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;

theorem Th29:
  for a being Element of G
  for f being Endomorphism of G st a is_inner_wrt f
  holds Ker f = (1).G
proof
  let a be Element of G;
  let f be Endomorphism of G;
  assume A1: for x being Element of G holds f.x = x |^ a;
  for x being Element of G holds x in Ker f implies x in (1).G
  proof
    let x be Element of G;
    assume x in Ker f;
    then 1_G = f.x by GROUP_6:41
            .= x |^ a by A1;
    then x = 1_G by GROUP_3:18;
    hence x in (1).G by GROUP_2:46;
  end;
  then A2: Ker f is Subgroup of (1).G by GROUP_2:58;
  (1).G is Subgroup of Ker f by GROUP_2:65;
  hence Ker f = (1).G by A2,GROUP_2:55;
end;
