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 Th80:
  for G being Group
  for H being Subgroup of G
  holds H is Subgroup of Normalizer H
proof
  let G be Group;
  let H be Subgroup of G;
  A1: for g being Element of G st g in H
  for x being Element of G st x in (carr H) |^ g holds x in carr H
  proof
    let g be Element of G;
    assume B1: g in H;
    let x be Element of G;
    assume x in (carr H) |^ g;
    then consider h being Element of G such that
    B2: x = h |^ g & h in carr H by GROUP_3:41;
    B3: h in H by B2;
    g" in H by B1,GROUP_2:51;
    then g" * h in H by B3, GROUP_2:50;
    then x in H by B1,B2, GROUP_2:50;
    hence x in carr H;
  end;

  for g being Element of G st g in H holds g in Normalizer H
  proof
    let g be Element of G;
    assume B1: g in H;
    for x being Element of G st x in carr H holds x in (carr H) |^ g
    proof
      let x be Element of G;
      thus x in carr H implies x in (carr H) |^ g
      proof
        assume x in carr H;
        then C1: x in H;
        set h = x |^ g";
        g" in H by B1,GROUP_2:51;
        then x * g" in H by C1,GROUP_2:50;
        then g * (x * g") in H by B1,GROUP_2:50;
        then C2: h in (carr H) by GROUP_1:def 3;
        h |^ g = x by GROUP_3:25;
        hence x in (carr H) |^ g by C2,GROUP_3:41;
      end;
    end;
    then (carr H) |^ g c= carr H & carr H c= (carr H) |^ g by A1,B1;
    then (carr H) |^ g = carr H by XBOOLE_0:def 10;
    hence g in Normalizer H by GROUP_3:129;
  end;
  hence H is Subgroup of Normalizer H by GROUP_2:58;
end;
