
theorem Th55:
  for C1,C2 being category, f1,g1 being morphism of C1,
      f2,g2 being morphism of C2 st f1 |> g1 & f2 |> g2 holds
  [f1,f2](*)[g1,g2] = [f1(*)g1,f2(*)g2]
  proof
    let C1,C2 be category;
    let f1,g1 be morphism of C1;
    let f2,g2 be morphism of C2;
    assume
A1: f1 |> g1;
    then
A2: C1 is non empty by CAT_6:1;
    assume
A3: f2 |> g2;
    then
A4: C2 is non empty by CAT_6:1;
A5: [f1,f2] |> [g1,g2] by A1,A3,Th54;
A6: pr1(C1,C2).([f1,f2](*)[g1,g2])
    = (pr1(C1,C2).[f1,f2])(*)(pr1(C1,C2).[g1,g2]) by A5,Th13
    .= f1(*)(pr1(C1,C2).[g1,g2]) by A2,A4,Def23
    .= f1(*)g1 by A2,A4,Def23;
    pr2(C1,C2).([f1,f2](*)[g1,g2])
    = (pr2(C1,C2).[f1,f2])(*)(pr2(C1,C2).[g1,g2]) by A5,Th13
    .= f2(*)(pr2(C1,C2).[g1,g2]) by A2,A4,Def23
    .= f2(*)g2 by A2,A4,Def23;
    hence [f1,f2](*)[g1,g2] = [f1(*)g1,f2(*)g2] by A2,A4,A6,Def23;
  end;
