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 Th16:
  Carrier (g+LG) = g + Carrier LG
  proof
    thus Carrier(g+LG)c=g+Carrier LG
    proof
      let x be object such that
      A1: x in Carrier(g+LG);
      reconsider w=x as Element of G by A1;
      A2: (g+LG).w <>0 by A1,RLVECT_2:19;
      A3: g+(w-g) = (w+-g)+g by RLVECT_1:def 11
                 .= w+(g+-g) by RLVECT_1:def 3
                 .= w+0.G by RLVECT_1:def 10
                 .= w;
      (g+LG).w=LG.(w-g) by Def1;
      then w-g in Carrier LG by A2;
      hence thesis by A3;
    end;
    let x be object;
    assume x in g+Carrier LG;
    then consider w be Element of G such that
    A4: x=g+w and
    A5: w in Carrier LG;
    g+w-g = g+w+-g by RLVECT_1:def 11
         .= (-g+g)+w by RLVECT_1:def 3
         .= 0.G+w by RLVECT_1:5
         .= w;
    then A6: (g+LG).(g+w)=LG.w by Def1;
    LG.w<>0 by A5,RLVECT_2:19;
    hence thesis by A4,A6;
  end;
