theorem Th32:
  G is commutative implies H is commutative
proof
  assume
A1: G is commutative;
  now
    let a,b be Element of H;
    carr(H) c= carr(G) by Th23;
    then reconsider a9 = a, b9 = b as Element of G;
    thus a*b = a9*b9 by Th25
      .= b9*a9 by A1
      .= b*a by Th25;
  end;
  hence thesis;
end;
