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

theorem Th11:
  Union A1 \+\ Union A2 c= Union (A1 (\+\) A2)
proof
A1: (Union A1 \ Union A2) c= Union (A1 (\) A2) & (Union A2 \ Union A1) c=
  Union (A2 (\) A1) by Th10;
  Union (A1 (\) A2) \/ Union (A2 (\) A1) = Union ((A1 (\) A2) (\/) (A2 (\)
  A1)) by Th9
    .= Union (A1 (\+\) A2) by Th3;
  hence thesis by A1,XBOOLE_1:13;
end;
