let V be RealLinearSpace; :: thesis: for W1, W2 being Subspace of V st V is_the_direct_sum_of W1,W2 holds
for v, v1, v2 being VECTOR of V st v |-- W1,W2 = [v1,v2] holds
v |-- W2,W1 = [v2,v1]
let W1, W2 be Subspace of V; :: thesis: ( V is_the_direct_sum_of W1,W2 implies for v, v1, v2 being VECTOR of V st v |-- W1,W2 = [v1,v2] holds
v |-- W2,W1 = [v2,v1] )
assume A1: V is_the_direct_sum_of W1,W2 ; :: thesis: for v, v1, v2 being VECTOR of V st v |-- W1,W2 = [v1,v2] holds
v |-- W2,W1 = [v2,v1]
let v, v1, v2 be VECTOR of V; :: thesis: ( v |-- W1,W2 = [v1,v2] implies v |-- W2,W1 = [v2,v1] )
assume v |-- W1,W2 = [v1,v2] ; :: thesis: v |-- W2,W1 = [v2,v1]
then A2: ( (v |-- W1,W2) `1 = v1 & (v |-- W1,W2) `2 = v2 ) by MCART_1:7;
then A3: ( v1 in W1 & v2 in W2 ) by A1, RLSUB_2:def 6;
v = v2 + v1 by A1, A2, RLSUB_2:def 6;
hence v |-- W2,W1 = [v2,v1] by A1, A3, Th15, RLSUB_2:46; :: thesis: verum