
theorem Th17:
  for C being non empty category, F1,F2 being covariant Functor of OrdC 2, C,
  f being morphism of OrdC 2 st f is non identity & F1.f = F2.f
  holds F1 = F2
  proof
    let C be non empty category;
    let F1,F2 be covariant Functor of OrdC 2, C;
    let f be morphism of OrdC 2;
    assume
A1: f is non identity;
    assume
A2: F1.f = F2.f;
A3: dom F1 = the carrier of OrdC 2 by FUNCT_2:def 1
    .= dom F2 by FUNCT_2:def 1;
    consider f1 be morphism of OrdC(2) such that
A4: f1 is not identity &
    Ob OrdC 2 = {dom f1, cod f1} & Mor OrdC 2 = {dom f1, cod f1, f1} &
    dom f1, cod f1, f1 are_mutually_distinct by CAT_7:39;
A5: f = dom f1 or f = cod f1 or f = f1 by A4,ENUMSET1:def 1;
    for x being object st x in dom F1 holds F1.x = F2.x
    proof
      let x be object;
      assume x in dom F1;
      then x in the carrier of OrdC 2;
      then
A6:   x in {dom f1, cod f1, f1} by A4,CAT_6:def 1;
      per cases by A6,ENUMSET1:def 1;
      suppose
A7:     x = dom f1;
        thus F1.x = dom(F2.f1) by A2,A5,A1,CAT_6:22,A7,CAT_6:32
        .= F2.x by A7,CAT_6:32;
      end;
      suppose
A8:     x = cod f1;
        thus F1.x = cod(F2.f1) by A2,A5,A1,CAT_6:22,A8,CAT_6:32
        .= F2.x by A8,CAT_6:32;
      end;
      suppose
A9:     x = f1;
        thus F1.x = F2.f1 by A2,A5,A1,CAT_6:22,A9,CAT_6:def 21
        .= F2.x by A9,CAT_6:def 21;
      end;
    end;
    hence F1 = F2 by A3,FUNCT_1:2;
  end;
