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
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 with_Linear_Compl Submodule 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 Th124;
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 VECTSP_5:def 6; :: thesis: verum