 reserve x,y,z for object,
   i,j,k,l,n,m for Nat,
   D,E for non empty set;
 reserve M for Matrix of D;
 reserve L for Matrix of E;
 reserve k,t,i,j,m,n for Nat,
   D for non empty set;
 reserve V for free Z_Module;
 reserve a for Element of INT.Ring,
   W for Element of V;
 reserve KL1,KL2,KL3 for Linear_Combination of V,
   X for Subset of V;
 reserve V for finite-rank free Z_Module,
   W for Element of V;
 reserve KL1,KL2,KL3 for Linear_Combination of V,
   X for Subset of V;
 reserve s for FinSequence,
   V1,V2,V3 for finite-rank free Z_Module,
   f,f1,f2 for Function of V1,V2,
   g for Function of V2,V3,
   b1 for OrdBasis of V1,
   b2 for OrdBasis of V2,
   b3 for OrdBasis of V3,
   v1,v2 for Vector of V2,
   v,w for Element of V1;
 reserve p2,F for FinSequence of V1,
   p1,d for FinSequence of INT.Ring,
   KL for Linear_Combination of V1;

theorem Th9:
  for a being Element of V1, F being FinSequence of V1,
  G being FinSequence of INT.Ring st len F = len G &
  for k for v being Element of INT.Ring st
  k in dom F & v = G.k holds F.k = v * a
  holds Sum(F) = Sum(G) * a
  proof
    let a be Element of V1;
    let F be FinSequence of V1;
    let G be FinSequence of INT.Ring;
    defpred P[Nat] means for H being FinSequence of V1,
    I being FinSequence of INT.Ring
    st len H = len I & len H = $1 &
    (for k for v be Element of INT.Ring st k in dom H & v = I.k holds
    H.k = v * a )
    holds Sum(H) = Sum(I) * a;
    A1: for n st P[n] holds P[n+1]
    proof
      let n;
      assume
      A2: for H being FinSequence of V1, I being FinSequence of INT.Ring
      st len H = len I & len H = n &
      (for k for v being Element of INT.Ring st k in dom H & v = I.k
      holds H.k = v * a)
      holds Sum(H) = Sum(I) * a;
      let H be FinSequence of V1, I be FinSequence of INT.Ring;
      assume that
      A3: len H = len I and
      A4: len H = n + 1 and
      A5: for k for v being Element of INT.Ring st k in dom H & v = I.k holds
      H.k = v * a;
      reconsider q = I | (Seg n) as FinSequence of INT.Ring by FINSEQ_1:18;
      reconsider p = H | (Seg n) as FinSequence of V1 by FINSEQ_1:18;
      A6: n <= n + 1 by NAT_1:12;
      then
      A7: len p = n by A4,FINSEQ_1:17;
      A8: dom p = Seg n by A4,A6,FINSEQ_1:17;
      A9: len q = n by A3,A4,A6,FINSEQ_1:17;
      A10: dom q = Seg n by A3,A4,A6,FINSEQ_1:17;
      A11:
      now
        len p <= len H by A4,A6,FINSEQ_1:17;
        then
        A12: dom p c= dom H by FINSEQ_3:30;
        let k;
        let v be Element of INT.Ring;
        assume that
        A13: k in dom p and
        A14: v = q.k;
        I.k = q.k by A8,A10,A13,FUNCT_1:47;
        then H.k = v * a by A5,A13,A14,A12;
        hence p.k = v * a by A13,FUNCT_1:47;
      end;
      reconsider n as Element of NAT by ORDINAL1:def 12;
      n + 1 in Seg(n + 1) by FINSEQ_1:4; then
      A15: n + 1 in dom H by A4,FINSEQ_1:def 3;
      then reconsider v1 = H.(n + 1) as Element of V1 by FINSEQ_2:11;
      reconsider v2 = I.(n + 1) as Element of INT.Ring by INT_1:def 2;
      A16: v1 = v2 * a by A5,A15;
      A17: I = q ^<*v2*> by FINSEQ_3:55,A3,A4;
      thus Sum(H) = Sum(p) + v1 by A4,A7,A8,RLVECT_1:38
      .= Sum(q) * a + v2 * a by A2,A7,A9,A11,A16
      .= (Sum(q) + v2) * a by VECTSP_1:def 15
      .= Sum(I) * a by A17,FVSUM_1:71;
    end;
    A17: P[0]
    proof
      let H be FinSequence of V1, I be FinSequence of INT.Ring;
      assume that
      A18: len H = len I and
      A19: len H = 0 and
      for k for v being Element of INT.Ring st k in dom H & v = I.k holds
      H.k = v * a;
      H = <*>(the carrier of V1) by A19;
      then
      A20: Sum(H) = 0.V1 by RLVECT_1:43;
      I = <*>(the carrier of INT.Ring) by A18,A19; then
      Sum I = 0.INT.Ring by RLVECT_1:43; then
      (Sum I) * a = 0.V1 by VECTSP_1:14;
      hence thesis by A20;
    end;
    for n holds P[n] from NAT_1:sch 2(A17,A1);
    hence thesis;
  end;
