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 Th24:
  r (*) (LR1+LR2) = (r(*)LR1) + (r(*)LR2)
  proof
    per cases;
    suppose A1: r = 0;
      set Z=ZeroLC(R);
      A2: now let v be Element of R;
            thus (Z+Z).v = Z.v+Z.v by RLVECT_2:def 10
                        .= Z.v+0 by RLVECT_2:20
                        .= Z.v;
          end;
      thus r(*)(LR1+LR2) = Z by A1,Def2
                        .= Z+Z by A2
                        .= (r(*)LR1)+Z by A1,Def2
                        .= (r(*)LR1)+(r(*)LR2) by A1,Def2;
    end;
    suppose A3: r<>0;
      now let v be Element of R;
        thus(r(*)(LR1+LR2)).v = (LR1+LR2).(r"*v) by A3,Def2
                             .= LR1.(r"*v)+LR2.(r"*v) by RLVECT_2:def 10
                             .= (r(*)LR1).v+LR2.(r"*v) by A3,Def2
                             .= (r(*)LR1).v+(r(*)LR2).v by A3,Def2
                             .= ((r(*)LR1)+(r(*)LR2)).v by RLVECT_2:def 10;
      end;
      hence thesis;
    end;
  end;
