let R be Ring; :: thesis: for V being RightMod of R
for a being Scalar of R
for A being Subset of V
for L being Linear_Combination of V st L is Linear_Combination of A holds
L * a is Linear_Combination of A
let V be RightMod of R; :: thesis: for a being Scalar of R
for A being Subset of V
for L being Linear_Combination of V st L is Linear_Combination of A holds
L * a is Linear_Combination of A
let a be Scalar of R; :: thesis: for A being Subset of V
for L being Linear_Combination of V st L is Linear_Combination of A holds
L * a is Linear_Combination of A
let A be Subset of V; :: thesis: for L being Linear_Combination of V st L is Linear_Combination of A holds
L * a is Linear_Combination of A
let L be Linear_Combination of V; :: thesis: ( L is Linear_Combination of A implies L * a is Linear_Combination of A )
assume L is Linear_Combination of A ; :: thesis: L * a is Linear_Combination of A
then ( Carrier (L * a) c= Carrier L & Carrier L c= A ) by Def7, Th58;
then Carrier (L * a) c= A by XBOOLE_1:1;
hence L * a is Linear_Combination of A by Def7; :: thesis: verum