
theorem Th24:
  OrdC 1 is with_terminal_objects
  proof
    consider f be morphism of OrdC 1 such that
A1: f is identity & Ob OrdC 1 = {f} & Mor OrdC 1 = {f} by Th15;
A2: for a,b being Object of OrdC 1, f1 being morphism of OrdC 1 holds
    f1 is Morphism of a,b
    proof
      let a,b be Object of OrdC 1;
      let f1 be morphism of OrdC 1;
A3:   dom f1 = f by A1,TARSKI:def 1 .= a by A1,TARSKI:def 1;
      cod f1 = f by A1,TARSKI:def 1 .= b by A1,TARSKI:def 1;
      then f1 in Hom(a,b) by A3,CAT_7:20;
      hence f1 is Morphism of a,b by CAT_7:def 3;
    end;
    reconsider a1 = f as Object of OrdC 1 by A1;
    take a1;
    let b1 be Object of OrdC 1;
    b1 = a1 by A1,TARSKI:def 1;
    hence Hom (b1,a1) <> {};
    reconsider f1 = f as Morphism of b1,a1 by A2;
    take f1;
    let g be Morphism of b1,a1;
    thus f1 = g by A1,TARSKI:def 1;
  end;
