reserve S, T, Y for non empty TopSpace,
  s, s1, s2, s3 for Point of S,
  t, t1, t2, t3 for Point of T,
  l1, l2 for Path of [s1,t1],[s2,t2],
  H for Homotopy of l1 ,l2;

theorem
  for S, T being non empty pathwise_connected TopSpace, s1, s2 being
Point of S, t1, t2 being Point of T holds pi_1([:S,T:],[s1,t1]), product <*pi_1
  (S,s2),pi_1(T,t2)*> are_isomorphic
proof
  let S, T be non empty pathwise_connected TopSpace, s1, s2 be Point of S, t1,
  t2 be Point of T;
  pi_1(S,s1),pi_1(S,s2) are_isomorphic & pi_1(T,t1),pi_1(T,t2)
  are_isomorphic by TOPALG_3:33;
  then
A1: product <*pi_1(S,s1),pi_1(T,t1)*>, product <*pi_1(S,s2),pi_1(T,t2)*>
  are_isomorphic by Th6;
  pi_1([:S,T:],[s1,t1]), product <*pi_1(S,s1),pi_1(T,t1)*> are_isomorphic
  by Th30;
  hence thesis by A1,GROUP_6:67;
end;
