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
for L being Linear_Combination of V holds
( L . v = 0. GF iff not v in Carrier L )

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
for L being Linear_Combination of V holds
( L . v = 0. GF iff not v in Carrier L )

let v be Element of V; :: thesis: for L being Linear_Combination of V holds
( L . v = 0. GF iff not v in Carrier L )

let L be Linear_Combination of V; :: thesis: ( L . v = 0. GF iff not v in Carrier L )
thus ( L . v = 0. GF implies not v in Carrier L ) :: thesis: ( not v in Carrier L implies L . v = 0. GF )
proof
assume A1: L . v = 0. GF ; :: thesis: not v in Carrier L
assume v in Carrier L ; :: thesis: contradiction
then ex u being Element of V st
( u = v & L . u <> 0. GF ) ;
hence contradiction by A1; :: thesis: verum
end;
assume not v in Carrier L ; :: thesis: L . v = 0. GF
hence L . v = 0. GF ; :: thesis: verum