
theorem Th49:
  for C1,C2 being category holds
  C1 [x] C2 is non empty iff (C1 is non empty & C2 is non empty)
  proof
    let C1,C2 be category;
    hereby
      assume
A1:   C1 [x] C2 is non empty;
      pr1(C1,C2) is covariant;
      hence C1 is non empty by A1,CAT_6:31;
      pr2(C1,C2) is covariant;
      hence C2 is non empty by A1,CAT_6:31;
    end;
    assume
A2: C1 is non empty & C2 is non empty;
    reconsider C01 = C1 as non empty category by A2;
    reconsider C02 = C2 as non empty category by A2;
    set D = OrdC 1;
    set G01 = the covariant Functor of D,C01;
    set G02 = the covariant Functor of D,C02;
    reconsider G1 = G01 as Functor of D,C1;
    reconsider G2 = G02 as Functor of D,C2;
    C1 [x] C2,pr1(C1,C2),pr2(C1,C2) is_product_of C1,C2 by Th48;
    then
    consider H be Functor of D,C1 [x] C2 such that
A3: H is covariant & pr1(C1,C2)(*)H = G1 & pr2(C1,C2)(*)H = G2 &
    (for H1 being Functor of D,C1 [x] C2 st
    H1 is covariant & pr1(C1,C2)(*)H1 = G1 & pr2(C1,C2)(*)H1 = G2 holds H = H1)
    by Def17;
    thus C1 [x] C2 is non empty by A3,CAT_6:31;
  end;
