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

for L being Linear_Compl of W

for v being VECTOR of V

for t being Element of [: the carrier of V, the carrier of V:] st (t `1) + (t `2) = v & t `1 in W & t `2 in L holds

t = v |-- (W,L)

let W be Subspace of V; :: thesis: for L being Linear_Compl of W

for v being VECTOR of V

for t being Element of [: the carrier of V, the carrier of V:] st (t `1) + (t `2) = v & t `1 in W & t `2 in L holds

t = v |-- (W,L)

let L be Linear_Compl of W; :: thesis: for v being VECTOR of V

for t being Element of [: the carrier of V, the carrier of V:] st (t `1) + (t `2) = v & t `1 in W & t `2 in L holds

t = v |-- (W,L)

V is_the_direct_sum_of W,L by Th35;

hence for v being VECTOR of V

for t being Element of [: the carrier of V, the carrier of V:] st (t `1) + (t `2) = v & t `1 in W & t `2 in L holds

t = v |-- (W,L) by Def6; :: thesis: verum

for L being Linear_Compl of W

for v being VECTOR of V

for t being Element of [: the carrier of V, the carrier of V:] st (t `1) + (t `2) = v & t `1 in W & t `2 in L holds

t = v |-- (W,L)

let W be Subspace of V; :: thesis: for L being Linear_Compl of W

for v being VECTOR of V

for t being Element of [: the carrier of V, the carrier of V:] st (t `1) + (t `2) = v & t `1 in W & t `2 in L holds

t = v |-- (W,L)

let L be Linear_Compl of W; :: thesis: for v being VECTOR of V

for t being Element of [: the carrier of V, the carrier of V:] st (t `1) + (t `2) = v & t `1 in W & t `2 in L holds

t = v |-- (W,L)

V is_the_direct_sum_of W,L by Th35;

hence for v being VECTOR of V

for t being Element of [: the carrier of V, the carrier of V:] st (t `1) + (t `2) = v & t `1 in W & t `2 in L holds

t = v |-- (W,L) by Def6; :: thesis: verum