reserve G for Group,
  a,b for Element of G,
  m, n for Nat,
  p for Prime;

theorem Th33:
  for G,H being Group holds G,H are_isomorphic &
  G is p-commutative-group-like implies H is p-commutative-group-like
proof
  let G,H be Group;
  assume that
A1: G,H are_isomorphic and
A2: G is p-commutative-group-like;
    let h1,h2 be Element of H;
    consider h being Homomorphism of G,H such that
A3: h is bijective by A1;
    consider a be Element of G such that
A4: h.a = h1 by A3,GROUP_6:58;
    consider b be Element of G such that
A5: h.b = h2 by A3,GROUP_6:58;
    h1 * h2 = h.(a * b) by A4,A5,GROUP_6:def 6;
    then (h1 * h2) |^ p = h.((a * b) |^ p) by GROUP_6:37
                       .= h.(a |^ p * (b |^ p)) by A2
                       .= h.(a |^ p) * h.(b |^ p) by GROUP_6:def 6
                       .= (h.a) |^ p * h.(b |^ p) by GROUP_6:37
                       .= h1 |^ p * (h2 |^ p) by A4,A5,GROUP_6:37;
    hence thesis;
end;
