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 v being Element of V
for L1, L2 being Linear_Combination of V holds (L1 - L2) . v = (L1 . v) - (L2 . v)
let V be non empty right_complementable VectSp-like Abelian add-associative right_zeroed VectSpStr of GF; :: thesis: for v being Element of V
for L1, L2 being Linear_Combination of V holds (L1 - L2) . v = (L1 . v) - (L2 . v)
let v be Element of V; :: thesis: for L1, L2 being Linear_Combination of V holds (L1 - L2) . v = (L1 . v) - (L2 . v)
let L1, L2 be Linear_Combination of V; :: thesis: (L1 - L2) . v = (L1 . v) - (L2 . v)
thus (L1 - L2) . v = (L1 . v) + ((- L2) . v) by Def11
.= (L1 . v) + (- (L2 . v)) by Th67
.= (L1 . v) - (L2 . v) by RLVECT_1:def 12 ; :: thesis: verum