
theorem Th3:
  for A,B being transitive with_units non empty AltCatStr, F1,F2
being Functor of A,B holds F1 is_transformable_to F2 implies for t1,
  t2 being transformation of F1,F2 st for a being Object of A holds t1!a = t2!a
  holds t1 = t2
proof
  let A,B be transitive with_units non empty AltCatStr, F1,F2 be
  Functor of A,B;
  assume
A1: F1 is_transformable_to F2;
  let t1,t2 be transformation of F1,F2;
  assume
A2: for a being Object of A holds t1!a = t2!a;
  now
    let a be object;
    assume a in the carrier of A;
    then reconsider o = a as Object of A;
    thus (t1 qua ManySortedSet of the carrier of A).a = t1!o by A1,Def4
      .= t2!o by A2
      .= (t2 qua ManySortedSet of the carrier of A).a by A1,Def4;
  end;
  hence thesis by PBOOLE:3;
end;
