reserve C for CatStr;
reserve f,g for Morphism of C;
reserve C for non void non empty CatStr,
  f,g for Morphism of C,
  a,b,c,d for Object of C;

theorem Th7:
  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 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 consider g being Morphism of C such that
A3:   x = g and
A4:   dom(g)=a & cod(g)=b;
      g is Morphism of a,b by A4,Th3;
      hence thesis by A2,A3;
    end;
    thus thesis by A1,Def3;
  end;
  hence thesis by TARSKI:def 1;
end;
