
theorem Th47:
  for C1,C2 being category holds (C1 ->OrdC1)[|x|](C2 ->OrdC1),
  pr1(C1 ->OrdC1,C2 ->OrdC1),pr2(C1 ->OrdC1,C2 ->OrdC1) is_product_of C1,C2
  proof
    let C1,C2 be category;
    set F1 = C1 ->OrdC1;
    set F2 = C2 ->OrdC1;
A1: pr1(F1,F2) is covariant by CAT_7:52;
A2: pr2(F1,F2) is covariant by CAT_7:52;
    for D1 being category, G1 being Functor of D1,C1, G2 being Functor of D1,C2
    st G1 is covariant & G2 is covariant holds
    ex H being Functor of D1,F1 [|x|] F2 st H is covariant &
    pr1(F1,F2) (*) H = G1 & pr2(F1,F2) (*) H = G2 &
    for H1 being Functor of D1,F1 [|x|] F2 st H1 is covariant &
    pr1(F1,F2) (*) H1 = G1 & pr2(F1,F2) (*) H1 = G2 holds H = H1
    proof
      let D1 be category;
      let G1 be Functor of D1,C1;
      let G2 be Functor of D1,C2;
      assume
A3:   G1 is covariant & G2 is covariant;
A4:  F1 [|x|] F2, pr1(F1,F2), pr2(F1,F2) is_pullback_of F1,F2 by CAT_7:52;
      F1 (*) G1 = F2 (*) G2 by A3,Th29;
      then consider H be Functor of D1,F1 [|x|] F2 such that
A5:   H is covariant & pr1(F1,F2) (*) H = G1 & pr2(F1,F2) (*) H = G2
      & for H1 being Functor of D1,F1 [|x|] F2
      st H1 is covariant & pr1(F1,F2) (*) H1 = G1 &
      pr2(F1,F2) (*) H1 = G2 holds H = H1 by A4,A3,A1,A2,CAT_7:def 20;
      take H;
      thus thesis by A5;
    end;
    hence thesis by A1,A2,Def17;
end;
