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 in W & (v |-- (W,L)) `2 in L )
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 in W & (v |-- (W,L)) `2 in L )
let L be Linear_Compl of W; :: thesis: for v being Vector of V holds
( (v |-- (W,L)) `1 in W & (v |-- (W,L)) `2 in L )
V is_the_direct_sum_of W,L by Th124;
hence for v being Vector of V holds
( (v |-- (W,L)) `1 in W & (v |-- (W,L)) `2 in L ) by VECTSP_5:def 6; :: thesis: verum