let R be Ring; :: thesis: 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; :: thesis: 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; :: thesis: 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; :: thesis: ( 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; :: thesis: V is_the_direct_sum_of W,L
hence V is_the_direct_sum_of W,L by VECTSP_5:41; :: thesis: verum