
theorem Th25:
  for C,D being terminal category holds C ~= D
  proof
    let C,D be terminal category;
    ex F being Functor of C,D, G being Functor of D,C st
    F is covariant & G is covariant & G (*) F = id C & F (*) G = id D
    proof
      consider F be Functor of C,D such that
A1:   F is covariant & for F1 being Functor of C,D st F1 is covariant holds
      F = F1 by Def4;
      consider G be Functor of D,C such that
A2:   G is covariant & for G1 being Functor of D,C st G1 is covariant holds
      G = G1 by Def4;
      take F,G;
      thus F is covariant & G is covariant by A1,A2;
      consider F1 be Functor of C,C such that
A3:   F1 is covariant & for F2 being Functor of C,C st F2 is covariant holds
      F1 = F2 by Def4;
      thus G (*) F = F1 by A1,A2,A3,CAT_6:35 .= id C by A3;
      consider G1 be Functor of D,D such that
A4:   G1 is covariant & for G2 being Functor of D,D st G2 is covariant holds
      G1 = G2 by Def4;
      thus F (*) G = G1 by A1,A2,A4,CAT_6:35 .= id D by A4;
    end;
    hence C ~= D by CAT_6:def 28;
  end;
