theorem
  f1 <--> f2 = <-> (f2 <--> f1)
proof
  set f = f2<-->f1;
A1: dom(f1<-->f2) = dom f1 /\ dom f2 & dom(f2<-->f1) = dom f2 /\ dom f1 by
Def46;
  hence
A2: dom(f1<-->f2) = dom<->f by Def33;
  let x be object;
  assume
A3: x in dom(f1<-->f2);
  hence (f1<-->f2).x = f1.x-f2.x by Def46
    .= -(f2.x-f1.x) by Th18
    .= -f.x by A1,A3,Def46
    .= (<->f).x by A2,A3,Def33;
end;
