
theorem Th7:
  for C being CategoryStr, a,b being Object of C
  for f being Morphism of a,b st Hom(a,b) <> {} & for g being
  Morphism of a,b holds f = g holds Hom(a,b) = {f}
proof
  let C be CategoryStr, a,b be Object of C;
  let f be Morphism of a,b such that
A1: Hom(a,b) <> {} and
A2: for g being Morphism of a,b holds f = g;
  for x being object holds x in Hom(a,b) iff x = f
  proof
    let x be object;
    thus x in Hom(a,b) implies x = f
    proof
      assume x in Hom(a,b);
      then x in {f where f is morphism of C : ex f1,f2 being morphism of C st
      a = f1 & b = f2 & f |> f1 & f2 |> f} by CAT_7:def 1;
      then consider g being morphism of C such that
A3:   x = g and
A4:   ex f1,f2 being morphism of C st
      a = f1 & b = f2 & g |> f1 & f2 |> g;
      g in {f where f is morphism of C : ex f1,f2 being morphism of C st
      a = f1 & b = f2 & f |> f1 & f2 |> f} by A4;
      then g in Hom(a,b) by CAT_7:def 1;
      then g is Morphism of a,b by CAT_7:def 3;
      hence thesis by A2,A3;
    end;
    thus thesis by A1,CAT_7:def 3;
  end;
  hence thesis by TARSKI:def 1;
end;
