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

theorem
  A (\+\) A1 = (A (\) A1) (\/) (A1 (\) A)
proof
  let n be Element of NAT;
  thus (A (\+\) A1).n = A \+\ A1.n by Def9
    .= (A (\) A1).n \/ (A1.n \ A) by Def7
    .= (A (\) A1).n \/ (A1 (\) A).n by Def8
    .= ((A (\) A1) (\/) (A1 (\) A)).n by Def2;
end;
