let V be RealLinearSpace; :: thesis: for v being VECTOR of V

for L1, L2 being Linear_Combination of V holds (L1 - L2) . v = (L1 . v) - (L2 . v)

let v be VECTOR 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 Def10

.= (L1 . v) + (- (L2 . v)) by Th49

.= (L1 . v) - (L2 . v) ; :: thesis: verum

for L1, L2 being Linear_Combination of V holds (L1 - L2) . v = (L1 . v) - (L2 . v)

let v be VECTOR 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 Def10

.= (L1 . v) + (- (L2 . v)) by Th49

.= (L1 . v) - (L2 . v) ; :: thesis: verum