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)]
A3: 0. V in W2 by RLSUB_1:25;
v + (0. V) = v by RLVECT_1:10;
hence v |-- W1,W2 = [v,(0. V)] by A1, A2, A3, Th15; :: thesis: verum