
theorem Th57:
  for C1,C2 being non empty category, D1,D2 being category,
      F1 being Functor of C1,D1, F2 being Functor of C2,D2,
      c1 being morphism of C1, c2 being morphism of C2
  st F1 is covariant & F2 is covariant holds (F1[x]F2).[c1,c2] = [F1.c1,F2.c2]
  proof
    let C1,C2 be non empty category;
    let D1,D2 be category;
    let F1 be Functor of C1,D1;
    let F2 be Functor of C2,D2;
    let c1 be morphism of C1;
    let c2 be morphism of C2;
    assume
A1: F1 is covariant & F2 is covariant;
A2: D1 is not empty & D2 is not empty by A1,CAT_6:31;
A3: F1 [x] F2 is covariant by A1,Def22;
A4: F1.c1 = F1.(pr1(C1,C2).[c1,c2]) by Def23
    .= (F1(*)pr1(C1,C2)).[c1,c2] by A1,CAT_6:34
    .= (pr1(D1,D2)(*)(F1 [x] F2)).[c1,c2] by A1,Def22
    .= pr1(D1,D2).((F1 [x] F2).[c1,c2]) by A3,CAT_6:34;
    F2.c2 = F2.(pr2(C1,C2).[c1,c2]) by Def23
    .= (F2(*)pr2(C1,C2)).[c1,c2] by A1,CAT_6:34
    .= (pr2(D1,D2)(*)(F1 [x] F2)).[c1,c2] by A1,Def22
    .= pr2(D1,D2).((F1 [x] F2).[c1,c2]) by A3,CAT_6:34;
    hence (F1 [x] F2).[c1,c2] = [F1.c1,F2.c2] by A4,A2,Def23;
  end;
