
theorem Th15:
  for X being set, A being Subset-Family of X holds UniCl UniCl A = UniCl A
proof
  let X be set, A be Subset-Family of X;
  hereby
    let x be object;
   reconsider xx=x as set by TARSKI:1;
    assume x in UniCl UniCl A;
    then consider Y being Subset-Family of X such that
A1: Y c= UniCl A and
A2: x = union Y by CANTOR_1:def 1;
    set Z = {y where y is Subset of X: y in A & y c= xx};
    Z c= bool X
    proof
      let a be object;
      assume a in Z;
      then ex y being Subset of X st a = y & y in A & y c= xx;
      hence thesis;
    end;
    then reconsider Z as Subset-Family of X;
A3: xx = union Z
    proof
      hereby
        let a be object;
        assume a in xx;
        then consider s being set such that
A4:     a in s and
A5:     s in Y by A2,TARSKI:def 4;
        consider t being Subset-Family of X such that
A6:     t c= A and
A7:     s = union t by A1,A5,CANTOR_1:def 1;
        consider u being set such that
A8:     a in u and
A9:     u in t by A4,A7,TARSKI:def 4;
        reconsider u as Subset of X by A9;
A10:    u c= s by A7,A9,ZFMISC_1:74;
        s c= xx by A2,A5,ZFMISC_1:74;
        then u c= xx by A10;
        then u in Z by A6,A9;
        hence a in union Z by A8,TARSKI:def 4;
      end;
      let a be object;
      assume a in union Z;
      then consider u being set such that
A11:  a in u and
A12:  u in Z by TARSKI:def 4;
      ex y being Subset of X st u = y & y in A & y c= xx by A12;
      hence thesis by A11;
    end;
    Z c= A
    proof
      let a be object;
      assume a in Z;
      then ex y being Subset of X st a = y & y in A & y c= xx;
      hence thesis;
    end;
    hence x in UniCl A by A3,CANTOR_1:def 1;
  end;
  thus thesis by CANTOR_1:1;
end;
