reserve x,y for set,
        r,s for Real,
        S for non empty addLoopStr,
        LS,LS1,LS2 for Linear_Combination of S,
        G for Abelian add-associative right_zeroed right_complementable
          non empty addLoopStr,
        LG,LG1,LG2 for Linear_Combination of G,
        g,h for Element of G,
        RLS for non empty RLSStruct,
        R for vector-distributive scalar-distributive scalar-associative
        scalar-unitalnon empty RLSStruct,
        AR for Subset of R,
        LR,LR1,LR2 for Linear_Combination of R,
        V for RealLinearSpace,
        v,v1,v2,w,p for VECTOR of V,
        A,B for Subset of V,
        F1,F2 for Subset-Family of V,
        L,L1,L2 for Linear_Combination of V;

theorem Th22:
  Carrier (r(*)LR) c= r*Carrier LR
  proof
    let x be object such that
    A1: x in Carrier(r(*)LR);
    reconsider v=x as Element of R by A1;
    A2: (r(*)LR).v<>0 by A1,RLVECT_2:19;
    (0 qua Real)(*)LR=ZeroLC(R) by Def2;
    then A3: r<>0 by A1,RLVECT_2:def 5;
    then (r(*)LR).v=LR.(r"*v) by Def2;
    then A4: r"*v in Carrier LR by A2;
    r*(r"*v) = (r*r")*v by RLVECT_1:def 7
            .= 1*v by A3,XCMPLX_0:def 7
            .= v by RLVECT_1:def 8;
    hence thesis by A4;
  end;
