theorem
  A1 \/ B1 = { x1 : x1 in A1 or x1 in B1 }
proof
  thus A1 \/ B1 c= { x1 : x1 in A1 or x1 in B1 }
  proof
    let a be object;
    assume
A1: a in A1 \/ B1;
    then reconsider x = a as Element of X1;
    x in A1 or x in B1 by A1,XBOOLE_0:def 3;
    hence thesis;
  end;
  let a be object;
  assume a in { x1 : x1 in A1 or x1 in B1 };
  then ex x1 st a = x1 & (x1 in A1 or x1 in B1);
  hence thesis by XBOOLE_0:def 3;
end;
