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 Th42:
  G={A,B,C,D,E,F,J} & A<>B & A<>C & A<>D & A<>E & A<>F & A<>J & B
<>C & B<>D & B<>E & B<>F & B<>J & C<>D & C<>E & C<>F & C<>J & D<>E & D<>F & D<>
J & E<>F & E<>J & F<>J implies CompF(A,G) = B '/\' C '/\' D '/\' E '/\' F '/\'
  J
proof
  assume that
A1: G={A,B,C,D,E,F,J} and
A2: A<>B and
A3: A<>C and
A4: A<>D & A<>E and
A5: A<>F & A<>J and
A6: B<>C & B<>D & B<>E & B<>F & B<>J & C<>D & C<>E & C<>F & C<>J & D<>E
  & D<>F & D<>J & E<>F & E<>J & F<>J;
A7: G \ {A}={A} \/ {B,C,D,E,F,J} \ {A} by A1,ENUMSET1:16;
  ( not D in {A})& not E in {A} by A4,TARSKI:def 1;
  then
A8: {D,E} \ {A} = {D,E} by ZFMISC_1:63;
A9: ( not F in {A})& not J in {A} by A5,TARSKI:def 1;
A10: not C in {A} by A3,TARSKI:def 1;
A11: not B in {A} by A2,TARSKI:def 1;
  {B,C,D,E,F,J} \ {A} = ({B} \/ {C,D,E,F,J}) \ {A} by ENUMSET1:11
    .= ({B} \ {A}) \/ ({C,D,E,F,J} \ {A}) by XBOOLE_1:42
    .= {B} \/ ({C,D,E,F,J} \ {A}) by A11,ZFMISC_1:59
    .= {B} \/ (({C} \/ {D,E,F,J}) \ {A}) by ENUMSET1:7
    .= {B} \/ (({C} \ {A}) \/ ({D,E,F,J} \ {A})) by XBOOLE_1:42
    .= {B} \/ (({C} \ {A}) \/ (({D,E} \/ {F,J}) \ {A})) by ENUMSET1:5
    .= {B} \/ (({C} \ {A}) \/ (({D,E} \ {A}) \/ ({F,J} \ {A}))) by XBOOLE_1:42
    .= {B} \/ (({C} \ {A}) \/ ({D,E} \/ {F,J})) by A9,A8,ZFMISC_1:63
    .= {B} \/ ({C} \/ ({D,E} \/ {F,J})) by A10,ZFMISC_1:59
    .= {B} \/ ({C} \/ {D,E,F,J}) by ENUMSET1:5
    .= {B} \/ {C,D,E,F,J} by ENUMSET1:7
    .= {B,C,D,E,F,J} by ENUMSET1:11;
  then
A12: G \ {A} = {A} \ {A} \/ {B,C,D,E,F,J} by A7,XBOOLE_1:42
    .= {} \/ {B,C,D,E,F,J} by XBOOLE_1:37;
A13: '/\' (G \ {A}) c= B '/\' C '/\' D '/\' E '/\' F '/\' J
  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
A14: dom h=(G \ {A}) and
A15: rng h = FF and
A16: for d being set st d in (G \ {A}) holds h.d in d and
A17: x=Intersect FF and
A18: x<>{} by BVFUNC_2:def 1;
A19: C in (G \ {A}) by A12,ENUMSET1:def 4;
    then
A20: h.C in C by A16;
    set mbcd=(h.B /\ h.C) /\ h.D;
A21: E in (G \ {A}) by A12,ENUMSET1:def 4;
    then
A22: h.E in rng h by A14,FUNCT_1:def 3;
    set mbc=h.B /\ h.C;
A23: B in (G \ {A}) by A12,ENUMSET1:def 4;
    then h.B in B by A16;
    then
A24: mbc in INTERSECTION(B,C) by A20,SETFAM_1:def 5;
A25: h.B in rng h by A14,A23,FUNCT_1:def 3;
    then
A26: Intersect FF = meet (rng h) by A15,SETFAM_1:def 9;
A27: h.C in rng h by A14,A19,FUNCT_1:def 3;
A28: F in (G \ {A}) by A12,ENUMSET1:def 4;
    then
A29: h.F in rng h by A14,FUNCT_1:def 3;
    set mbcdef=((h.B /\ h.C) /\ h.D) /\ h.E /\ h.F;
    set mbcde=(h.B /\ h.C) /\ h.D /\ h.E;
A30: not x in {{}} by A18,TARSKI:def 1;
A31: J in (G \ {A}) by A12,ENUMSET1:def 4;
    then
A32: h.J in rng h by A14,FUNCT_1:def 3;
A33: D in (G \ {A}) by A12,ENUMSET1:def 4;
    then
A34: h.D in rng h by A14,FUNCT_1:def 3;
A35: xx c= ((((h.B /\ h.C) /\ h.D) /\ h.E) /\ h.F) /\ h.J
    proof
      let m be object;
      assume
A36:  m in xx;
      then m in h.B & m in h.C by A17,A25,A27,A26,SETFAM_1:def 1;
      then
A37:  m in h.B /\ h.C by XBOOLE_0:def 4;
      m in h.D by A17,A34,A26,A36,SETFAM_1:def 1;
      then
A38:  m in h.B /\ h.C /\ h.D by A37,XBOOLE_0:def 4;
      m in h.E by A17,A22,A26,A36,SETFAM_1:def 1;
      then
A39:  m in h.B /\ h.C /\ h.D /\ h.E by A38,XBOOLE_0:def 4;
      m in h.F by A17,A29,A26,A36,SETFAM_1:def 1;
      then
A40:  m in h.B /\ h.C /\ h.D /\ h.E /\ h.F by A39,XBOOLE_0:def 4;
      m in h. J by A17,A32,A26,A36,SETFAM_1:def 1;
      hence thesis by A40,XBOOLE_0:def 4;
    end;
    then mbcd<>{} by A18;
    then
A41: not mbcd in {{}} by TARSKI:def 1;
    ((((h.B /\ h.C) /\ h.D) /\ h.E) /\ h.F) /\ h.J c= xx
    proof
A42:  rng h c= {h.B,h.C,h.D,h.E,h.F,h.J}
      proof
        let u be object;
        assume u in rng h;
        then consider x1 being object such that
A43:    x1 in dom h and
A44:    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 by A12,A14,A43,
ENUMSET1:def 4;
        hence thesis by A44,ENUMSET1:def 4;
      end;
      let m be object;
      assume
A45:  m in ((((h.B /\ h.C) /\ h.D) /\ h.E) /\ h.F) /\ h.J;
      then
A46:  m in h. J by XBOOLE_0:def 4;
A47:  m in h.B /\ h.C /\ h.D /\ h.E /\ h.F by A45,XBOOLE_0:def 4;
      then
A48:  m in h.B /\ h.C /\ h.D /\ h.E by XBOOLE_0:def 4;
      then
A49:  m in h.E by XBOOLE_0:def 4;
A50:  m in h.B /\ h.C /\ h.D by A48,XBOOLE_0:def 4;
      then
A51:  m in h.D by XBOOLE_0:def 4;
      m in h.B /\ h.C by A50,XBOOLE_0:def 4;
      then
A52:  m in h.B & m in h.C by XBOOLE_0:def 4;
      m in h.F by A47,XBOOLE_0:def 4;
      then for y being set holds y in rng h implies m in y by A52,A51,A49,A46
,A42,ENUMSET1:def 4;
      hence thesis by A17,A25,A26,SETFAM_1:def 1;
    end;
    then
A53: ((((h.B /\ h.C) /\ h.D) /\ h.E) /\ h.F) /\ h.J = x by A35,XBOOLE_0:def 10;
    mbc<>{} by A18,A35;
    then not mbc in {{}} by TARSKI:def 1;
    then mbc in INTERSECTION(B,C) \ {{}} by A24,XBOOLE_0:def 5;
    then
A54: mbc in B '/\' C by PARTIT1:def 4;
    h.D in D by A16,A33;
    then mbcd in INTERSECTION(B '/\' C,D) by A54,SETFAM_1:def 5;
    then mbcd in INTERSECTION(B '/\' C,D) \ {{}} by A41,XBOOLE_0:def 5;
    then
A55: mbcd in B '/\' C '/\' D by PARTIT1:def 4;
    mbcde<>{} by A18,A35;
    then
A56: not mbcde in {{}} by TARSKI:def 1;
    h.E in E by A16,A21;
    then mbcde in INTERSECTION(B '/\' C '/\' D,E) by A55,SETFAM_1:def 5;
    then mbcde in INTERSECTION(B '/\' C '/\' D,E) \ {{}} by A56,XBOOLE_0:def 5;
    then
A57: mbcde in (B '/\' C '/\' D '/\' E) by PARTIT1:def 4;
    mbcdef<>{} by A18,A35;
    then
A58: not mbcdef in {{}} by TARSKI:def 1;
    h.F in F by A16,A28;
    then mbcdef in INTERSECTION(B '/\' C '/\' D '/\' E,F) by A57,SETFAM_1:def 5
;
    then mbcdef in INTERSECTION(B '/\' C '/\' D '/\' E,F) \ {{}} by A58,
XBOOLE_0:def 5;
    then
A59: mbcdef in (B '/\' C '/\' D '/\' E '/\' F) by PARTIT1:def 4;
    h.J in J by A16,A31;
    then x in INTERSECTION(B '/\' C '/\' D '/\' E '/\' F,J) by A53,A59,
SETFAM_1:def 5;
    then x in INTERSECTION(B '/\' C '/\' D '/\' E '/\' F,J) \ {{}} by A30,
XBOOLE_0:def 5;
    hence thesis by PARTIT1:def 4;
  end;
A60: B '/\' C '/\' D '/\' E '/\' F '/\' J c= '/\' (G \ {A})
  proof
    let x be object;
     reconsider xx=x as set by TARSKI:1;
    assume
A61: x in B '/\' C '/\' D '/\' E '/\' F '/\' J;
    then
A62: x<>{} by EQREL_1:def 4;
    x in INTERSECTION(B '/\' C '/\' D '/\' E '/\' F,J) \ {{}} by A61,
PARTIT1:def 4;
    then consider bcdef,j being set such that
A63: bcdef in B '/\' C '/\' D '/\' E '/\' F and
A64: j in J and
A65: x = bcdef /\ j by SETFAM_1:def 5;
    bcdef in INTERSECTION(B '/\' C '/\' D '/\' E,F) \ {{}} by A63,PARTIT1:def 4
;
    then consider bcde,f being set such that
A66: bcde in B '/\' C '/\' D '/\' E and
A67: f in F and
A68: bcdef = bcde /\ f by SETFAM_1:def 5;
    bcde in INTERSECTION(B '/\' C '/\' D,E) \ {{}} by A66,PARTIT1:def 4;
    then consider bcd,e being set such that
A69: bcd in B '/\' C '/\' D and
A70: e in E and
A71: bcde = bcd /\ e by SETFAM_1:def 5;
    bcd in INTERSECTION(B '/\' C,D) \ {{}} by A69,PARTIT1:def 4;
    then consider bc,d being set such that
A72: bc in B '/\' C and
A73: d in D and
A74: bcd = bc /\ d by SETFAM_1:def 5;
    bc in INTERSECTION(B,C) \ {{}} by A72,PARTIT1:def 4;
    then consider b,c being set such that
A75: b in B & c in C and
A76: bc = b /\ c by SETFAM_1:def 5;
    set h = (B .--> b) +* (C .--> c) +* (D .--> d) +* (E .--> e) +* (F .--> f)
    +* (J .--> j);
A77: h.B = b by A6,Th37;
A78: dom h = {J,B,C,D,E,F} by Th38
      .= {J} \/ {B,C,D,E,F} by ENUMSET1:11
      .= {B,C,D,E,F,J} by ENUMSET1:15;
    then D in dom h by ENUMSET1:def 4;
    then
A79: h.D in rng h by FUNCT_1:def 3;
A80: 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 by A12,ENUMSET1:def 4;
      hence thesis by A6,A64,A67,A70,A73,A75,Th37;
    end;
    E in dom h by A78,ENUMSET1:def 4;
    then
A81: h.E in rng h by FUNCT_1:def 3;
    C in dom h by A78,ENUMSET1:def 4;
    then
A82: h.C in rng h by FUNCT_1:def 3;
A83: h.C = c by A6,Th37;
A84: rng h c= {h.B,h.C,h.D,h.E,h.F,h.J}
    proof
      let t be object;
      assume t in rng h;
      then consider x1 being object such that
A85:  x1 in dom h and
A86:  t = h.x1 by FUNCT_1:def 3;
      x1=B or x1=C or x1=D or x1=E or x1=F or x1=J by A78,A85,ENUMSET1:def 4;
      hence thesis by A86,ENUMSET1:def 4;
    end;
    J in dom h by A78,ENUMSET1:def 4;
    then
A87: h.J in rng h by FUNCT_1:def 3;
    F in dom h by A78,ENUMSET1:def 4;
    then
A88: h.F in rng h by FUNCT_1:def 3;
    B in dom h by A78,ENUMSET1:def 4;
    then
A89: h.B in rng h by FUNCT_1:def 3;
    {h.B,h.C,h.D,h.E,h.F,h.J} c= rng h
    by A79,A89,A82,A81,A88,A87,ENUMSET1:def 4;
    then
A90: rng h = {h.B,h.C,h.D,h.E,h.F,h.J} by A84,XBOOLE_0:def 10;
A91: h.J = j by A6,Th37;
A92: h.F = f by A6,Th37;
A93: h.E = e by A6,Th37;
A94: h.D = d by A6,Th37;
    rng h c= bool Y
    proof
      let t be object;
      assume t in rng h;
      then t=h.D or t=h.B or t=h.C or t=h.E or t=h.F or t=h.J by A84,
ENUMSET1:def 4;
      hence thesis by A64,A67,A70,A73,A75,A94,A77,A83,A93,A92,A91;
    end;
    then reconsider FF=rng h as Subset-Family of Y;
A95: dom ((B .--> b) +* (C .--> c) +* (D .--> d) +* (E .--> e) +* (F .-->
    f) +* (J .--> j)) = {J,B,C,D,E,F} by Th38
      .= {J} \/ {B,C,D,E,F} by ENUMSET1:11
      .= {B,C,D,E,F,J} by ENUMSET1:15;
    reconsider h as Function;
A96: xx c= Intersect FF
    proof
      let u be object;
      assume
A97:  u in xx;
      for y be set holds y in FF implies u in y
      proof
        let y be set;
        assume
A98:    y in FF;
        now
          per cases by A84,A98,ENUMSET1:def 4;
          case
A99:        y=h.D;
            u in (d /\ ((b /\ c) /\ e)) /\ f /\ j by A65,A68,A71,A74,A76,A97,
XBOOLE_1:16;
            then u in (d /\ ((b /\ c) /\ e /\ f)) /\ j by XBOOLE_1:16;
            then u in d /\ (((b /\ c) /\ e /\ f) /\ j) by XBOOLE_1:16;
            hence thesis by A94,A99,XBOOLE_0:def 4;
          end;
          case
A100:       y=h.B;
            u in (c /\ (d /\ b)) /\ e /\ f /\ j by A65,A68,A71,A74,A76,A97,
XBOOLE_1:16;
            then u in c /\ ((d /\ b) /\ e) /\ f /\ j by XBOOLE_1:16;
            then u in c /\ ((d /\ e) /\ b) /\ f /\ j by XBOOLE_1:16;
            then u in c /\ (((d /\ e) /\ b) /\ f) /\ j by XBOOLE_1:16;
            then u in c /\ ((((d /\ e) /\ b) /\ f) /\ j) by XBOOLE_1:16;
            then u in c /\ (((d /\ e) /\ (f /\ b)) /\ j) by XBOOLE_1:16;
            then u in c /\ ((d /\ e) /\ ((f /\ b) /\ j)) by XBOOLE_1:16;
            then u in c /\ ((d /\ e) /\ (f /\ (j /\ b))) by XBOOLE_1:16;
            then u in (c /\ (d /\ e)) /\ (f /\ (j /\ b)) by XBOOLE_1:16;
            then u in ((c /\ (d /\ e)) /\ f) /\ (j /\ b) by XBOOLE_1:16;
            then u in (((c /\ (d /\ e)) /\ f) /\ j) /\ b by XBOOLE_1:16;
            hence thesis by A77,A100,XBOOLE_0:def 4;
          end;
          case
A101:       y=h.C;
            u in (c /\ (d /\ b)) /\ e /\ f /\ j by A65,A68,A71,A74,A76,A97,
XBOOLE_1:16;
            then u in c /\ ((d /\ b) /\ e) /\ f /\ j by XBOOLE_1:16;
            then u in c /\ ((d /\ e) /\ b) /\ f /\ j by XBOOLE_1:16;
            then u in c /\ (((d /\ e) /\ b) /\ f) /\ j by XBOOLE_1:16;
            then u in c /\ ((((d /\ e) /\ b) /\ f) /\ j) by XBOOLE_1:16;
            hence thesis by A83,A101,XBOOLE_0:def 4;
          end;
          case
A102:       y=h.E;
            u in ((b /\ c) /\ d) /\ (f /\ e) /\ j by A65,A68,A71,A74,A76,A97,
XBOOLE_1:16;
            then u in ((b /\ c) /\ d) /\ ((f /\ e) /\ j) by XBOOLE_1:16;
            then u in ((b /\ c) /\ d) /\ ((f /\ j) /\ e) by XBOOLE_1:16;
            then u in (((b /\ c) /\ d) /\ (f /\ j)) /\ e by XBOOLE_1:16;
            hence thesis by A93,A102,XBOOLE_0:def 4;
          end;
          case
A103:       y=h.F;
            u in (((b /\ c) /\ d) /\ e) /\ j /\ f by A65,A68,A71,A74,A76,A97,
XBOOLE_1:16;
            hence thesis by A92,A103,XBOOLE_0:def 4;
          end;
          case
            y=h.J;
            hence thesis by A65,A91,A97,XBOOLE_0:def 4;
          end;
        end;
        hence thesis;
      end;
      then u in meet FF by A90,SETFAM_1:def 1;
      hence thesis by A90,SETFAM_1:def 9;
    end;
A104: Intersect FF = meet (rng h) by A79,SETFAM_1:def 9;
    Intersect FF c= xx
    proof
      let t be object;
      assume
A105: t in Intersect FF;
      h.C in rng h by A90,ENUMSET1:def 4;
      then
A106: t in c by A83,A104,A105,SETFAM_1:def 1;
      h.B in rng h by A90,ENUMSET1:def 4;
      then t in b by A77,A104,A105,SETFAM_1:def 1;
      then
A107: t in b /\ c by A106,XBOOLE_0:def 4;
      h.D in rng h by A90,ENUMSET1:def 4;
      then t in d by A94,A104,A105,SETFAM_1:def 1;
      then
A108: t in (b /\ c) /\ d by A107,XBOOLE_0:def 4;
      h.E in rng h by A90,ENUMSET1:def 4;
      then t in e by A93,A104,A105,SETFAM_1:def 1;
      then
A109: t in (b /\ c) /\ d /\ e by A108,XBOOLE_0:def 4;
      h.F in rng h by A90,ENUMSET1:def 4;
      then t in f by A92,A104,A105,SETFAM_1:def 1;
      then
A110: t in (b /\ c) /\ d /\ e /\ f by A109,XBOOLE_0:def 4;
      h.J in rng h by A90,ENUMSET1:def 4;
      then t in j by A91,A104,A105,SETFAM_1:def 1;
      hence thesis by A65,A68,A71,A74,A76,A110,XBOOLE_0:def 4;
    end;
    then x = Intersect FF by A96,XBOOLE_0:def 10;
    hence thesis by A12,A95,A80,A62,BVFUNC_2:def 1;
  end;
  CompF(A,G)='/\' (G \ {A}) by BVFUNC_2:def 7;
  hence thesis by A60,A13,XBOOLE_0:def 10;
end;
