
theorem Th52:
  for C1,C2 be category, f being morphism of C1 [x] C2
  ex f1 being morphism of C1, f2 being morphism of C2 st f = [f1,f2]
  proof
    let C1,C2 be category;
    let f be morphism of C1 [x] C2;
    per cases;
    suppose
A1:  C1 is non empty & C2 is non empty;
      take pr1(C1,C2).f, pr2(C1,C2).f;
      thus f = [pr1(C1,C2).f, pr2(C1,C2).f] by A1,Def23;
    end;
    suppose
A2:   C1 is empty or C2 is empty;
      set f1 = the morphism of C1;
      set f2 = the morphism of C2;
      take f1,f2;
      f = {} by A2,SUBSET_1:def 1
      .= the morphism of C1 [x] C2 by A2,SUBSET_1:def 1;
      hence thesis by A2,Def23;
    end;
  end;
