let V be Z_Module; :: thesis: for W being with_Linear_Compl Submodule of V
for L being Linear_Compl of W
for v being VECTOR of V holds ((v |-- (W,L)) `1) + ((v |-- (W,L)) `2) = v
let W be with_Linear_Compl Submodule of V; :: thesis: for L being Linear_Compl of W
for v being VECTOR of V holds ((v |-- (W,L)) `1) + ((v |-- (W,L)) `2) = v
let L be Linear_Compl of W; :: thesis: for v being VECTOR of V holds ((v |-- (W,L)) `1) + ((v |-- (W,L)) `2) = v
V is_the_direct_sum_of W,L by Th124;
hence for v being VECTOR of V holds ((v |-- (W,L)) `1) + ((v |-- (W,L)) `2) = v by Def20; :: thesis: verum