
theorem Th21:
  for X being set, A being Subset-Family of X holds
  UniCl FinMeetCl UniCl A = UniCl FinMeetCl A
proof
  let X be set, A be Subset-Family of X;
  per cases;
  suppose
A1: A = {};
    then
A2: FinMeetCl A = {X} by Th17;
    UniCl A = {{}} by A1,Th16;
    then
A3: FinMeetCl UniCl A = {{},X} by Th11;
    UniCl FinMeetCl A = {X,{}} by A2,Th11;
    hence thesis by A3,Th18;
  end;
  suppose A <> {};
    then reconsider A as non empty Subset-Family of X;
A4: UniCl FinMeetCl UniCl A c= UniCl UniCl FinMeetCl A
    proof
      let x be object;
      assume x in UniCl FinMeetCl UniCl A;
      then consider Y being Subset-Family of X such that
A5:   Y c= FinMeetCl UniCl A and
A6:   x = union Y by CANTOR_1:def 1;
      Y c= UniCl FinMeetCl A
      proof
        let y be object;
   reconsider yy=y as set by TARSKI:1;
        assume y in Y;
        then consider Z being Subset-Family of X such that
A7:     Z c= UniCl A and
A8:     Z is finite and
A9:     y = Intersect Z by A5,CANTOR_1:def 3;
        per cases;
        suppose Z = {};
          then y = X by A9,SETFAM_1:def 9;
          then
A10:      y in FinMeetCl A by CANTOR_1:8;
          FinMeetCl A c= UniCl FinMeetCl A by CANTOR_1:1;
          hence thesis by A10;
        end;
        suppose
A11:      Z <> {};
          then
A12:      y = meet Z by A9,SETFAM_1:def 9;
          set G = {meet rng f where f is Element of Funcs(Z,A):
          for z being set st z in Z holds f.z c= z};
A13:      G c= FinMeetCl A
          proof
            let a be object;
            assume a in G;
            then consider f being Element of Funcs(Z,A) such that
A14:        a = meet rng f and for z being set st z in Z holds f.z c= z;
            reconsider B = rng f as Subset-Family of X by XBOOLE_1:1;
            reconsider B as Subset-Family of X;
            Intersect B = a by A11,A14,SETFAM_1:def 9;
            hence thesis by A8,CANTOR_1:def 3;
          end;
          then reconsider G as Subset-Family of X by XBOOLE_1:1;
          reconsider G as Subset-Family of X;
          union G = yy
          proof
            hereby
              let a be object;
              assume a in union G;
              then consider b being set such that
A15:          a in b and
A16:          b in G by TARSKI:def 4;
              consider f being Element of Funcs(Z,A) such that
A17:          b = meet rng f and
A18:          for z being set st z in Z holds f.z c= z by A16;
A19:          dom f = Z by FUNCT_2:def 1;
              reconsider B = rng f as Subset-Family of X by XBOOLE_1:1;
              reconsider B as Subset-Family of X;
              b c= yy
              proof
                let c be object;
                assume
A20:            c in b;
                now
                  let d be set;
                  assume
A21:              d in Z;
                  then f.d in B by A19,FUNCT_1:def 3;
                  then
A22:              b c= f.d by A17,SETFAM_1:3;
A23:              f.d c= d by A18,A21;
                  c in f.d by A20,A22;
                  hence c in d by A23;
                end;
                hence thesis by A11,A12,SETFAM_1:def 1;
              end;
              hence a in yy by A15;
            end;
            let a be object;
            assume
A24:        a in yy;
            defpred P[object,object] means
             ex A,B being set st A = $1 & B = $2 & a in B & B c= A;
A25:        now
              let z be object;
              assume
A26:          z in Z;
              reconsider zz=z as set by TARSKI:1;
A27:          a in zz by A12,A24,SETFAM_1:def 1,A26;
              consider C being Subset-Family of X such that
A28:          C c= A and
A29:          z = union C by A7,A26,CANTOR_1:def 1;
              consider w being set such that
A30:          a in w and
A31:          w in C by A27,A29,TARSKI:def 4;
               reconsider w as object;
              take w;
              thus w in A by A28,A31;
              thus P[z, w] by A29,A30,A31,ZFMISC_1:74;
            end;
            consider f being Function such that
A32:        dom f = Z & rng f c= A and
A33:        for z being object st z in Z holds P[z, f.z]
                 from FUNCT_1:sch 6(A25);
            reconsider f as Element of Funcs(Z,A) by A32,FUNCT_2:def 2;
            for z being set st z in Z holds f.z c= z
             proof let z be set;
              assume z in Z;
               then P[z, f.z] by A33;
              hence thesis;
             end;
            then
A34:        meet rng f in G;
            now
              thus rng f <> {} by A11;
              let y be set;
              assume y in rng f;
              then consider z being object such that
A35:               z in Z & y = f.z by A32,FUNCT_1:def 3;
               P[z,f.z] by A33,A35;
              hence a in y by A35;
            end;
            then a in meet rng f by SETFAM_1:def 1;
            hence thesis by A34,TARSKI:def 4;
          end;
          hence thesis by A13,CANTOR_1:def 1;
        end;
      end;
      hence thesis by A6,CANTOR_1:def 1;
    end;
    FinMeetCl A c= FinMeetCl UniCl A by CANTOR_1:1,14;
    then
A36: UniCl FinMeetCl A c= UniCl FinMeetCl UniCl A by CANTOR_1:9;
    UniCl UniCl FinMeetCl A = UniCl FinMeetCl A by Th15;
    hence thesis by A4,A36;
  end;
end;
