let V be RealLinearSpace; :: 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, Th19, RLSUB_2:46;
hence v |-- W1,W2 = [(0. V),v] by A1, Th18, RLSUB_2:46; :: thesis: verum