
theorem Th8:
  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 v |-- (W2,W1) = [v2,v1]
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,v1,v2 be Vector of V;
  assume
A2: v |-- (W1,W2) = [v1,v2];
  then
A3: (v |-- (W1,W2))`1 = v1;
  then
A4: v1 in W1 by A1,VECTSP_5:def 6;
A5: (v |-- (W1,W2))`2 = v2 by A2;
  then
A6: v2 in W2 by A1,VECTSP_5:def 6;
  v = v2 + v1 by A1,A3,A5,VECTSP_5:def 6;
  hence thesis by A1,A4,A6,Th5,VECTSP_5:41;
end;
