let GF be non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr ; :: thesis: for V being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF
for v being Element of V holds (ZeroLC V) . v = 0. GF
let V be non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF; :: thesis: for v being Element of V holds (ZeroLC V) . v = 0. GF
let v be Element of V; :: thesis: (ZeroLC V) . v = 0. GF
( Carrier (ZeroLC V) = {} & not v in {} ) by Def3;
hence (ZeroLC V) . v = 0. GF ; :: thesis: verum