theorem
  dom ((x --> y)+*(x .-->z)) = succ x
proof
  thus dom ((x --> y)+*(x .-->z))
      = dom (x --> y) \/ dom (x .-->z) by Def1
    .= x \/ dom (x .-->z)
    .= x \/ {x}
    .= succ x by ORDINAL1:def 1;
end;
