reserve n,m,k for Nat,
  x,X for set,
  A for Subset of X,
  A1,A2 for SetSequence of X;

theorem Th41:
  Union (A1 (\) A) = Union A1 \ A
proof
  thus Union (A1 (\) A) c= Union A1 \ A
  proof
    let x be object;
    assume
A1: x in Union (A1 (\) A);
A2: ex k st x in A1.k & not x in A
    proof
      consider k such that
A3:   x in (A1 (\) A).k by A1,PROB_1:12;
      x in A1.k \ A by A3,Def8;
      then x in A1.k & not x in A by XBOOLE_0:def 5;
      hence thesis;
    end;
    then x in Union A1 by PROB_1:12;
    hence thesis by A2,XBOOLE_0:def 5;
  end;
  let x be object;
  assume
A4: x in Union A1 \ A;
  then
A5: not x in A by XBOOLE_0:def 5;
A6: x in Union A1 by A4,XBOOLE_0:def 5;
  ex k st x in (A1 (\) A).k
  proof
    consider k such that
A7: x in A1.k by A6,PROB_1:12;
    x in A1.k \ A by A5,A7,XBOOLE_0:def 5;
    then x in (A1 (\) A).k by Def8;
    hence thesis;
  end;
  hence thesis by PROB_1:12;
end;
