let V be RealLinearSpace; :: thesis: for W being Subspace 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 Subspace 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 Th35;

hence for v being VECTOR of V holds

( (v |-- (W,L)) `1 in W & (v |-- (W,L)) `2 in L ) by Def6; :: thesis: verum

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 Subspace 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 Th35;

hence for v being VECTOR of V holds

( (v |-- (W,L)) `1 in W & (v |-- (W,L)) `2 in L ) by Def6; :: thesis: verum