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,Def6A;
    hence thesis by A1;
  end;
  given h such that
A3: x = h and
A4: H * h = H;
  carr H * h = carr H by A4,Def6A;
  hence thesis by A3,Th129;
end;
