let K be non empty right_complementable Abelian add-associative right_zeroed well-unital distributive associative doubleLoopStr ; :: thesis: for V being VectSp of K
for W1, W2 being Subspace of V st V is_the_direct_sum_of W1,W2 holds
for v being Vector of V st v in W2 holds
v |-- (W1,W2) = [(0. V),v]
let V be VectSp of K; :: thesis: for W1, W2 being Subspace of V st V is_the_direct_sum_of W1,W2 holds
for v being Vector of V st v in W2 holds
v |-- (W1,W2) = [(0. V),v]
let W1, W2 be Subspace of V; :: thesis: ( V is_the_direct_sum_of W1,W2 implies for v being Vector of V st v in W2 holds
v |-- (W1,W2) = [(0. V),v] )
assume A1: V is_the_direct_sum_of W1,W2 ; :: thesis: for v being Vector of V st v in W2 holds
v |-- (W1,W2) = [(0. V),v]
let v be Vector of V; :: thesis: ( v in W2 implies v |-- (W1,W2) = [(0. V),v] )
assume v in W2 ; :: thesis: v |-- (W1,W2) = [(0. V),v]
then v |-- (W2,W1) = [v,(0. V)] by A1, Th9, VECTSP_5:41;
hence v |-- (W1,W2) = [(0. V),v] by A1, Th8, VECTSP_5:41; :: thesis: verum