theorem Th134:
  for H being strict Subgroup of G holds x in Normalizer H iff
  ex h st x = h & H |^ h = H
proof
  let H be strict Subgroup of G;
  thus x in Normalizer H implies ex h st x = h & H |^ h = H
  proof
    assume x in Normalizer H;
    then consider a such that
A1: x = a and
A2: carr H |^ a = carr H by Th129;
    H |^ a = H by A2,Def6;
    hence thesis by A1;
  end;
  given h such that
A3: x = h and
A4: H |^ h = H;
  carr H |^ h = carr H by A4,Def6;
  hence thesis by A3,Th129;
end;
