let V be Z_Module; :: thesis: for L1, L2 being Z_Linear_Combination of V
for A being Subset of V st L1 is Z_Linear_Combination of A & L2 is Z_Linear_Combination of A holds
L1 + L2 is Z_Linear_Combination of A
let L1, L2 be Z_Linear_Combination of V; :: thesis: for A being Subset of V st L1 is Z_Linear_Combination of A & L2 is Z_Linear_Combination of A holds
L1 + L2 is Z_Linear_Combination of A
let A be Subset of V; :: thesis: ( L1 is Z_Linear_Combination of A & L2 is Z_Linear_Combination of A implies L1 + L2 is Z_Linear_Combination of A )
assume ( L1 is Z_Linear_Combination of A & L2 is Z_Linear_Combination of A ) ; :: thesis: L1 + L2 is Z_Linear_Combination of A
then ( Carrier L1 c= A & Carrier L2 c= A ) by Def21;
then A1: (Carrier L1) \/ (Carrier L2) c= A by XBOOLE_1:8;
Carrier (L1 + L2) c= (Carrier L1) \/ (Carrier L2) by Th26;
hence Carrier (L1 + L2) c= A by A1; :: according to ZMODUL02:def 21 :: thesis: verum