let V be RealLinearSpace; :: thesis: for L1, L2 being Linear_Combination of V holds Sum (L1 - L2) = (Sum L1) - (Sum L2)

let L1, L2 be Linear_Combination of V; :: thesis: Sum (L1 - L2) = (Sum L1) - (Sum L2)

thus Sum (L1 - L2) = (Sum L1) + (Sum (- L2)) by Th1

.= (Sum L1) + (- (Sum L2)) by Th3

.= (Sum L1) - (Sum L2) by RLVECT_1:def 11 ; :: thesis: verum

let L1, L2 be Linear_Combination of V; :: thesis: Sum (L1 - L2) = (Sum L1) - (Sum L2)

thus Sum (L1 - L2) = (Sum L1) + (Sum (- L2)) by Th1

.= (Sum L1) + (- (Sum L2)) by Th3

.= (Sum L1) - (Sum L2) by RLVECT_1:def 11 ; :: thesis: verum