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

theorem Th3:
  A1 (\+\) A2 = (A1 (\) A2) (\/) (A2 (\) A1)
proof
  let n be Element of NAT;
  thus (A1 (\+\) A2).n = A1.n \+\ A2.n by Def4
    .= (A1 (\) A2).n \/ (A2.n \ A1.n) by Def3
    .= (A1 (\) A2).n \/ (A2 (\) A1).n by Def3
    .= ((A1 (\) A2) (\/) (A2 (\) A1)).n by Def2;
end;
