
theorem Th69:
  OrdC 1 is with_exponential_objects
  proof
    set C = OrdC 1;
    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 o1,o2 being Object of C, f1 being morphism of C holds
    f1 is Morphism of o1,o2
    proof
      let o1,o2 be Object of C;
      let f1 be morphism of C;
A3:   dom f1 = f by A1,TARSKI:def 1 .= o1 by A1,TARSKI:def 1;
      cod f1 = f by A1,TARSKI:def 1 .= o2 by A1,TARSKI:def 1;
      then f1 in Hom(o1,o2) by A3,CAT_7:20;
      hence f1 is Morphism of o1,o2 by CAT_7:def 3;
    end;
    for a,b being Object of C holds ex c being Object of C,
    e being Morphism of c [x] a,b st
    Hom(c [x] a, b)<>{} & c,e is_exponent_of a,b
    proof
      let a,b be Object of C;
      set c = a;
      take c;
      reconsider e = f as Morphism of c [x] a,b by A2;
      take e;
      c [x] a = f by A1,TARSKI:def 1 .= b by A1,TARSKI:def 1;
      hence
A4:   Hom(c [x] a, b)<>{};
      for d being Object of C, f1 being Morphism of d [x] a,b
      st Hom(d [x] a,b)<>{} holds
      Hom(d,c) <> {} & ex h being Morphism of d,c st f1 = e * (h [x] id- a) &
      for h1 being Morphism of d,c st f1 = e * (h1[x]id- a) holds h = h1
      proof
        let d be Object of C;
        let f1 be Morphism of d [x] a,b;
        assume Hom(d [x] a,b)<>{};
        reconsider h = f as Morphism of d,a by A2;
        d = f by A1,TARSKI:def 1 .= a by A1,TARSKI:def 1;
        hence Hom(d,c) <> {};
        take h;
        thus f1 = f by A1,TARSKI:def 1 .= e * (h [x] id- a) by A1,TARSKI:def 1;
        let h1 be Morphism of d,c;
        assume f1 = e * (h1[x]id- a);
        thus h = h1 by A1,TARSKI:def 1;
      end;
      hence c,e is_exponent_of a,b by A4,Def29;
    end;
    hence thesis;
  end;
