let R be Ring; :: thesis: for V being RightMod of R
for L being Linear_Combination of V holds L * (0. R) = ZeroLC V
let V be RightMod of R; :: thesis: for L being Linear_Combination of V holds L * (0. R) = ZeroLC V
let L be Linear_Combination of V; :: thesis: L * (0. R) = ZeroLC V
let v be Vector of V; :: according to RMOD_4:def 8 :: thesis: (L * (0. R)) . v = (ZeroLC V) . v
thus (L * (0. R)) . v = (L . v) * (0. R) by Def10
.= 0. R
.= (ZeroLC V) . v by Th18 ; :: thesis: verum