let GF be non empty right_complementable associative well-unital distributive Abelian add-associative right_zeroed doubleLoopStr ; :: thesis: for V being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed VectSpStr of GF
for L being Linear_Combination of V holds Sum (- L) = - (Sum L)
let V be non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed VectSpStr of GF; :: thesis: for L being Linear_Combination of V holds Sum (- L) = - (Sum L)
let L be Linear_Combination of V; :: thesis: Sum (- L) = - (Sum L)
(Sum L) + (Sum (- L)) = Sum (L - L) by Th77
.= Sum (ZeroLC V) by Th76
.= 0. V by Lm1 ;
hence Sum (- L) = - (Sum L) by VECTSP_1:16; :: thesis: verum