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 = - (L . v)
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 = - (L . v)
let v be Element of V; :: thesis: for L being Linear_Combination of V holds (- L) . v = - (L . v)
let L be Linear_Combination of V; :: thesis: (- L) . v = - (L . v)
thus (- L) . v = (- (1. GF)) * (L . v) by Def9
.= - (L . v) by Lm2 ; :: thesis: verum