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 A1: Carrier L c= A by Def5;
Carrier (L * a) c= Carrier L by Th42;
then Carrier (L * a) c= A by A1;
hence L * a is Linear_Combination of A by Def5; :: thesis: verum