A24: d in D and
A25: bcd = bc /\ d by SETFAM_1:def 5;
    bc in INTERSECTION(B,C) \ {{}} by A23,PARTIT1:def 4;
    then consider b,c being set such that
A26: b in B and
A27: c in C and
A28: bc = b /\ c by SETFAM_1:def 5;
    set h = (B .--> b) +* (C .--> c) +* (D .--> d) +* (E .--> e) +* (F .--> f);
A29: h.B = b by A6,Th26;
A30: h.E = e by A6,Th26;
A31: h.F = f by A6,Th26;
A32: dom ((B .--> b) +* (C .--> c) +* (D .--> d) +* (E .--> e) +* (F .-->
    f)) = {F,B,C,D,E} by Th27
      .= {F} \/ {B,C,D,E} by ENUMSET1:7
      .= {B,C,D,E,F} by ENUMSET1:10;
    then
A33: C in dom h by ENUMSET1:def 3;
A34: F in dom h by A32,ENUMSET1:def 3;
A35: E in dom h by A32,ENUMSET1:def 3;
A36: h.C = c by A6,Th26;
A37: rng h c= {h.D,h.B,h.C,h.E,h.F}
    proof
      let t be object;
      assume t in rng h;
      then consider x1 being object such that
A38:  x1 in dom h and
A39:  t = h.x1 by FUNCT_1:def 3;
      now
        per cases by A32,A38,ENUMSET1:def 3;
        case
          x1=D;
          hence thesis by A39,ENUMSET1:def 3;
        end;
        case
          x1=B;
          hence thesis by A39,ENUMSET1:def 3;
        end;
        case
          x1=C;
          hence thesis by A39,ENUMSET1:def 3;
        end;
        case
          x1=E;
          hence thesis by A39,ENUMSET1:def 3;
        end;
        case
          x1=F;
          hence thesis by A39,ENUMSET1:def 3;
        end;
      end;
      hence thesis;
    end;
A40: h.D = d by A6,Th26;
    rng h c= bool Y
    proof
      let t be object;
      assume
A41:  t in rng h;
      now
        per cases by A37,A41,ENUMSET1:def 3;
        case
          t=h.D;
          hence thesis by A24,A40;
        end;
        case
          t=h.B;
          hence thesis by A26,A29;
        end;
        case
          t=h.C;
          hence thesis by A27,A36;
        end;
        case
          t=h.E;
          hence thesis by A21,A30;
        end;
        case
          t=h.F;
          hence thesis by A18,A31;
        end;
      end;
      hence thesis;
    end;
    then reconsider FF=rng h as Subset-Family of Y;
A42: D in dom h by A32,ENUMSET1:def 3;
    then h.D in rng h by FUNCT_1:def 3;
    then
A43: Intersect FF = meet (rng h) by SETFAM_1:def 9;
A44: B in dom h by A32,ENUMSET1:def 3;
    {h.D,h.B,h.C,h.E,h.F} c= rng h
    proof
      let t be object;
      assume
A45:  t in {h.D,h.B,h.C,h.E,h.F};
      now
        per cases by A45,ENUMSET1:def 3;
        case
          t=h.D;
          hence thesis by A42,FUNCT_1:def 3;
        end;
        case
          t=h.B;
          hence thesis by A44,FUNCT_1:def 3;
        end;
        case
          t=h.C;
          hence thesis by A33,FUNCT_1:def 3;
        end;
        case
          t=h.E;
          hence thesis by A35,FUNCT_1:def 3;
        end;
        case
          t=h.F;
          hence thesis by A34,FUNCT_1:def 3;
        end;
      end;
      hence thesis;
    end;
    then
A46: rng h = {h.D,h.B,h.C,h.E,h.F} by A37,XBOOLE_0:def 10;
A47: xx c= Intersect FF
    proof
      let u be object;
      assume
A48:  u in xx;
      for y be set holds y in FF implies u in y
      proof
        let y be set;
        assume
A49:    y in FF;
        now
          per cases by A37,A49,ENUMSET1:def 3;
          case
A50:        y=h.D;
            u in (d /\ ((b /\ c) /\ e)) /\ f by A19,A22,A25,A28,A48,XBOOLE_1:16
;
            then
A51:        u in d /\ ((b /\ c) /\ e /\ f) by XBOOLE_1:16;
            y=d by A6,A50,Th26;
            hence thesis by A51,XBOOLE_0:def 4;
          end;
          case
A52:        y=h.B;
            u in (c /\ (d /\ b)) /\ e /\ f by A19,A22,A25,A28,A48,XBOOLE_1:16;
            then u in c /\ ((d /\ b) /\ e) /\ f by XBOOLE_1:16;
            then u in c /\ ((d /\ e) /\ b) /\ f by XBOOLE_1:16;
            then u in c /\ (((d /\ e) /\ b) /\ f) by XBOOLE_1:16;
            then u in c /\ ((d /\ e) /\ (f /\ b)) by XBOOLE_1:16;
            then u in (c /\ (d /\ e)) /\ (f /\ b) by XBOOLE_1:16;
            then
A53:        u in ((c /\ (d /\ e)) /\ f) /\ b by XBOOLE_1:16;
            y=b by A6,A52,Th26;
            hence thesis by A53,XBOOLE_0:def 4;
          end;
          case
A54:        y=h.C;
            u in ((c /\ (b /\ d)) /\ e) /\ f by A19,A22,A25,A28,A48,XBOOLE_1:16
;
            then u in (c /\ ((b /\ d) /\ e)) /\ f by XBOOLE_1:16;
            then
A55:        u in c /\ (((b /\ d) /\ e) /\ f) by XBOOLE_1:16;
            y=c by A6,A54,Th26;
            hence thesis by A55,XBOOLE_0:def 4;
          end;
          case
            y=h.E;
            then
A56:        y=e by A6,Th26;
            u in (((b /\ c) /\ d) /\ f) /\ e by A19,A22,A25,A28,A48,XBOOLE_1:16
;
            hence thesis by A56,XBOOLE_0:def 4;
          end;
          case
            y=h.F;
            hence thesis by A19,A31,A48,XBOOLE_0:def 4;
          end;
        end;
        hence thesis;
      end;
      then u in meet FF by A46,SETFAM_1:def 1;
      hence thesis by A46,SETFAM_1:def 9;
    end;
A57: for p being set st p in (G \ {A}) holds h.p in p
    proof
      let p be set;
      assume
A58:  p in (G \ {A});
      now
        per cases by A7,A12,A58,ENUMSET1:def 3;
        case
          p=D;
          hence thesis by A6,A24,Th26;
        end;
        case
          p=B;
          hence thesis by A6,A26,Th26;
        end;
        case
          p=C;
          hence thesis by A6,A27,Th26;
        end;
        case
          p=E;
          hence thesis by A6,A21,Th26;
        end;
        case
          p=F;
          hence thesis by A6,A18,Th26;
        end;
      end;
      hence thesis;
    end;
    Intersect FF c= xx
    proof
      let t be object;
      assume
A59:  t in Intersect FF;
      h.C in rng h by A46,ENUMSET1:def 3;
      then
A60:  t in c by A36,A43,A59,SETFAM_1:def 1;
      h.B in rng h by A46,ENUMSET1:def 3;
      then t in b by A29,A43,A59,SETFAM_1:def 1;
      then
A61:  t in b /\ c by A60,XBOOLE_0:def 4;
      h.D in rng h by A46,ENUMSET1:def 3;
      then t in d by A40,A43,A59,SETFAM_1:def 1;
      then
A62:  t in (b /\ c) /\ d by A61,XBOOLE_0:def 4;
      h.E in rng h by A46,ENUMSET1:def 3;
      then t in e by A30,A43,A59,SETFAM_1:def 1;
      then
A63:  t in (b /\ c) /\ d /\ e by A62,XBOOLE_0:def 4;
      h.F in rng h by A46,ENUMSET1:def 3;
      then t in f by A31,A43,A59,SETFAM_1:def 1;
      hence thesis by A19,A22,A25,A28,A63,XBOOLE_0:def 4;
    end;
    then x = Intersect FF by A47,XBOOLE_0:def 10;
    hence thesis by A7,A13,A12,A32,A57,A16,BVFUNC_2:def 1;
  end;
A64: '/\' (G \ {A}) c= B '/\' C '/\' D '/\' E '/\' F
  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
A65: dom h=(G \ {A}) and
A66: rng h = FF and
A67: for d being set st d in (G \ {A}) holds h.d in d and
A68: x=Intersect FF and
A69: x<>{} by BVFUNC_2:def 1;
A70: C in (G \ {A}) by A7,A13,A12,ENUMSET1:def 3;
    then
A71: h.C in C by A67;
    set mbc=h.B /\ h.C;
A72: B in (G \ {A}) by A7,A13,A12,ENUMSET1:def 3;
    then h.B in B by A67;
    then
A73: mbc in INTERSECTION(B,C) by A71,SETFAM_1:def 5;
    set mbcd=(h.B /\ h.C) /\ h.D;
A74: E in (G \ {A}) by A7,A13,A12,ENUMSET1:def 3;
    then
A75: h.E in rng h by A65,FUNCT_1:def 3;
A76: h.B in rng h by A65,A72,FUNCT_1:def 3;
    then
A77: Intersect FF = meet (rng h) by A66,SETFAM_1:def 9;
A78: h.C in rng h by A65,A70,FUNCT_1:def 3;
A79: F in (G \ {A}) by A7,A13,A12,ENUMSET1:def 3;
    then
A80: h.F in rng h by A65,FUNCT_1:def 3;
A81: D in (G \ {A}) by A7,A13,A12,ENUMSET1:def 3;
    then
A82: h.D in rng h by A65,FUNCT_1:def 3;
A83: xx c= h.B /\ h.C /\ h.D /\ h.E /\ h.F
    proof
      let m be object;
      assume
A84:  m in xx;
      then m in h.B & m in h.C by A68,A76,A78,A77,SETFAM_1:def 1;
      then
A85:  m in h.B /\ h.C by XBOOLE_0:def 4;
      m in h.D by A68,A82,A77,A84,SETFAM_1:def 1;
      then
A86:  m in h.B /\ h.C /\ h.D by A85,XBOOLE_0:def 4;
      m in h.E by A68,A75,A77,A84,SETFAM_1:def 1;
      then
A87:  m in h.B /\ h.C /\ h.D /\ h.E by A86,XBOOLE_0:def 4;
      m in h.F by A68,A80,A77,A84,SETFAM_1:def 1;
      hence thesis by A87,XBOOLE_0:def 4;
    end;
    then mbcd<>{} by A69;
    then
A88: not mbcd in {{}} by TARSKI:def 1;
A89: rng h <> {} by A65,A72,FUNCT_1:3;
    h.B /\ h.C /\ h.D /\ h.E /\ h.F c= xx
    proof
      let m be object;
      assume
A90:  m in h.B /\ h.C /\ h.D /\ h.E /\ h.F;
      then
A91:  m in h.B /\ h.C /\ h.D /\ h.E by XBOOLE_0:def 4;
      then
A92:  m in h.B /\ h.C /\ h.D by XBOOLE_0:def 4;
      then
A93:  m in h.B /\ h.C by XBOOLE_0:def 4;
A94:  rng h c= {h.B,h.C,h.D,h.E,h.F}
      proof
        let u be object;
        assume u in rng h;
        then consider x1 being object such that
A95:    x1 in dom h and
A96:    u = h.x1 by FUNCT_1:def 3;
        now
          per cases by A7,A12,A65,A95,ENUMSET1:def 3;
          case
            x1=B;
            hence thesis by A96,ENUMSET1:def 3;
          end;
          case
            x1=C;
            hence thesis by A96,ENUMSET1:def 3;
          end;
          case
            x1=D;
            hence thesis by A96,ENUMSET1:def 3;
          end;
          case
            x1=E;
            hence thesis by A96,ENUMSET1:def 3;
          end;
          case
            x1=F;
            hence thesis by A96,ENUMSET1:def 3;
          end;
        end;
        hence thesis;
      end;
      for y being set holds y in rng h implies m in y
      proof
        let y be set;
        assume
A97:    y in rng h;
        now
          per cases by A94,A97,ENUMSET1:def 3;
          case
            y=h.B;
            hence thesis by A93,XBOOLE_0:def 4;
          end;
          case
            y=h.C;
            hence thesis by A93,XBOOLE_0:def 4;
          end;
          case
            y=h.D;
            hence thesis by A92,XBOOLE_0:def 4;
          end;
          case
            y=h.E;
            hence thesis by A91,XBOOLE_0:def 4;
          end;
          case
            y=h.F;
            hence thesis by A90,XBOOLE_0:def 4;
          end;
        end;
        hence thesis;
      end;
      hence thesis by A68,A89,A77,SETFAM_1:def 1;
    end;
    then
A98: ((h.B /\ h.C) /\ h.D) /\ h.E /\ h.F = x by A83,XBOOLE_0:def 10;
    mbc<>{} by A69,A83;
    then not mbc in {{}} by TARSKI:def 1;
    then mbc in INTERSECTION(B,C) \ {{}} by A73,XBOOLE_0:def 5;
    then
A99: mbc in B '/\' C by PARTIT1:def 4;
    h.D in D by A67,A81;
    then mbcd in INTERSECTION(B '/\' C,D) by A99,SETFAM_1:def 5;
    then mbcd in INTERSECTION(B '/\' C,D) \ {{}} by A88,XBOOLE_0:def 5;
    then
A100: mbcd in B '/\' C '/\' D by PARTIT1:def 4;
    set mbcde=(h.B /\ h.C) /\ h.D /\ h.E;
A101: not x in {{}} by A69,TARSKI:def 1;
    mbcde<>{} by A69,A83;
    then
A102: not mbcde in {{}} by TARSKI:def 1;
    h.E in E by A67,A74;
    then mbcde in INTERSECTION(B '/\' C '/\' D,E) by A100,SETFAM_1:def 5;
    then mbcde in INTERSECTION(B '/\' C '/\' D,E) \ {{}} by A102,XBOOLE_0:def 5
;
    then
A103: mbcde in (B '/\' C '/\' D '/\' E) by PARTIT1:def 4;
    h.F in F by A67,A79;
    then x in INTERSECTION(B '/\' C '/\' D '/\' E,F) by A98,A103,SETFAM_1:def 5
;
    then x in INTERSECTION(B '/\' C '/\' D '/\' E,F) \ {{}} by A101,
XBOOLE_0:def 5;
    hence thesis by PARTIT1:def 4;
  end;
  CompF(A,G)='/\' (G \ {A}) by BVFUNC_2:def 7;
  hence thesis by A14,A64,XBOOLE_0:def 10;
end;
