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 Th14:
  for F be FinSequence of V holds (r*L) * F = r * (L*F)
  proof
   let p be FinSequence of V;
   A1: len((r*L)*p)=len p by FINSEQ_2:33;
   A2: len(L*p)=len p by FINSEQ_2:33;
   then reconsider rLp=(r*L)*p,Lp=L*p as Element of len p-tuples_on REAL
     by A1,FINSEQ_2:92;
   A3: now let k be Nat;
         assume A4: 1<=k & k<=len p;
         then k in dom Lp by A2,FINSEQ_3:25;
         then A5: Lp.k=L.(p.k) & p.k in dom L by FUNCT_1:11,12;
         k in dom rLp by A1,A4,FINSEQ_3:25;
         then A6: rLp.k=(r*L).(p.k) by FUNCT_1:12;
         (r*Lp).k=r*(Lp.k) & dom L=the carrier of V
           by FUNCT_2:def 1,RVSUM_1:44;
         hence rLp.k=(r*Lp).k by A5,A6,RLVECT_2:def 11;
       end;
   len(r*Lp)=len p by CARD_1:def 7;
   hence thesis by A1,A3;
 end;
