reserve X for set;
reserve G for Group;
reserve H for Subgroup of G;
reserve h,x,y for object;
reserve f for Endomorphism of G;
reserve phi for Automorphism of G;
reserve K for characteristic Subgroup of G;

theorem Th64:
  for G being Group
  for a,x being Element of G
  holds x in Normalizer a iff ex h being Element of G st x = h & a |^ h = a
proof
  let G be Group;
  let a,x be Element of G;
  A1: x in Normalizer{a} iff ex h being Element of G st x = h & {a} |^ h = {a}
  by GROUP_3:129;
  {a} |^ x = {a |^ x} by GROUP_3:37;
  then x in Normalizer{a} iff a |^ x = a by A1,ZFMISC_1:3;
  hence thesis;
end;
