theorem
  c is_a_product_wrt p1,p2 & d is_a_product_wrt q1,q2 & cod p1 = cod q1
  & cod p2 = cod q2 implies c,d are_isomorphic
proof
  assume that
A1: c is_a_product_wrt p1,p2 and
A2: d is_a_product_wrt q1,q2 and
A3: cod p1 = cod q1 & cod p2 = cod q2;
  set I = {0,{0}}, F = (0,{0})-->(p1,p2), F9 = (0,{0})-->(q1,q2);
A4: c is_a_product_wrt F & d is_a_product_wrt F9 by A1,A2,Th54;
  cods F = (0,{0})-->(cod q1,cod q2) by A3,Th7
    .= cods F9 by Th7;
  hence thesis by A4,Th48;
end;
