
theorem Th19:
  for X,Y being set, A being Subset-Family of X, B being Subset-Family of Y
  holds (A c= B implies UniCl A c= UniCl B) &
  (A = B implies UniCl A = UniCl B)
proof
  let X,Y be set, A be Subset-Family of X, B be Subset-Family of Y;
A1: now
    let X,Y be set;
    let A be Subset-Family of X, B be Subset-Family of Y such that
A2: A c= B;
    thus 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;
      y c= B by A2,A3;
      then y is Subset-Family of Y by XBOOLE_1:1;
      then ex y being Subset-Family of Y st y c= B & x = union y by A2,A3,A4,
XBOOLE_1:1;
      hence thesis by CANTOR_1:def 1;
    end;
  end;
  hence A c= B implies UniCl A c= UniCl B;
  assume A = B;
  hence UniCl A c= UniCl B & UniCl B c= UniCl A by A1;
end;
