
theorem Th35:
  for K be add-associative right_zeroed right_complementable
Abelian associative well-unital distributive non empty doubleLoopStr for V be
VectSp of K, f be linear-Functional of V for A be Vector of VectQuot(V,Ker f),
  v be Vector of V st A = v+Ker f holds (CQFunctional(f)).A = f.v
proof
  let K be add-associative right_zeroed right_complementable Abelian
associative well-unital distributive non empty doubleLoopStr, V be VectSp of
  K, f be linear-Functional of V;
  let A be Vector of VectQuot(V,Ker f), v be Vector of V;
  assume
A1: A = v+Ker f;
  the carrier of (Ker f) = ker f by Def11;
  hence thesis by A1,Def12;
end;
