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 Th79:
  for G being Group
  for H being Subgroup of G
  for h,n being Element of G
  st h in H & n in Normalizer H
  holds h |^ n in H
proof
  let G be Group;
  let H be Subgroup of G;
  let h,n be Element of G;
  assume A1: h in H;
  assume n in Normalizer H;
  then consider g being Element of G such that
  A2: n" = g & (carr H) |^ g = carr H by GROUP_2:51,GROUP_3:129;
  consider h1 being Element of G such that
  A3: h = h1 |^ g & h1 in carr H by A1,A2,GROUP_3:41;
  h |^ n = (h1 |^ (n")) |^ n by A2,A3
        .= h1 by GROUP_3:25;
  hence h |^ n in H by A3;
end;
