
theorem Th9:
  for K be add-associative right_zeroed right_complementable
associative Abelian well-unital distributive non empty doubleLoopStr for V be
VectSp of K for W1,W2 be Subspace of V st V is_the_direct_sum_of W1,W2 for v be
  Vector of V st v in W1 holds v |-- (W1,W2) = [v,0.V]
proof
  let K be add-associative right_zeroed right_complementable associative
  Abelian well-unital distributive non empty doubleLoopStr, V be VectSp of K;
  let W1,W2 be Subspace of V such that
A1: V is_the_direct_sum_of W1,W2;
  let v be Vector of V such that
A2: v in W1;
  0.V in W2 & v + 0.V = v by RLVECT_1:4,VECTSP_4:17;
  hence thesis by A1,A2,Th5;
end;
