
theorem Th1:
  for X being set, A,B being Subset-Family of X st B = A \ {{}} or
  A = B \/ {{}} holds UniCl A = UniCl B
proof
  let X be set;
  let A,B be Subset-Family of X;
  assume
A1: B = A \ {{}} or A = B \/ {{}};
A2: UniCl A c= UniCl B
  proof
    let x be object;
    assume x in UniCl A;
    then consider Y being Subset-Family of X such that
A3: Y c= A and
A4: x = union Y by CANTOR_1:def 1;
A5: Y \ {{}} c= B
    proof
      let w be object;
      assume
A6:   w in Y \ {{}};
      per cases by A1;
      suppose
A7:     B = A \ {{}};
        w in Y & not w in {{}} by A6,XBOOLE_0:def 5;
        hence thesis by A3,A7,XBOOLE_0:def 5;
      end;
      suppose
A8:     A = B \/ {{}};
        w in Y & not w in {{}} by A6,XBOOLE_0:def 5;
        hence thesis by A3,A8,XBOOLE_0:def 3;
      end;
    end;
    reconsider Z = Y \ {{}} as Subset-Family of X;
    Z \/ {{}} = Y \/ {{}} by XBOOLE_1:39;
    then union (Z \/ {{}}) = union Y \/ union {{}} by ZFMISC_1:78
      .= union Y \/ {} by ZFMISC_1:25
      .= union Y;
    then x = union Z \/ union {{}} by A4,ZFMISC_1:78
      .= union Z \/ {} by ZFMISC_1:25
      .= union Z;
    hence thesis by A5,CANTOR_1:def 1;
  end;
  UniCl B c= UniCl A by A1,CANTOR_1:9,XBOOLE_1:7,36;
  hence thesis by A2;
end;
