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 W1 holds
v |-- (W1,W2) = [v,(0. 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 W1 holds
v |-- (W1,W2) = [v,(0. V)] )
assume A1: V is_the_direct_sum_of W1,W2 ; :: thesis: for v being VECTOR of V st v in W1 holds
v |-- (W1,W2) = [v,(0. V)]
let v be VECTOR of V; :: thesis: ( v in W1 implies v |-- (W1,W2) = [v,(0. V)] )
assume A2: v in W1 ; :: thesis: v |-- (W1,W2) = [v,(0. V)]
( 0. V in W2 & v + (0. V) = v ) by RLSUB_1:17;
hence v |-- (W1,W2) = [v,(0. V)] by A1, A2, Th2; :: thesis: verum