theorem Th12:
  for o being Object of A st G1 is_transformable_to G2 holds (q*F1
  )!o = q!(F1.o)
proof
  let o be Object of A;
  assume
A1: G1 is_transformable_to G2;
  then G1*F1 is_transformable_to G2*F1 by Th10;
  hence (q*F1)!o = (q*F1).o by FUNCTOR2:def 4
    .= q!(F1.o) by A1,Def2;
end;
