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

theorem Th40:
  A \ Union A1 c= Union (A (\) A1)
proof
  let x be object;
  assume
A1: x in A \ Union A1;
  then
A2: not x in Union A1 by XBOOLE_0:def 5;
  now
    let k;
    x in A & not x in A1.k by A1,A2,PROB_1:12,XBOOLE_0:def 5;
    then x in A \ A1.k by XBOOLE_0:def 5;
    hence x in (A (\) A1).k by Def7;
  end;
  then x in (A (\) A1).1;
  hence thesis by PROB_1:12;
end;
