The Mizar article:

Five Variable Predicate Calculus for Boolean Valued Functions. Part I

by
Shunichi Kobayashi

Received December 27, 1999

Copyright (c) 1999 Association of Mizar Users

MML identifier: BVFUNC22
[ MML identifier index ]


environ

 vocabulary PARTIT1, EQREL_1, BVFUNC_2, BOOLE, SETFAM_1, CAT_1, FUNCT_4,
      RELAT_1, FUNCT_1, CANTOR_1, T_1TOPSP;
 notation TARSKI, XBOOLE_0, ENUMSET1, ZFMISC_1, SUBSET_1, SETFAM_1, RELAT_1,
      FUNCT_1, CQC_LANG, FUNCT_4, EQREL_1, PARTIT1, CANTOR_1, BVFUNC_1,
      BVFUNC_2;
 constructors BVFUNC_2, SETWISEO, CANTOR_1, FUNCT_7, BVFUNC_1;
 clusters SUBSET_1, PARTIT1, CQC_LANG, XBOOLE_0;
 requirements SUBSET, BOOLE;
 definitions TARSKI;
 theorems EQREL_1, ZFMISC_1, T_1TOPSP, PARTIT1, BVFUNC_2, BVFUNC11, BVFUNC14,
      FUNCT_4, CQC_LANG, TARSKI, ENUMSET1, FUNCT_1, CANTOR_1, SETFAM_1,
      XBOOLE_0, XBOOLE_1;

begin :: Chap. 1  Preliminaries

reserve Y for non empty set,

        G for Subset of PARTITIONS(Y),
        A, B, C, D, E for a_partition of Y;

theorem Th1:
 G={A,B,C,D,E} & A<>B & A<>C & A<>D & A<>E
 implies CompF(A,G) = B '/\' C '/\' D '/\' E
proof
  assume A1: G={A,B,C,D,E} & A<>B & A<>C & A<>D & A<>E;
  per cases;
  suppose
A2: B = C;
   then G = {A,B,B,D} \/ {E} by A1,ENUMSET1:50
         .= {B,B,A,D} \/ {E} by ENUMSET1:110
         .= {B,B,A,D,E} by ENUMSET1:50
         .= {B,A,D,E} by ENUMSET1:72
         .={A,B,D,E} by ENUMSET1:108;
  hence CompF(A,G) = B '/\' D '/\' E by A1,BVFUNC14:7
           .= B '/\' C '/\' D '/\' E by A2,PARTIT1:15;
  suppose
A3: B = D;
   then G = {A,B,C,B} \/ {E} by A1,ENUMSET1:50
         .= {B,B,A,C} \/ {E} by ENUMSET1:112
         .= {B,B,A,C,E} by ENUMSET1:50
         .= {B,A,C,E} by ENUMSET1:72
         .={A,B,C,E} by ENUMSET1:108;
  hence CompF(A,G) = B '/\' C '/\' E by A1,BVFUNC14:7
           .= B '/\' D '/\' C '/\' E by A3,PARTIT1:15
           .= B '/\' C '/\' D '/\' E by PARTIT1:16;
  suppose
A4: B = E;
   then G = {A} \/ {B,C,D,B} by A1,ENUMSET1:47
         .= {A} \/ {B,B,C,D} by ENUMSET1:105
         .= {A} \/ {B,C,D} by ENUMSET1:71
         .={A,B,C,D} by ENUMSET1:44;
  hence CompF(A,G) = B '/\' C '/\' D by A1,BVFUNC14:7
           .= B '/\' E '/\' C '/\' D by A4,PARTIT1:15
           .= B '/\' E '/\' (C '/\' D) by PARTIT1:16
           .= B '/\' (C '/\' D) '/\' E by PARTIT1:16
           .= B '/\' C '/\' D '/\' E by PARTIT1:16;
  suppose
A5: C = D;
   then G = {A,B,C,C} \/ {E} by A1,ENUMSET1:50
         .= {C,C,A,B} \/ {E} by ENUMSET1:118
         .= {C,A,B} \/ {E} by ENUMSET1:71
         .= {C,A,B,E} by ENUMSET1:46
         .={A,B,C,E} by ENUMSET1:110;
  hence CompF(A,G) = B '/\' C '/\' E by A1,BVFUNC14:7
           .= B '/\' (C '/\' D) '/\' E by A5,PARTIT1:15
           .= B '/\' C '/\' D '/\' E by PARTIT1:16;
  suppose
A6: C = E;
   then G = {A} \/ {B,C,D,C} by A1,ENUMSET1:47
         .= {A} \/ {C,C,B,D} by ENUMSET1:117
         .= {A} \/ {C,B,D} by ENUMSET1:71
         .= {A,C,B,D} by ENUMSET1:44
         .={A,B,C,D} by ENUMSET1:104;
  hence CompF(A,G) = B '/\' C '/\' D by A1,BVFUNC14:7
           .= B '/\' (C '/\' E) '/\' D by A6,PARTIT1:15
           .= B '/\' (C '/\' E '/\' D) by PARTIT1:16
           .= B '/\' (C '/\' D '/\' E) by PARTIT1:16
           .= B '/\' (C '/\' D) '/\' E by PARTIT1:16
           .= B '/\' C '/\' D '/\' E by PARTIT1:16;
  suppose
A7: D = E;
   then G = {A} \/ {B,C,D,D} by A1,ENUMSET1:47
         .= {A} \/ {D,D,B,C} by ENUMSET1:118
         .= {A} \/ {D,B,C} by ENUMSET1:71
         .= {A,D,B,C} by ENUMSET1:44
         .={A,B,C,D} by ENUMSET1:105;
  hence CompF(A,G) = B '/\' C '/\' D by A1,BVFUNC14:7
           .= B '/\' C '/\' (D '/\' E) by A7,PARTIT1:15
           .= B '/\' (C '/\' (D '/\' E)) by PARTIT1:16
           .= B '/\' (C '/\' D '/\' E) by PARTIT1:16
           .= B '/\' (C '/\' D) '/\' E by PARTIT1:16
           .= B '/\' C '/\' D '/\' E by PARTIT1:16;
  suppose
A8: B<>C & B<>D & B<>E & C<>D & C<>E & D<>E;
     G \ {A}={A} \/ {B,C,D,E} \ {A} by A1,ENUMSET1:47;
  then A9:G \ {A} = ({A} \ {A}) \/ ({B,C,D,E} \ {A}) by XBOOLE_1:42;
  A10:not B in {A} by A1,TARSKI:def 1;
  A11:not C in {A} by A1,TARSKI:def 1;
  A12:not D in {A} by A1,TARSKI:def 1;
  A13:not E in {A} by A1,TARSKI:def 1;
    {B,C,D,E} \ {A} = ({B} \/ {C,D,E}) \ {A} by ENUMSET1:44
  .= ({B} \ {A}) \/ ({C,D,E} \ {A}) by XBOOLE_1:42
  .= {B} \/ ({C,D,E} \ {A}) by A10,ZFMISC_1:67
  .= {B} \/ (({C} \/ {D,E}) \ {A}) by ENUMSET1:42
  .= {B} \/ (({C} \ {A}) \/ ({D,E} \ {A})) by XBOOLE_1:42
  .= {B} \/ (({C} \ {A}) \/ {D,E}) by A12,A13,ZFMISC_1:72
  .= {B} \/ ({C} \/ {D,E}) by A11,ZFMISC_1:67
  .= {B} \/ {C,D,E} by ENUMSET1:42;
  then A14:G \ {A} = ({A} \ {A}) \/ {B,C,D,E} by A9,ENUMSET1:44;
    A in {A} by TARSKI:def 1;
  then A15: {A} \ {A}={} by ZFMISC_1:68;
  A16:B '/\' C '/\' D '/\' E c= '/\' (G \ {A})
  proof
    let x be set;
    assume A17:x in B '/\' C '/\' D '/\' E;
    then x in INTERSECTION(B '/\' C '/\' D,E) \ {{}} by PARTIT1:def 4;
    then x in INTERSECTION(B '/\' C '/\' D,E) & not x in {{}} by XBOOLE_0:def 4
;
    then consider bcd,e being set such that
A18:  bcd in B '/\' C '/\' D & e in E & x = bcd /\ e by SETFAM_1:def 5;
      bcd in INTERSECTION(B '/\' C,D) \ {{}} by A18,PARTIT1:def 4;
    then bcd in INTERSECTION(B '/\' C,D) & not bcd in {{}} by XBOOLE_0:def 4;
    then consider bc,d being set such that
A19:  bc in B '/\' C & d in D & bcd = bc /\ d by SETFAM_1:def 5;
      bc in INTERSECTION(B,C) \ {{}} by A19,PARTIT1:def 4;
    then bc in INTERSECTION(B,C) & not bc in {{}} by XBOOLE_0:def 4;
    then consider b,c being set such that
A20:  b in B & c in C & bc = b /\ c by SETFAM_1:def 5;
    set h = (B .--> b) +* (C .--> c) +* (D .--> d) +* (E .--> e);
      dom ((B .--> b) +* (C .--> c))
      = dom (B .--> b) \/ dom (C .--> c) by FUNCT_4:def 1;
then dom ((B .--> b) +* (C .--> c) +* (D .--> d))
      = dom (B .--> b) \/ dom (C .--> c) \/ dom (D .--> d) by FUNCT_4:def 1;
then A21:dom ((B .--> b) +* (C .--> c) +* (D .--> d) +* (E .--> e))
      = dom (B .--> b) \/ dom (C .--> c) \/ dom (D .--> d) \/ dom (E .--> e)
      by FUNCT_4:def 1;
A22:  dom (B .--> b) = {B} by CQC_LANG:5;
A23:  dom (C .--> c) = {C} by CQC_LANG:5;
A24:  dom (D .--> d) = {D} by CQC_LANG:5;
A25:  dom (E .--> e) = {E} by CQC_LANG:5;
A26:      dom ((B .--> b) +* (C .--> c) +* (D .--> d) +* (E .--> e))
      = {B} \/ {C} \/ {D} \/ {E} by A21,A22,A23,A24,CQC_LANG:5
     .= {B,C} \/ {D} \/ {E} by ENUMSET1:41
     .= {B,C,D} \/ {E} by ENUMSET1:43
     .= {B,C,D,E} by ENUMSET1:46;
A27:  h.D = d
      proof
          not D in dom (E .--> e) by A8,A25,TARSKI:def 1;
        then A28:h.D=((B .--> b) +* (C .--> c) +* (D .--> d)).D
         by FUNCT_4:12;
          D in dom (D .--> d) by A24,TARSKI:def 1;
        then h.D = (D .--> d).D by A28,FUNCT_4:14;
        hence thesis by CQC_LANG:6;
      end;
A29:  h.B = b
      proof
        A30:not B in dom (D .--> d) by A8,A24,TARSKI:def 1;
          not B in dom (E .--> e) by A8,A25,TARSKI:def 1;
        then h.B=((B .--> b) +* (C .--> c) +* (D .--> d)).B
         by FUNCT_4:12;
        then A31:h.B=((B .--> b) +* (C .--> c)).B by A30,FUNCT_4:12;
          not B in dom (C .--> c) by A8,A23,TARSKI:def 1;
        then h.B=(B .--> b).B by A31,FUNCT_4:12;
        hence thesis by CQC_LANG:6;
      end;
A32:  h.C = c
      proof
        A33:not C in dom (D .--> d) by A8,A24,TARSKI:def 1;
          not C in dom (E .--> e) by A8,A25,TARSKI:def 1;
        then h.C=((B .--> b) +* (C .--> c) +* (D .--> d)).C
         by FUNCT_4:12;
        then A34:h.C=((B .--> b) +* (C .--> c)).C by A33,FUNCT_4:12;
          C in dom (C .--> c) by A23,TARSKI:def 1;
        then h.C=(C .--> c).C by A34,FUNCT_4:14;
        hence thesis by CQC_LANG:6;
      end;
A35:  h.E = e
      proof
          E in dom (E .--> e) by A25,TARSKI:def 1;
        then h.E = (E .--> e).E by FUNCT_4:14;
        hence thesis by CQC_LANG:6;
      end;
A36:  for p being set st p in (G \ {A}) holds h.p in p
      proof
        let p be set;
        assume A37: p in (G \ {A});
          now per cases by A14,A15,A37,ENUMSET1:18;
        case p=D;
        hence thesis by A19,A27;
        case p=B;
        hence thesis by A20,A29;
        case p=C;
        hence thesis by A20,A32;
        case p=E;
        hence thesis by A18,A35;
        end;
        hence thesis;
      end;
A38:      D in dom h by A26,ENUMSET1:19;
A39:      B in dom h by A26,ENUMSET1:19;
A40:      C in dom h by A26,ENUMSET1:19;
A41:      E in dom h by A26,ENUMSET1:19;
      A42:rng h c= {h.D,h.B,h.C,h.E}
      proof
        let t be set;
        assume t in rng h;
        then consider x1 being set such that
A43:      x1 in dom h & t = h.x1 by FUNCT_1:def 5;
          now per cases by A26,A43,ENUMSET1:18;
        case x1=D;
        hence thesis by A43,ENUMSET1:19;
        case x1=B;
        hence thesis by A43,ENUMSET1:19;
        case x1=C;
        hence thesis by A43,ENUMSET1:19;
        case x1=E;
        hence thesis by A43,ENUMSET1:19;
        end;
        hence thesis;
      end;
A44:    {h.D,h.B,h.C,h.E} c= rng h
      proof
        let t be set;
        assume A45: t in {h.D,h.B,h.C,h.E};
          now per cases by A45,ENUMSET1:18;
        case t=h.D;
        hence thesis by A38,FUNCT_1:def 5;
        case t=h.B;
        hence thesis by A39,FUNCT_1:def 5;
        case t=h.C;
        hence thesis by A40,FUNCT_1:def 5;
        case t=h.E;
        hence thesis by A41,FUNCT_1:def 5;
        end;
        hence thesis;
      end;
then A46:  {h.D,h.B,h.C,h.E} = rng h by A42,XBOOLE_0:def 10;
        rng h c= bool Y
      proof
        let t be set;
        assume A47: t in rng h;
          now per cases by A42,A47,ENUMSET1:18;
        case t=h.D;
        hence thesis by A19,A27;
        case t=h.B;
        hence thesis by A20,A29;
        case t=h.C;
        hence thesis by A20,A32;
        case t=h.E;
        hence thesis by A18,A35;
        end;
        hence thesis;
      end;
      then reconsider F=rng h as Subset-Family of Y by SETFAM_1:def 7;
      reconsider h as Function;
A48:  rng h <> {} by A38,FUNCT_1:12;
    A49:x c= Intersect F
    proof
      let u be set;
      assume A50:u in x;
A51:  h.D in {h.D,h.B,h.C,h.E} by ENUMSET1:19;
        for y be set holds y in F implies u in y
      proof
        let y be set;
        assume A52: y in F;
          now per cases by A42,A52,ENUMSET1:18;
        case A53: y=h.D;
          u in d /\ ((b /\ c) /\ e) by A18,A19,A20,A50,XBOOLE_1:16;
        hence thesis by A27,A53,XBOOLE_0:def 3;
        case A54: y=h.B;
          u in (c /\ (d /\ b)) /\ e by A18,A19,A20,A50,XBOOLE_1:16;
        then u in c /\ ((d /\ b) /\ e) by XBOOLE_1:16;
        then u in c /\ ((d /\ e) /\ b) by XBOOLE_1:16;
        then u in (c /\ (d /\ e)) /\ b by XBOOLE_1:16;
        hence thesis by A29,A54,XBOOLE_0:def 3;
        case A55: y=h.C;
          u in (c /\ (b /\ d)) /\ e by A18,A19,A20,A50,XBOOLE_1:16;
        then u in c /\ (b /\ d /\ e) by XBOOLE_1:16;
        hence thesis by A32,A55,XBOOLE_0:def 3;
        case y=h.E;
        hence thesis by A18,A35,A50,XBOOLE_0:def 3;
        end;
        hence thesis;
      end;
      then u in meet F by A44,A51,SETFAM_1:def 1;
      hence thesis by A44,A51,CANTOR_1:def 3;
    end;
      Intersect F c= x
    proof
      let t be set;
      assume t in Intersect F;
then A56:  t in meet (rng h) by A48,CANTOR_1:def 3;
        h.B in rng h & h.C in rng h & h.D in rng h & h.E in rng h
          by A46,ENUMSET1:19;
then A57:  t in h.B & t in h.C & t in h.D & t in h.E by A56,SETFAM_1:def 1;
      then t in b /\ c by A29,A32,XBOOLE_0:def 3;
      then t in (b /\ c) /\ d by A27,A57,XBOOLE_0:def 3;
      hence thesis by A18,A19,A20,A35,A57,XBOOLE_0:def 3;
    end;
    then A58:x = Intersect F by A49,XBOOLE_0:def 10;
      x<>{} by A17,EQREL_1:def 6;
    hence thesis by A14,A15,A26,A36,A58,BVFUNC_2:def 1;
  end;
    '/\' (G \ {A}) c= B '/\' C '/\' D '/\' E
  proof
    let x be set;
    assume x in '/\' (G \ {A});
    then consider h being Function, F being Subset-Family of Y such that
A59:  dom h=(G \ {A}) & rng h = F &
      (for d being set st d in (G \ {A}) holds h.d in d) &
      x=Intersect F & x<>{} by BVFUNC_2:def 1;
      B in (G \ {A}) & C in (G \ {A}) & D in (G \ {A}) & E in (G \ {A})
      by A14,A15,ENUMSET1:19;
    then A60:h.B in B & h.C in C & h.D in D & h.E in E by A59;
      B in dom h & C in dom h & D in dom h & E in dom h by A14,A15,A59,ENUMSET1
:19;
    then A61:h.B in rng h & h.C in rng h & h.D in rng h & h.E in rng h
    by FUNCT_1:def 5;
    A62:not (x in {{}}) by A59,TARSKI:def 1;
    A63:h.B /\ h.C /\ h.D /\ h.E c= x
    proof
      let m be set;
      assume A64: m in h.B /\ h.C /\ h.D /\ h.E;
then A65:    m in h.B /\ h.C /\ h.D & m in h.E by XBOOLE_0:def 3;
then A66:      m in h.B /\ h.C & m in h.D by XBOOLE_0:def 3;
      A67:rng h c= {h.B,h.C,h.D,h.E}
      proof
        let u be set;
        assume u in rng h;
        then consider x1 being set such that
A68:      x1 in dom h & u = h.x1 by FUNCT_1:def 5;
          now per cases by A14,A15,A59,A68,ENUMSET1:18;
        case x1=B;
        hence thesis by A68,ENUMSET1:19;
        case x1=C;
        hence thesis by A68,ENUMSET1:19;
        case x1=D;
        hence thesis by A68,ENUMSET1:19;
        case x1=E;
        hence thesis by A68,ENUMSET1:19;
        end;
        hence thesis;
      end;
        for y being set holds y in rng h implies m in y
      proof
        let y be set;
        assume A69: y in rng h;
          now per cases by A67,A69,ENUMSET1:18;
         case y=h.B;
          hence thesis by A66,XBOOLE_0:def 3;
         case y=h.C;
          hence thesis by A66,XBOOLE_0:def 3;
         case y=h.D;
          hence thesis by A65,XBOOLE_0:def 3;
         case y=h.E;
          hence thesis by A64,XBOOLE_0:def 3;
        end;
        hence thesis;
      end;
      then m in meet (rng h) by A61,SETFAM_1:def 1;
      hence thesis by A59,A61,CANTOR_1:def 3;
    end;
    A70: x c= h.B /\ h.C /\ h.D /\ h.E
    proof
      let m be set;
      assume m in x;
      then m in meet (rng h) by A59,A61,CANTOR_1:def 3;
      then A71:m in h.B & m in h.C & m in h.D & m in h.E by A61,SETFAM_1:def 1;
      then m in h.B /\ h.C by XBOOLE_0:def 3;
      then m in h.B /\ h.C /\ h.D by A71,XBOOLE_0:def 3;
      hence thesis by A71,XBOOLE_0:def 3;
    end;
    then A72:((h.B /\ h.C) /\ h.D) /\ h.E = x by A63,XBOOLE_0:def 10;
    set mbc=h.B /\ h.C;
    set mbcd=(h.B /\ h.C) /\ h.D;
     mbcd<>{} by A59,A63,A70,XBOOLE_0:def 10;
    then A73:not (mbcd in {{}}) by TARSKI:def 1;
      mbc<>{} by A59,A63,A70,XBOOLE_0:def 10;
    then A74:not (mbc in {{}}) by TARSKI:def 1;
      mbc in INTERSECTION(B,C) by A60,SETFAM_1:def 5;
    then mbc in INTERSECTION(B,C) \ {{}} by A74,XBOOLE_0:def 4;
    then mbc in B '/\' C by PARTIT1:def 4;
    then mbcd in INTERSECTION(B '/\' C,D) by A60,SETFAM_1:def 5;
    then mbcd in INTERSECTION(B '/\' C,D) \ {{}} by A73,XBOOLE_0:def 4;
    then mbcd in B '/\' C '/\' D by PARTIT1:def 4;
    then x in INTERSECTION(B '/\' C '/\' D,E) by A60,A72,SETFAM_1:def 5;
    then x in INTERSECTION(B '/\' C '/\' D,E) \ {{}} by A62,XBOOLE_0:def 4;
    hence thesis by PARTIT1:def 4;
  end;
  then '/\' (G \ {A}) = B '/\' C '/\' D '/\' E by A16,XBOOLE_0:def 10;
  hence thesis by BVFUNC_2:def 7;
end;

theorem Th2:
G is independent & G={A,B,C,D,E} &
A<>B & A<>C & A<>D & A<>E & B<>C & B<>D & B<>E & C<>D & C<>E & D<>E
implies
CompF(B,G) = A '/\' C '/\' D '/\' E
proof
  assume A1:G is independent & G={A,B,C,D,E} &
  A<>B & A<>C & A<>D & A<>E & B<>C & B<>D & B<>E & C<>D & C<>E & D<>E;
    {A,B,C,D,E}={A,B} \/ {C,D,E} by ENUMSET1:48;
  then G={B,A,C,D,E} by A1,ENUMSET1:48;
  hence thesis by A1,Th1;
end;

theorem Th3:
G is independent & G={A,B,C,D,E} &
A<>B & A<>C & A<>D & A<>E & B<>C & B<>D & B<>E & C<>D & C<>E & D<>E
implies
CompF(C,G) = A '/\' B '/\' D '/\' E
proof
  assume A1:G is independent & G={A,B,C,D,E} &
  A<>B & A<>C & A<>D & A<>E & B<>C & B<>D & B<>E & C<>D & C<>E & D<>E;
    {A,B,C,D,E}={A,B,C} \/ {D,E} by ENUMSET1:49;
  then {A,B,C,D,E}={A} \/ {B,C} \/ {D,E} by ENUMSET1:42;
  then {A,B,C,D,E}={A,C,B} \/ {D,E} by ENUMSET1:42;
  then {A,B,C,D,E}={A,C} \/ {B} \/ {D,E} by ENUMSET1:43;
  then {A,B,C,D,E}={C,A,B} \/ {D,E} by ENUMSET1:43;
  then G={C,A,B,D,E} by A1,ENUMSET1:49;
  hence thesis by A1,Th1;
end;

theorem Th4:
G is independent & G={A,B,C,D,E} &
A<>B & A<>C & A<>D & A<>E & B<>C & B<>D & B<>E & C<>D & C<>E & D<>E
implies
CompF(D,G) = A '/\' B '/\' C '/\' E
proof
  assume A1:G is independent & G={A,B,C,D,E} &
  A<>B & A<>C & A<>D & A<>E & B<>C & B<>D & B<>E & C<>D & C<>E & D<>E;
    {A,B,C,D,E}={A,B} \/ {C,D,E} by ENUMSET1:48;
  then {A,B,C,D,E}={A,B} \/ ({C,D} \/ {E}) by ENUMSET1:43;
  then {A,B,C,D,E}={A,B} \/ {D,C,E} by ENUMSET1:43;
  then G={A,B,D,C,E} by A1,ENUMSET1:48;
  hence thesis by A1,Th3;
end;

theorem
  G is independent & G={A,B,C,D,E} &
A<>B & A<>C & A<>D & A<>E & B<>C & B<>D & B<>E & C<>D & C<>E & D<>E
implies
CompF(E,G) = A '/\' B '/\' C '/\' D
proof
  assume A1:G is independent & G={A,B,C,D,E} &
  A<>B & A<>C & A<>D & A<>E & B<>C & B<>D & B<>E & C<>D & C<>E & D<>E;
    {A,B,C,D,E}={A,B,C} \/ {D,E} by ENUMSET1:49;
  then G={A,B,C,E,D} by A1,ENUMSET1:49;
  hence thesis by A1,Th4;
end;

theorem Th6:
for A,B,C,D,E being set, h being Function,
A',B',C',D',E' being set
st A<>B & A<>C & A<>D & A<>E & B<>C & B<>D & B<>E & C<>D & C<>E & D<>E &
h = (B .--> B') +* (C .--> C') +*
    (D .--> D') +* (E .--> E') +* (A .--> A')
holds h.A = A' & h.B = B' & h.C = C' & h.D = D' & h.E = E'
proof
  let A,B,C,D,E be set;
  let h be Function;
  let A',B',C',D',E' be set;
  assume A1: A<>B & A<>C & A<>D & A<>E & B<>C & B<>D & B<>E & C<>D &
   C<>E & D<>E &
h = (B .--> B') +* (C .--> C') +*
    (D .--> D') +* (E .--> E') +* (A .--> A');
A2:  dom (C .--> C') = {C} by CQC_LANG:5;
A3:  dom (D .--> D') = {D} by CQC_LANG:5;
A4:  dom (E .--> E') = {E} by CQC_LANG:5;
A5:  dom (A .--> A') = {A} by CQC_LANG:5;
A6:  h.A = A'
      proof
          A in dom (A .--> A') by A5,TARSKI:def 1;
        then h.A = (A .--> A').A by A1,FUNCT_4:14;
        hence thesis by CQC_LANG:6;
      end;
          not E in dom (A .--> A') by A1,A5,TARSKI:def 1;
        then A7:h.E=((B .--> B') +* (C .--> C') +*
                 (D .--> D') +* (E .--> E')).E
         by A1,FUNCT_4:12;
A8:  h.B = B'
      proof
          not B in dom (A .--> A') by A1,A5,TARSKI:def 1;
        then A9:h.B=((B .--> B') +* (C .--> C') +*
                 (D .--> D') +* (E .--> E')).B by A1,FUNCT_4:12;
          not B in dom (E .--> E') by A1,A4,TARSKI:def 1;
        then A10:h.B=((B .--> B') +* (C .--> C') +*
                 (D .--> D')).B by A9,FUNCT_4:12;
          not B in dom (D .--> D') by A1,A3,TARSKI:def 1;
then A11:        ((B .--> B') +* (C .--> C') +* (D .--> D')).B=
            ((B .--> B') +* (C .--> C')).B by FUNCT_4:12;
          not B in dom (C .--> C') by A1,A2,TARSKI:def 1;
        then h.B=(B .--> B').B by A10,A11,FUNCT_4:12;
        hence thesis by CQC_LANG:6;
      end;
A12:  h.C = C'
      proof
          not C in dom (A .--> A') by A1,A5,TARSKI:def 1;
        then A13:h.C=((B .--> B') +* (C .--> C') +*
                 (D .--> D') +* (E .--> E')).C by A1,FUNCT_4:12;
          not C in dom (E .--> E') by A1,A4,TARSKI:def 1;
        then A14:h.C=((B .--> B') +* (C .--> C') +* (D .--> D')).C
         by A13,FUNCT_4:12;
          not C in dom (D .--> D') by A1,A3,TARSKI:def 1;
then A15:        ((B .--> B') +* (C .--> C') +* (D .--> D')).C=
            ((B .--> B') +* (C .--> C')).C by FUNCT_4:12;
          C in dom (C .--> C') by A2,TARSKI:def 1;
        then h.C=(C .--> C').C by A14,A15,FUNCT_4:14;
        hence thesis by CQC_LANG:6;
      end;
A16:  h.D = D'
      proof
          not D in dom (A .--> A') by A1,A5,TARSKI:def 1;
        then A17:h.D=((B .--> B') +* (C .--> C') +*
                 (D .--> D') +* (E .--> E')).D by A1,FUNCT_4:12;
          not D in dom (E .--> E') by A1,A4,TARSKI:def 1;
        then A18:h.D=((B .--> B') +* (C .--> C') +* (D .--> D')).D
         by A17,FUNCT_4:12;
          D in dom (D .--> D') by A3,TARSKI:def 1;
        then h.D=(D .--> D').D by A18,FUNCT_4:14;
        hence thesis by CQC_LANG:6;
      end;
        h.E = E'
      proof
          E in dom (E .--> E') by A4,TARSKI:def 1;
        then h.E=(E .--> E').E by A7,FUNCT_4:14;
        hence thesis by CQC_LANG:6;
      end;
  hence thesis by A6,A8,A12,A16;
end;

theorem Th7:
for A,B,C,D,E being set, h being Function,
A',B',C',D',E' being set
st h = (B .--> B') +* (C .--> C') +*
    (D .--> D') +* (E .--> E') +* (A .--> A')
holds dom h = {A,B,C,D,E}
proof
  let A,B,C,D,E be set;
  let h be Function;
  let A',B',C',D',E' be set;
assume
  A1: h =
  (B .--> B') +* (C .--> C') +* (D .--> D') +* (E .--> E') +* (A .--> A');
    dom ((B .--> B') +* (C .--> C'))
      = dom (B .--> B') \/ dom (C .--> C') by FUNCT_4:def 1;
then dom ((B .--> B') +* (C .--> C') +* (D .--> D'))
      = dom (B .--> B') \/ dom (C .--> C') \/
        dom (D .--> D') by FUNCT_4:def 1;
then dom ((B .--> B') +* (C .--> C') +* (D .--> D') +* (E .--> E'))
      = dom (B .--> B') \/ dom (C .--> C') \/ dom (D .--> D') \/
        dom (E .--> E') by FUNCT_4:def 1;
then A2:  dom ((B .--> B') +* (C .--> C') +* (D .--> D') +* (E .--> E') +*
           (A .--> A'))
   = dom (B .--> B') \/ dom (C .--> C') \/ dom (D .--> D') \/ dom (E .--> E')
\/
     dom (A .--> A') by FUNCT_4:def 1;
A3:  dom (B .--> B') = {B} by CQC_LANG:5;
A4:  dom (C .--> C') = {C} by CQC_LANG:5;
A5:  dom (D .--> D') = {D} by CQC_LANG:5;
    dom (E .--> E') = {E} by CQC_LANG:5;
      then dom ((B .--> B') +* (C .--> C') +* (D .--> D') +* (E .--> E') +*
           (A .--> A'))
      = {A} \/ (({B} \/ {C}) \/ {D} \/ {E}) by A2,A3,A4,A5,CQC_LANG:5
     .= {A} \/ ({B,C} \/ {D} \/ {E}) by ENUMSET1:41
     .= {A} \/ ({B,C,D} \/ {E}) by ENUMSET1:43
     .= {A} \/ {B,C,D,E} by ENUMSET1:46
    .= {A,B,C,D,E} by ENUMSET1:47;
  hence thesis by A1;
end;

theorem Th8:
for A,B,C,D,E being set, h being Function,
A',B',C',D',E' being set
st h = (B .--> B') +* (C .--> C') +*
    (D .--> D') +* (E .--> E') +* (A .--> A')
holds rng h = {h.A,h.B,h.C,h.D,h.E}
proof
  let A,B,C,D,E be set;
  let h be Function;
  let A',B',C',D',E' be set;
assume
  h = (B .--> B') +* (C .--> C') +* (D .--> D') +* (E .--> E') +* (A .--> A');
then A1:  dom h = {A,B,C,D,E} by Th7;
      then A2: A in dom h by ENUMSET1:24;
A3:      B in dom h by A1,ENUMSET1:24;
A4:      C in dom h by A1,ENUMSET1:24;
A5:      D in dom h by A1,ENUMSET1:24;
A6:      E in dom h by A1,ENUMSET1:24;
      A7:rng h c= {h.A,h.B,h.C,h.D,h.E}
      proof
        let t be set;
        assume t in rng h;
        then consider x1 being set such that
A8:      x1 in dom h & t = h.x1 by FUNCT_1:def 5;
          now per cases by A1,A8,ENUMSET1:23;
        case x1=A;
        hence thesis by A8,ENUMSET1:24;
        case x1=B;
        hence thesis by A8,ENUMSET1:24;
        case x1=C;
        hence thesis by A8,ENUMSET1:24;
        case x1=D;
        hence thesis by A8,ENUMSET1:24;
        case x1=E;
        hence thesis by A8,ENUMSET1:24;
        end;
        hence thesis;
      end;
        {h.A,h.B,h.C,h.D,h.E} c= rng h
      proof
        let t be set;
        assume A9: t in {h.A,h.B,h.C,h.D,h.E};
          now per cases by A9,ENUMSET1:23;
        case t=h.A;
        hence thesis by A2,FUNCT_1:def 5;
        case t=h.B;
        hence thesis by A3,FUNCT_1:def 5;
        case t=h.C;
        hence thesis by A4,FUNCT_1:def 5;
        case t=h.D;
        hence thesis by A5,FUNCT_1:def 5;
        case t=h.E;
        hence thesis by A6,FUNCT_1:def 5;
        end;
        hence thesis;
      end;
  hence thesis by A7,XBOOLE_0:def 10;
end;

theorem
  for G being Subset of PARTITIONS(Y),
A,B,C,D,E being a_partition of Y, z,u being Element of Y, h being Function
st G is independent & G={A,B,C,D,E} &
A<>B & A<>C & A<>D & A<>E & B<>C & B<>D & B<>E & C<>D & C<>E & D<>E
holds
EqClass(u,B '/\' C '/\' D '/\' E) meets EqClass(z,A)
proof
  let G be Subset of PARTITIONS(Y);
  let A,B,C,D,E be a_partition of Y;
  let z,u be Element of Y;
  let h be Function;
  assume A1:G is independent & G={A,B,C,D,E} &
  A<>B & A<>C & A<>D & A<>E & B<>C & B<>D & B<>E & C<>D & C<>E & D<>E;
      set GG=EqClass(u,B '/\' C '/\' D '/\' E);
      set h = (B .--> EqClass(u,B)) +* (C .--> EqClass(u,C)) +*
              (D .--> EqClass(u,D)) +* (E .--> EqClass(u,E)) +*
              (A .--> EqClass(z,A));
A2:  dom h = G by A1,Th7;
A3:  h.A = EqClass(z,A) by A1,Th6;
A4:  h.B = EqClass(u,B) by A1,Th6;
A5:  h.C = EqClass(u,C) by A1,Th6;
A6:  h.D = EqClass(u,D) by A1,Th6;
A7:  h.E = EqClass(u,E) by A1,Th6;
A8:  for d being set st d in G holds h.d in d
      proof
        let d be set;
        assume A9: d in G;
          now per cases by A1,A9,ENUMSET1:23;
        case A10:d=A;
          h.A=EqClass(z,A) by A1,Th6;
        hence thesis by A10;
        case A11:d=B;
          h.B=EqClass(u,B) by A1,Th6;
        hence thesis by A11;
        case A12:d=C;
          h.C=EqClass(u,C) by A1,Th6;
        hence thesis by A12;
        case A13:d=D;
          h.D=EqClass(u,D) by A1,Th6;
        hence thesis by A13;
        case A14:d=E;
          h.E=EqClass(u,E) by A1,Th6;
        hence thesis by A14;
        end;
        hence thesis;
      end;
        A in dom h by A1,A2,ENUMSET1:24;
then A15:  h.A in rng h by FUNCT_1:def 5;
        B in dom h by A1,A2,ENUMSET1:24;
then A16:  h.B in rng h by FUNCT_1:def 5;
        C in dom h by A1,A2,ENUMSET1:24;
then A17:  h.C in rng h by FUNCT_1:def 5;
        D in dom h by A1,A2,ENUMSET1:24;
then A18:  h.D in rng h by FUNCT_1:def 5;
        E in dom h by A1,A2,ENUMSET1:24;
then A19:  h.E in rng h by FUNCT_1:def 5;
      A20:rng h = {h.A,h.B,h.C,h.D,h.E} by Th8;
        rng h c= bool Y
      proof
        let t be set;
        assume A21: t in rng h;
          now per cases by A20,A21,ENUMSET1:23;
        case t=h.A;
        then t=EqClass(z,A) by A1,Th6;
        hence thesis;
        case t=h.B;
        hence thesis by A4;
        case t=h.C;
        hence thesis by A5;
        case t=h.D;
        hence thesis by A6;
        case t=h.E;
        hence thesis by A7;
        end;
        hence thesis;
      end;
      then reconsider FF=rng h as Subset-Family of Y by SETFAM_1:def 7;
        (Intersect FF)<>{} by A1,A2,A8,BVFUNC_2:def 5;
      then consider m being set such that
A22:    m in Intersect FF by XBOOLE_0:def 1;
        m in meet FF by A15,A22,CANTOR_1:def 3;
      then A23:
 m in h.A & m in h.B & m in h.C & m in h.D & m in h.E
   by A15,A16,A17,A18,A19,SETFAM_1:def 1;
        GG = EqClass(u,B '/\' C '/\' D) /\ EqClass(u,E) by BVFUNC14:1;
then A24:      GG = EqClass(u,B '/\' C) /\ EqClass(u,D) /\ EqClass(u,E)
        by BVFUNC14:1;
        m in EqClass(u,B) /\ EqClass(u,C) by A4,A5,A23,XBOOLE_0:def 3;
      then m in EqClass(u,B) /\ EqClass(u,C) /\ EqClass(u,D) by A6,A23,XBOOLE_0
:def 3;
      then m in EqClass(u,B) /\ EqClass(u,C) /\ EqClass(u,D) /\ EqClass(u,E)
        by A7,A23,XBOOLE_0:def 3;
      then m in EqClass(u,B) /\ EqClass(u,C) /\ EqClass(u,D) /\ EqClass(u,E)
        /\ EqClass(z,A) by A3,A23,XBOOLE_0:def 3;
      then EqClass(u,B) /\ EqClass(u,C) /\ EqClass(u,D) /\ EqClass(u,E) meets
        EqClass(z,A) by XBOOLE_0:4;
  hence thesis by A24,BVFUNC14:1;
end;

theorem
  for G being Subset of PARTITIONS(Y),
A,B,C,D,E being a_partition of Y, z,u being Element of Y
st G is independent & G={A,B,C,D,E} &
A<>B & A<>C & A<>D & A<>E & B<>C & B<>D & B<>E & C<>D & C<>E & D<>E &
EqClass(z,C '/\' D '/\' E)=EqClass(u,C '/\' D '/\' E)
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 be a_partition of Y;
  let z,u be Element of Y;
  assume A1: G is independent & G={A,B,C,D,E} &
  A<>B & A<>C & A<>D & A<>E & B<>C & B<>D & B<>E & C<>D & C<>E & D<>E &
             EqClass(z,C '/\' D '/\' E)=EqClass(u,C '/\' D '/\' E);
      then A2:CompF(B,G) = A '/\' C '/\' D '/\' E by Th2;
      set H=EqClass(z,CompF(B,G));
      set I=EqClass(z,A); set GG=EqClass(u,(((B '/\' C) '/\' D) '/\' E));
      set h = (B .--> EqClass(u,B)) +* (C .--> EqClass(u,C)) +*
              (D .--> EqClass(u,D)) +* (E .--> EqClass(u,E)) +*
              (A .--> EqClass(z,A));
A3:  dom h = G by A1,Th7;
A4:  h.A = EqClass(z,A) by A1,Th6;
A5:  h.B = EqClass(u,B) by A1,Th6;
A6:  h.C = EqClass(u,C) by A1,Th6;
A7:  h.D = EqClass(u,D) by A1,Th6;
A8:  h.E = EqClass(u,E) by A1,Th6;
A9:  for d being set st d in G holds h.d in d
      proof
        let d be set;
        assume A10: d in G;
          now per cases by A1,A10,ENUMSET1:23;
        case A11:d=A;
          h.A=EqClass(z,A) by A1,Th6;
        hence thesis by A11;
        case A12:d=B;
          h.B=EqClass(u,B) by A1,Th6;
        hence thesis by A12;
        case A13:d=C;
          h.C=EqClass(u,C) by A1,Th6;
        hence thesis by A13;
        case A14:d=D;
          h.D=EqClass(u,D) by A1,Th6;
        hence thesis by A14;
        case A15:d=E;
          h.E=EqClass(u,E) by A1,Th6;
        hence thesis by A15;
        end;
        hence thesis;
      end;
        A in dom h by A1,A3,ENUMSET1:24;
then A16:  h.A in rng h by FUNCT_1:def 5;
        B in dom h by A1,A3,ENUMSET1:24;
then A17:  h.B in rng h by FUNCT_1:def 5;
        C in dom h by A1,A3,ENUMSET1:24;
then A18:  h.C in rng h by FUNCT_1:def 5;
        D in dom h by A1,A3,ENUMSET1:24;
then A19:  h.D in rng h by FUNCT_1:def 5;
        E in dom h by A1,A3,ENUMSET1:24;
then A20:  h.E in rng h by FUNCT_1:def 5;
      A21:rng h = {h.A,h.B,h.C,h.D,h.E} by Th8;
        rng h c= bool Y
      proof
        let t be set;
        assume A22: t in rng h;
          now per cases by A21,A22,ENUMSET1:23;
        case t=h.A;
        then t=EqClass(z,A) by A1,Th6;
        hence thesis;
        case t=h.B;
        hence thesis by A5;
        case t=h.C;
        hence thesis by A6;
        case t=h.D;
        hence thesis by A7;
        case t=h.E;
        hence thesis by A8;
        end;
        hence thesis;
      end;
      then reconsider FF=rng h as Subset-Family of Y by SETFAM_1:def 7;
        (Intersect FF)<>{} by A1,A3,A9,BVFUNC_2:def 5;
      then consider m being set such that
A23:    m in Intersect FF by XBOOLE_0:def 1;
        m in meet FF by A16,A23,CANTOR_1:def 3;
      then A24:
 m in h.A & m in h.B & m in h.C & m in h.D & m in h.E
   by A16,A17,A18,A19,A20,SETFAM_1:def 1;
        GG = EqClass(u,B '/\' C '/\' D) /\ EqClass(u,E) by BVFUNC14:1;
then A25:      GG = EqClass(u,B '/\' C) /\ EqClass(u,D) /\ EqClass(u,E)
        by BVFUNC14:1;
        m in EqClass(u,B) /\ EqClass(u,C) by A5,A6,A24,XBOOLE_0:def 3;
      then m in EqClass(u,B) /\ EqClass(u,C) /\ EqClass(u,D) by A7,A24,XBOOLE_0
:def 3;
      then m in EqClass(u,B) /\ EqClass(u,C) /\ EqClass(u,D) /\ EqClass(u,E)
        by A8,A24,XBOOLE_0:def 3;
      then m in EqClass(u,B) /\ EqClass(u,C) /\ EqClass(u,D) /\
          EqClass(u,E) /\ EqClass(z,A)
        by A4,A24,XBOOLE_0:def 3;
      then GG /\ I <> {} by A25,BVFUNC14:1;
      then consider p being set such that
A26:   p in GG /\ I by XBOOLE_0:def 1;
      reconsider p as Element of Y by A26;
      set K=EqClass(p,C '/\' D '/\' E);
      A27:p in K & K in C '/\' D '/\' E by T_1TOPSP:def 1;
A28:      p in GG & p in I by A26,XBOOLE_0:def 3;
      then p in I /\ K by A27,XBOOLE_0:def 3;
      then I /\ K in INTERSECTION(A,C '/\' D '/\' E) & not (I /\ K in {{}})
by SETFAM_1:def 5,TARSKI:def 1;
then A29: I /\ K in INTERSECTION(A,C '/\' D '/\' E) \ {{}} by XBOOLE_0:def 4;
      set L=EqClass(z,C '/\' D '/\' E);
        GG = EqClass(u,((B '/\' (C '/\' D)) '/\' E)) by PARTIT1:16;
      then GG = EqClass(u,B '/\' (C '/\' D '/\' E)) by PARTIT1:16;
then A30: GG c= L by A1,BVFUNC11:3;
A31: p in GG by A26,XBOOLE_0:def 3;
        p in EqClass(p,C '/\' D '/\' E) by T_1TOPSP:def 1;
      then K meets L by A30,A31,XBOOLE_0:3;
      then K=L by T_1TOPSP:9;
      then A32:z in K by T_1TOPSP:def 1;
        z in I by T_1TOPSP:def 1;
      then A33:z in I /\ K by A32,XBOOLE_0:def 3;
A34:z in H by T_1TOPSP:def 1;
        A '/\' (C '/\' D '/\' E) = A '/\' (C '/\' D) '/\' E by PARTIT1:16;
      then A '/\' (C '/\' D '/\' E) = A '/\' C '/\' D '/\' E by PARTIT1:16;
      then I /\ K in CompF(B,G) by A2,A29,PARTIT1:def 4;
      then I /\ K = H or I /\ K misses H by EQREL_1:def 6;
      then p in H by A27,A28,A33,A34,XBOOLE_0:3,def 3;
      then p in GG /\ H by A31,XBOOLE_0:def 3;
      then GG meets H by XBOOLE_0:4;
  hence thesis by A1,Th1;
end;

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