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 Th13:
  for F be FinSequence of S holds (LS1+LS2) * F = (LS1*F) + (LS2*F)
  proof
    let p be FinSequence of S;
    A1: len(LS1*p)=len p by FINSEQ_2:33;
    A2: len(LS2*p)=len p by FINSEQ_2:33;
    then reconsider L1p=LS1*p,L2p=LS2*p as Element of len p-tuples_on REAL
      by A1,FINSEQ_2:92;
    A3: len((LS1+LS2)*p)=len p by FINSEQ_2:33;
    A4: now let k be Nat;
          assume A5: 1<=k & k<=len p;
          then k in dom((LS1+LS2)*p) by A3,FINSEQ_3:25;
          then A6: ((LS1+LS2)*p).k=(LS1+LS2).(p.k) by FUNCT_1:12;
          k in dom L1p by A1,A5,FINSEQ_3:25;
          then A7: p.k in dom LS1 & L1p.k=LS1.(p.k) by FUNCT_1:11,12;
          k in dom L2p by A2,A5,FINSEQ_3:25;
          then A8: L2p.k=LS2.(p.k) by FUNCT_1:12;
          dom LS1=the carrier of S by FUNCT_2:def 1;
          hence ((LS1+LS2)*p).k = L1p.k+L2p.k by A6,A8,A7,RLVECT_2:def 10
                                   .= (L1p+L2p).k by RVSUM_1:11;
        end;
    len(L1p+L2p)=len p by CARD_1:def 7;
    hence thesis by A3,A4;
  end;
