theorem Th129:
  x in Normalizer A iff ex h st x = h & A |^ h = A
proof
  thus x in Normalizer A implies ex h st x = h & A |^ h = A
  proof
    assume x in Normalizer A;
    then x in the carrier of Normalizer A;
    then x in {h : A |^ h = A} by Def14;
    hence thesis;
  end;
  given h such that
A1: x = h & A |^ h = A;
  x in {b : A |^ b = A} by A1;
  then x in the carrier of Normalizer A by Def14;
  hence thesis;
end;
