 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 LmSign1X:
  for F being FinSequence of F_Real,
  G being FinSequence of INT.Ring
  st F = G
  holds Sum F = Sum G
  proof
    defpred P[Nat] means
    for F being FinSequence of F_Real,
    G being FinSequence of INT.Ring
    st len F = $1 & F = G holds Sum F = Sum G;
    P1: P[0]
    proof
      let F be FinSequence of F_Real,
      G be FinSequence of INT.Ring;
      assume AS1: len F = 0 & F = G;
      then F = <*> the carrier of F_Real;
      then
      P1: Sum F = 0.F_Real by RLVECT_1:43
      .= 0;
      G = <*> REAL by AS1; then
      G = <*>the carrier of INT.Ring; then
      Sum G = 0.INT.Ring by RLVECT_1:43;
      hence thesis by P1;
    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,
      G be FinSequence of INT.Ring;
      assume AS2: len F = n+1 & F = G;
      reconsider F0 = F| n as FinSequence of F_Real;
      n+1 in Seg (n+1) by FINSEQ_1:4;
      then 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;
      F0 = F | dom F0 by P4,FINSEQ_1:def 16; then
      P3: Sum F = Sum F0 + af by AS2,A9,RLVECT_1:38;
      reconsider G0 = G| n as FinSequence of INT.Ring;
      n+1 in Seg (n+1) by FINSEQ_1:4;
      then n+1 in dom G by AS2,FINSEQ_1:def 3;
      then G.(n+1) in rng G by FUNCT_1:3;
      then reconsider bf = G.(n+1) as Element of INT.Ring;
      len G = n + 1 & G0 = G | (Seg n) by AS2,FINSEQ_1:def 16; then
      G = G0^<*bf*> by FINSEQ_3:55;
      then Sum G = Sum G0 + bf by FVSUM_1:71;
      hence Sum F = Sum G by AS1,AS2,P1,P3;
    end;
    X1: for n being Nat holds P[n] from NAT_1:sch 2(P1,P2);
    let F be FinSequence of F_Real,
    G be FinSequence of INT.Ring;
    assume X2: F = G;
    len F is Nat;
    hence thesis by X1,X2;
  end;
