
theorem Th7:
  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,v1
  ,v2 be Vector of V st v |-- (W1,W2) = [v1,v2] holds v1 in W1 & v2 in W2
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,v1,v2 be Vector of V;
  assume v |-- (W1,W2) = [v1,v2];
  then (v |-- (W1,W2))`1 = v1 & (v |-- (W1,W2))`2 = v2;
  hence thesis by A1,VECTSP_5:def 6;
end;
