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 A2: v |-- (W1,W2) = [v1,v2] ; :: thesis: v |-- (W2,W1) = [v2,v1]
then A3: (v |-- (W1,W2)) `1 = v1 ;
then A4: v1 in W1 by A1, RLSUB_2:def 6;
A5: (v |-- (W1,W2)) `2 = v2 by A2;
then A6: v2 in W2 by A1, RLSUB_2:def 6;
v = v2 + v1 by A1, A3, A5, RLSUB_2:def 6;
hence v |-- (W2,W1) = [v2,v1] by A1, A4, A6, Th2, RLSUB_2:38; :: thesis: verum