theorem
  f1 <++> f2 = f2 <++> f1
proof
  dom(f1<++>f2) = dom f1 /\ dom f2 by Def45;
  hence
A1: dom(f1<++>f2) = dom(f2<++>f1) by Def45;
  let x be object;
  assume
A2: x in dom(f1<++>f2);
  hence (f1<++>f2).x = f1.x + f2.x by Def45
    .= (f2<++>f1).x by A1,A2,Def45;
end;
