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

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