let R be Ring; for V being LeftMod of R
for W being with_Linear_Compl Submodule of V
for L being Linear_Compl of W holds
( V is_the_direct_sum_of L,W & V is_the_direct_sum_of W,L )
let V be LeftMod of R; for W being with_Linear_Compl Submodule of V
for L being Linear_Compl of W holds
( V is_the_direct_sum_of L,W & V is_the_direct_sum_of W,L )
let W be with_Linear_Compl Submodule of V; for L being Linear_Compl of W holds
( V is_the_direct_sum_of L,W & V is_the_direct_sum_of W,L )
let L be Linear_Compl of W; ( V is_the_direct_sum_of L,W & V is_the_direct_sum_of W,L )
thus
V is_the_direct_sum_of L,W
by Def19; V is_the_direct_sum_of W,L
hence
V is_the_direct_sum_of W,L
by VECTSP_5:41; verum