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
  r * (s(*)L) = s (*) (r*L)
  proof
    per cases;
    suppose A1: s=0;
      hence r*(s(*)L) = r*ZeroLC(V) by Def2
                         .= r*(0*L) by RLVECT_2:43
                         .= (r*0)*L by RLVECT_2:47
                         .= ZeroLC(V) by RLVECT_2:43
                         .= s(*)(r*L) by A1,Def2;
    end;
    suppose A2: s<>0;
      now let v;
        thus(r*(s(*)L)).v = r*((s(*)L).v) by RLVECT_2:def 11
                         .= r*(L.(s"*v)) by A2,Def2
                         .= (r*L).(s"*v) by RLVECT_2:def 11
                         .= (s(*)(r*L)).v by A2,Def2;
      end;
      hence thesis;
    end;
  end;
