let V be RealLinearSpace; :: thesis: for W1, W2 being Subspace of V st V is_the_direct_sum_of W1,W2 holds
W2 is Linear_Compl of W1
let W1, W2 be Subspace of V; :: thesis: ( V is_the_direct_sum_of W1,W2 implies W2 is Linear_Compl of W1 )
assume V is_the_direct_sum_of W1,W2 ; :: thesis: W2 is Linear_Compl of W1
then ( V is_the_direct_sum_of W2,W1 & RLSStruct(# the carrier of V,the U2 of V,the addF of V,the Mult of V #) = W1 + W2 ) by Def4, Lm16;
hence W2 is Linear_Compl of W1 by Def5; :: thesis: verum