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