
theorem Th15:
  ex f being morphism of OrdC 1 st f is identity &
  Ob OrdC 1 = {f} & Mor OrdC 1 = {f}
  proof
    consider C be strict preorder category such that
A1: Ob C = 1 and
    for o1,o2 being Object of C st o1 in o2 holds Hom(o1,o2) = {[o1,o2]} and
A2: RelOb C = RelIncl 1 and
A3: Mor C = 1 \/ {[o1,o2] where o1,o2 is Element of 1: o1 in o2}
    by CAT_7:37;
A4: C is 1-ordered by A2,WELLORD1:38,CAT_7:def 14;
    then
A5: C ~= OrdC 1 by CAT_7:38;
    consider F be Functor of C, OrdC 1,
             G be Functor of OrdC 1,C such that
A6: F is covariant & G is covariant and
A7: G (*) F = id C & F (*) G = id OrdC 1 by A4,CAT_7:38,CAT_6:def 28;
A8: 0 in Ob C by A1,CARD_1:49,TARSKI:def 1;
    then reconsider g = 0 as morphism of C;
A9: C is non empty by A1;
    then
A10: g is identity by A8,CAT_6:22;
    set f = F.g;
    take f;
    thus
A11: f is identity by A6,A10,CAT_6:def 22,def 25;
    card Ob OrdC 1 = card 1 by A1,A5,CAT_7:14;
    then consider x be object such that
A12: Ob OrdC 1 = {x} by CARD_2:42;
    f is Object of OrdC 1 by A11,CAT_6:22;
    hence
A13: Ob OrdC 1 = {f} by A12,TARSKI:def 1;
    for x being object holds x in Mor OrdC 1 iff x in {f}
    proof
      let x be object;
      hereby
        assume
A14:    x in Mor OrdC 1;
        then
A15:     x in the carrier of OrdC 1 by CAT_6:def 1;
        reconsider f1 = x as morphism of OrdC 1 by A14;
        per cases;
        suppose f1 is identity;
          then f1 is Object of OrdC 1 by CAT_6:22;
          hence x in {f} by A13;
        end;
        suppose
A16:       f1 is not identity;
A17:       (id (the carrier of OrdC 1)).x = x by A15,FUNCT_1:18;
A18:       F.(G.f1) = (F(*)G).f1 by A6,CAT_6:34
          .= (id the carrier of OrdC 1).f1 by A7,STRUCT_0:def 4
          .= f1 by A17,CAT_6:def 21;
           G.f1 is not identity by A18,A16,CAT_6:def 22,A6,CAT_6:def 25;
          then not G.f1 in 1 by A1,A9,CAT_6:22;
          then G.f1 in {[o1,o2] where o1,o2 is Element of 1: o1 in o2}
          by A3,XBOOLE_0:def 3;
          then consider o1,o2 be Element of 1 such that
A19:      G.f1 = [o1,o2] & o1 in o2;
A20:      o1 = 0 by CARD_1:49,TARSKI:def 1;
          o2 = 0 by CARD_1:49,TARSKI:def 1;
          hence x in {f} by A19,A20;
        end;
      end;
      assume x in {f};
      hence x in Mor OrdC 1 by A13;
    end;
    hence Mor OrdC 1 = {f} by TARSKI:2;
  end;
