
theorem Th10:
  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 W2 holds v |-- (W1,W2) = [0.V,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;
  assume
A1: V is_the_direct_sum_of W1,W2;
  let v be Vector of V;
  assume v in W2;
  then v |-- (W2,W1) = [v,0.V] by A1,Th9,VECTSP_5:41;
  hence thesis by A1,Th8,VECTSP_5:41;
end;
