let K be non empty right_complementable Abelian add-associative right_zeroed associative well-unital distributive 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, Th10, VECTSP_5:51;
hence v |-- W1,W2 = [(0. V),v] by A1, Th9, VECTSP_5:51; :: thesis: verum