reserve I for set,
  x,x1,x2,y,z for set,
  A for non empty set;
reserve C,D for Category;
reserve a,b,c,d for Object of C;
reserve f,g,h,i,j,k,p1,p2,q1,q2,i1,i2,j1,j2 for Morphism of C;
reserve f for Morphism of a,b,
        g for Morphism of b,a;
reserve g for Morphism of b,c;
reserve f,g for Morphism of C;

theorem
 for f being Morphism of d,c
  st c is_a_product_wrt p1,p2 & dom f = d & cod f = c & f is invertible
 holds d is_a_product_wrt p1(*)f,p2(*)f
proof
 let f be Morphism of d,c such that
A1: c is_a_product_wrt p1,p2 and
A2: dom f = d & cod f = c & f is invertible;
  c is_a_product_wrt (0,{0})-->(p1,p2) by A1,Th54;
  then d is_a_product_wrt ((0,{0})-->(p1,p2))*f by A2,Th51;
  then d is_a_product_wrt (0,{0})-->(p1(*)f,p2(*)f) by Th14;
  hence thesis by Th54;
end;
