
theorem Th1:
  for G be Group, H be normal Subgroup of G, x,y be Element of G st y in H
  holds x * y * x" in H & x * (y * x") in H
  proof
    let G be Group,
        H be normal Subgroup of G,
        x,y be Element of G;
    assume
    A1: y in H;
    x * H = H * x by GROUP_3:117; then
    consider g be Element of G such that
    A2: x * y = g * x & g in H by A1,GROUP_2:103,104;
    (x * y) * x" = g by A2,GROUP_3:1;
    hence thesis by A2,GROUP_1:def 3;
  end;
