let R be Ring; :: thesis: for V being LeftMod of R
for W1, W2 being Submodule of V st V is_the_direct_sum_of W1,W2 holds
W2 is Linear_Compl of W1
let V be LeftMod of R; :: thesis: for W1, W2 being Submodule of V st V is_the_direct_sum_of W1,W2 holds
W2 is Linear_Compl of W1
let W1, W2 be Submodule 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 A1: V is_the_direct_sum_of W2,W1 by VECTSP_5:41;
then W1 is with_Linear_Compl ;
hence W2 is Linear_Compl of W1 by Def19, A1; :: thesis: verum