let V be Z_Module; :: thesis: for W being with_Linear_Compl Submodule of V
for L being Linear_Compl of W holds W + L = Z_ModuleStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #)
let W be with_Linear_Compl Submodule of V; :: thesis: for L being Linear_Compl of W holds W + L = Z_ModuleStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #)
let L be Linear_Compl of W; :: thesis: W + L = Z_ModuleStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #)
V is_the_direct_sum_of W,L by Th124;
hence W + L = Z_ModuleStruct(# the carrier of V, the ZeroF of V, the addF of V, the Mult of V #) ; :: thesis: verum