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 VectSp-like Abelian add-associative right_zeroed VectSpStr of GF
for L being Linear_Combination of V holds L - L = ZeroLC V
let V be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr of GF; :: thesis: for L being Linear_Combination of V holds L - L = ZeroLC V
let L be Linear_Combination of V; :: thesis: L - L = ZeroLC V
let v be Element of V; :: according to VECTSP_6:def 10 :: thesis: (L - L) . v = (ZeroLC V) . v
thus (L - L) . v = (L . v) - (L . v) by Th73
.= 0. GF by RLVECT_1:28
.= (ZeroLC V) . v by Th22 ; :: thesis: verum