 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 LmSign1C:
  for F be FinSequence of F_Real
  st for i being Nat st i in dom F holds F.i in INT
  holds Sum F in INT
  proof
    defpred P[Nat] means
    for F being FinSequence of F_Real
    st len F = $1 &
    for i being Nat st i in dom F holds F.i in INT
    holds Sum F in INT;
    P1: P[0]
    proof
      let F be FinSequence of F_Real;
      assume AS1: len F = 0 &
      for i being Nat st i in dom F holds F.i in INT;
      F = <*> the carrier of F_Real by AS1;
      then Sum F = 0.F_Real by RLVECT_1:43
      .= 0;
      hence Sum F in INT by INT_1:def 2;
    end;
    P2: for n being Nat st P[n] holds P[n+1]
    proof
      let n be Nat;
      assume AS1: P[n];
      let F be FinSequence of F_Real;
      assume AS2: len F = n+1 &
      for i being Nat st i in dom F holds F.i in INT;
      reconsider  F0 = F| n as FinSequence of F_Real;
      n+1 in Seg (n+1) by FINSEQ_1:4;
      then
      A70: n+1 in dom F by AS2,FINSEQ_1:def 3;
      then F.(n+1) in rng F by FUNCT_1:3;
      then reconsider af = F.(n+1) as Element of F_Real;
      P1: len F0 = n by FINSEQ_1:59,AS2,NAT_1:11;
      then
      P4: dom F0 = Seg n by FINSEQ_1:def 3;
      A9: len F = (len F0) + 1 by AS2,FINSEQ_1:59,NAT_1:11;
      A11: F0 = F | dom F0 by P4,FINSEQ_1:def 16;
      then
      P3: Sum F = Sum F0 + af by AS2,A9,RLVECT_1:38;
      for i being Nat st i in dom F0 holds F0.i in INT
      proof
        let i be Nat;
        assume P40: i in dom F0;
        dom F = Seg (n+1) by AS2,FINSEQ_1:def 3;
        then dom F0 c= dom F by P4,FINSEQ_1:5,NAT_1:11;
        then F.i in INT by AS2,P40;
        hence thesis by A11,P40,FUNCT_1:47;
      end;
      then Sum F0 in INT by P1,AS1;
      then reconsider i1 = Sum F0 as Integer;
      F.(n+1) in INT by A70,AS2;
      then reconsider i2 = af as Integer;
      Sum F = i1 + i2 by P3;
      hence Sum F in INT by INT_1:def 2;
    end;
    X1: for n being Nat holds P[n] from NAT_1:sch 2(P1,P2);
    let F be FinSequence of F_Real;
    assume X2: for i being Nat st i in dom F holds F.i in INT;
    len F is Nat;
    hence Sum F in INT by X1,X2;
  end;
