let GF be non empty right_complementable well-unital distributive Abelian add-associative right_zeroed associative doubleLoopStr ; :: thesis: for V being non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF
for A being Subset of V
for L1, L2 being Linear_Combination of V st L1 is Linear_Combination of A & L2 is Linear_Combination of A holds
L1 + L2 is Linear_Combination of A
let V be non empty right_complementable vector-distributive scalar-distributive scalar-associative scalar-unital Abelian add-associative right_zeroed ModuleStr over GF; :: thesis: for A being Subset of V
for L1, L2 being Linear_Combination of V st L1 is Linear_Combination of A & L2 is Linear_Combination of A holds
L1 + L2 is Linear_Combination of A
let A be Subset of V; :: thesis: for L1, L2 being Linear_Combination of V st L1 is Linear_Combination of A & L2 is Linear_Combination of A holds
L1 + L2 is Linear_Combination of A
let L1, L2 be Linear_Combination of V; :: thesis: ( L1 is Linear_Combination of A & L2 is Linear_Combination of A implies L1 + L2 is Linear_Combination of A )
assume ( L1 is Linear_Combination of A & L2 is Linear_Combination of A ) ; :: thesis: L1 + L2 is Linear_Combination of A
then ( Carrier L1 c= A & Carrier L2 c= A ) by Def4;
then A1: (Carrier L1) \/ (Carrier L2) c= A by XBOOLE_1:8;
Carrier (L1 + L2) c= (Carrier L1) \/ (Carrier L2) by Th23;
hence Carrier (L1 + L2) c= A by A1; :: according to VECTSP_6:def 4 :: thesis: verum