theorem Th137:
  for G being strict addGroup, H being strict Subgroup of G holds
  H is normal Subgroup of G iff Normalizer H = G
proof
  let G be strict addGroup, H be strict Subgroup of G;
  thus H is normal Subgroup of G implies Normalizer H = G
  proof
    assume
A1: H is normal Subgroup of G;
    now
      let a be Element of G;
      H * a = H by A1,Def13;
      hence a in Normalizer H by Th134;
    end;
    hence thesis by Th62;
  end;
  assume
A2: Normalizer H = G;
  H is normal
  proof
    let a be Element of G;
    a in Normalizer H by A2;
    then ex b being Element of G st b = a & H * b = H by Th134;
    hence thesis;
  end;
  hence thesis;
end;
