let V be RealLinearSpace; :: 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 VECTOR of V; :: according to RLVECT_2:def 9 :: thesis: (L - L) . v = (ZeroLC V) . v

thus (L - L) . v = (L . v) - (L . v) by Th54

.= (ZeroLC V) . v by Th20 ; :: thesis: verum

let L be Linear_Combination of V; :: thesis: L - L = ZeroLC V

let v be VECTOR of V; :: according to RLVECT_2:def 9 :: thesis: (L - L) . v = (ZeroLC V) . v

thus (L - L) . v = (L . v) - (L . v) by Th54

.= (ZeroLC V) . v by Th20 ; :: thesis: verum