let R be Ring; :: thesis: for V being RightMod of R
for a being Scalar of R
for L being Linear_Combination of V holds Carrier (L * a) c= Carrier L
let V be RightMod of R; :: thesis: for a being Scalar of R
for L being Linear_Combination of V holds Carrier (L * a) c= Carrier L
let a be Scalar of R; :: thesis: for L being Linear_Combination of V holds Carrier (L * a) c= Carrier L
let L be Linear_Combination of V; :: thesis: Carrier (L * a) c= Carrier L
set T = { u where u is Vector of V : (L * a) . u <> 0. R } ;
set S = { v where v is Vector of V : L . v <> 0. R } ;
{ u where u is Vector of V : (L * a) . u <> 0. R } c= { v where v is Vector of V : L . v <> 0. R }

proof

let x be object ; :: according to TARSKI:def 3 :: thesis: ( not x in { u where u is Vector of V : (L * a) . u <> 0. R } or x in { v where v is Vector of V : L . v <> 0. R } )
assume x in { u where u is Vector of V : (L * a) . u <> 0. R } ; :: thesis: x in { v where v is Vector of V : L . v <> 0. R }
then consider u being Vector of V such that
A1: x = u and
A2: (L * a) . u <> 0. R ;
(L * a) . u = (L . u) * a by Def10;
then L . u <> 0. R by A2;
hence x in { v where v is Vector of V : L . v <> 0. R } by A1; :: thesis: verum

end;

hence Carrier (L * a) c= Carrier L ; :: thesis: verum