
theorem Th54:
  for C1,C2 being category, f1,g1 being morphism of C1,
      f2,g2 being morphism of C2
  holds [f1,f2] |> [g1,g2] iff f1 |> g1 & f2 |> g2
  proof
    let C1,C2 be category;
    let f1,g1 be morphism of C1;
    let f2,g2 be morphism of C2;
    per cases;
    suppose
A1:   C1 is non empty & C2 is non empty;
      hereby
        assume
A2:    [f1,f2] |> [g1,g2];
A3:     pr1(C1,C2).[f1,f2] = f1 & pr1(C1,C2).[g1,g2] = g1 by A1,Def23;
        pr1(C1,C2) is multiplicative by CAT_6:def 25;
        hence f1 |> g1 by A3,A2,CAT_6:def 23;
A4:     pr2(C1,C2).[f1,f2] = f2 & pr2(C1,C2).[g1,g2] = g2 by A1,Def23;
        pr2(C1,C2) is multiplicative by CAT_6:def 25;
        hence f2 |> g2 by A4,A2,CAT_6:def 23;
      end;
      assume f1 |> g1 & f2 |> g2;
      hence [f1,f2] |> [g1,g2] by A1,Lm2;
    end;
    suppose C1 is empty or C2 is empty; hence thesis by CAT_6:1; end;
  end;
