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

theorem Th10:
  Union A1 \ Union A2 c= Union (A1 (\) A2)
proof
  let x be object;
  assume
A1: x in Union A1 \ Union A2;
  then x in Union A1 by XBOOLE_0:def 5;
  then consider n1 being Nat such that
A2: x in A1.n1 by PROB_1:12;
  not x in Union A2 by A1,XBOOLE_0:def 5;
  then not x in A2.n1 by PROB_1:12;
  then x in A1.n1 \ A2.n1 by A2,XBOOLE_0:def 5;
  then x in (A1 (\) A2).n1 by Def3;
  hence thesis by PROB_1:12;
end;
