reserve i,i1,i2,i3,i4,i5,j,r,a,b,x,y for Integer,
  d,e,k,n for Nat,
  fp,fk for FinSequence of INT,
  f,f1,f2 for FinSequence of REAL,
  p for Prime;
reserve fr for FinSequence of REAL;

theorem Th6:
  len fp = n+2 implies for a being Integer holds ex fr being
  FinSequence of INT, r being Integer st len fr = n+1 & (for x being Element of
  INT holds (Poly-INT fp).x = (x-a)*(Poly-INT fr).x + r) & fp.(n+2) = fr.(n+1)
proof
  assume
A1: len fp = n+2;
  (n+1)+1 in Seg ((n+1)+1) by FINSEQ_1:4;
  then n+2 in dom fp by A1,FINSEQ_1:def 3;
  then reconsider A = fp.(n+2) as Element of INT by FINSEQ_2:11;
  reconsider n1=n+1 as Element of NAT;
  let a be Integer;
  defpred P[Nat,Integer,set] means $3 = fp.(n+2-$1) +a*$2;
A2: for d being Nat st 1 <= d & d < n1 holds for x being Element
  of INT ex y being Element of INT st P[d,x,y]
  proof
    let d be Nat;
    assume that
    1 <= d and
    d < n1;
    let x be Element of INT;
    set y = fp.(n+2-d) +a*x;
    reconsider y as Element of INT by INT_1:def 2;
    take y;
    thus thesis;
  end;
  consider p being FinSequence of INT such that
A3: len p = n1 & (p.1 = A or n1 = 0) & for d being Nat st 1
  <= d & d < n1 holds P[d,p.d,p.(d+1)] from RECDEF_1:sch 4(A2);
  take fr = Rev p;
  take r = fp.1 + a*fr.1;
A4: len fr = n+1 by A3,FINSEQ_5:def 3;
  for x being Element of INT holds (Poly-INT fp).x = (x-a)*(Poly-INT fr).x + r
  proof
    let x be Element of INT;
    deffunc F(Nat) = (fr.$1)*x|^$1;
    deffunc FF(Nat) = a*(fr.$1) * x|^($1-'1);
    consider f1 being FinSequence of INT such that
A5: len f1 = len fp and
A6: for d st d in dom f1 holds f1.d = (fp.d) * x|^(d-'1) and
A7: (Poly-INT fp).x = Sum f1 by Def1;
A8: f1 <> {} by A1,A5;
    then n+2 in dom f1 by A1,A5,FINSEQ_5:6;
    then f1.(n+2)=(fp.(n+2))*x|^((n+1)+1-'1) by A6;
    then
A9: f1.(n+2)=(fp.(n+2))*x|^(n+1) by NAT_D:34;
    f1.1 = (fp.1)*x|^(1-'1) by A6,A8,FINSEQ_5:6;
    then f1.1=(fp.1)*x|^0 by XREAL_1:232;
    then
A10: f1.1 = (fp.1)*1 by NEWTON:4;
    reconsider n as Element of NAT by ORDINAL1:def 12;
    consider f4 being FinSequence such that
A11: len f4 = n+1 & for d being Nat st d in dom f4 holds f4.d=F(d)
    from FINSEQ_1:sch 2;
A12: for d being Nat st d in dom f4 holds f4.d in INT
    proof
      let d be Nat;
      reconsider d1 = d as Element of NAT by ORDINAL1:def 12;
      assume d in dom f4;
      then f4.d1 = (fr.d1)*x|^d1 by A11;
      hence thesis by INT_1:def 2;
    end;
    f4 <> {} by A11;
    then n+1 in dom f4 by A11,FINSEQ_5:6;
    then f4.(n+1)=(fr.(n+1))*x|^(n+1) by A11;
    then
A13: f4.(n+1) = (fp.(n+2))*x|^(n+1) by A3,FINSEQ_5:62;
    reconsider f4 as FinSequence of INT by A12,FINSEQ_2:12;
    consider f5 being FinSequence such that
A14: len f5 = n+1 & for d being Nat st d in dom f5 holds f5.d=FF(d)
    from FINSEQ_1:sch 2;
A15: for d being Nat st d in dom f5 holds f5.d in INT
    proof
      let d be Nat;
      assume d in dom f5;
      then f5.d = a*(fr.d) * x|^(d-'1) by A14;
      hence thesis by INT_1:def 2;
    end;
    f5 <> {} by A14;
    then 1 in dom f5 by FINSEQ_5:6;
    then f5.1 = a*(fr.1) * x|^(1-'1) by A14;
    then f5.1 = a*(fr.1)*x|^0 by XREAL_1:232;
    then
A16: f5.1 = a*(fr.1)*1 by NEWTON:4;
    reconsider f5 as FinSequence of INT by A15,FINSEQ_2:12;
A17: f4 is FinSequence of REAL by FINSEQ_3:117;
    consider f2 being FinSequence of INT such that
A18: len f2 = len fr and
A19: for d st d in dom f2 holds f2.d = (fr.d) * x|^(d-'1) and
A20: (Poly-INT fr).x = Sum f2 by Def1;
    set f3 = (x-a)*f2;
A21: dom f3 = dom f2 by VALUED_1:def 5;
    then
A22: len f3 = len f2 by FINSEQ_3:29;
A23: dom f3 = dom f4 by A4,A18,A11,A21,FINSEQ_3:29;
A24: for k being Element of NAT st k in dom f3 holds f3.k = (fr.k)*x|^k -
    a*(fr.k) * x|^(k-'1)
    proof
      let k be Element of NAT;
      assume
A25:  k in dom f3;
      then
A26:  k >= 1 by FINSEQ_3:25;
A27:  k in dom f2 by A25,VALUED_1:def 5;
      thus f3.k = (x-a)*(f2.k) by A25,VALUED_1:def 5
        .= (x-a)*((fr.k) * x|^(k-'1)) by A19,A27
        .= (fr.k) * (x|^(k-'1)*x) - a*(fr.k) * x|^(k-'1)
        .= (fr.k)*x|^(k-'1+1) - a*(fr.k) * x|^(k-'1) by NEWTON:6
        .= (fr.k)*x|^k - a*(fr.k) * x|^(k-'1) by A26,XREAL_1:235;
    end;
A28: dom f3 = dom f5 by A4,A18,A14,A21,FINSEQ_3:29;
A29: for d st d in dom f3 holds f3.d = f4.d - f5.d
    proof
      let d;
      assume
A30:  d in dom f3;
      then
A31:  f5.d = a*(fr.d) * x|^(d-'1) by A14,A28;
      f4.d = (fr.d)*x|^d by A11,A23,A30;
      hence thesis by A24,A30,A31;
    end;
    f5 is FinSequence of REAL by FINSEQ_3:117;
    then consider f6 being FinSequence of REAL such that
A32: len f6 = len f3 - 1 and
A33: for d st d in dom f6 holds f6.d = f4.d - f5.(d + 1) and
A34: Sum f3 = Sum f6 + f4.(n+1) - f5.1 by A4,A18,A11,A14,A22,A29,A17,Th5;
A35: len f6 <= len f3 by A4,A18,A22,A32,XREAL_1:145;
    then
A36: dom f6 c= dom f3 by FINSEQ_3:30;
A37: for d being Element of NAT st d in dom f6 holds f6.d = f1.(d+1)
    proof
      let d be Element of NAT;
A38:  dom f6 c= dom p by A3,A4,A18,A22,A35,FINSEQ_3:30;
      assume
A39:  d in dom f6;
      then
A40:  d in Seg n by A4,A18,A22,A32,FINSEQ_1:def 3;
      then
A41:  d <= n by FINSEQ_1:1;
      then
A42:  n-d >= 0 by XREAL_1:48;
      then reconsider d9=n-d+1 as Element of NAT by INT_1:3;
      d >= 1 by A40,FINSEQ_1:1;
      then n-d <= n-1 by XREAL_1:10;
      then d9 <= (n-1)+1 by XREAL_1:6;
      then
A43:  d9 < n+1 by XREAL_1:145;
      d9 >= 0+1 by A42,XREAL_1:6;
      then
A44:  p.(d9+1) = fp.(n+2-d9) +a*(p.d9) by A3,A43;
      d < n+1 by A41,XREAL_1:145;
      then
A45:  d+1 in Seg (n+1) by FINSEQ_3:11;
      then
A46:  d+1 in dom f5 by A14,FINSEQ_1:def 3;
      d+0 < n+2 by A41,XREAL_1:8;
      then d+1 in Seg (n+2) by FINSEQ_3:11;
      then
A47:  d+1 in dom f1 by A1,A5,FINSEQ_1:def 3;
A48:  d+1 in dom p by A3,A45,FINSEQ_1:def 3;
      thus f6.d = f4.d - f5.(d + 1) by A33,A39
        .= (fr.d)*x|^d - f5.(d+1) by A11,A23,A36,A39
        .= (fr.d)*x|^d - a*fr.(d+1) * x|^(d+1-'1) by A14,A46
        .= (fr.d)*x|^d - a*(fr.(d+1)) * x|^d by NAT_D:34
        .= ((fr.d) - a*fr.(d+1))* x|^d
        .= (p.((n+1)-d+1) - a*fr.(d+1))* x|^d by A3,A39,A38,FINSEQ_5:58
        .= (p.((n-d+1)+1)-a*p.((n+1)-(d+1)+1))* x|^d by A3,A48,FINSEQ_5:58
        .= fp.(d+1)* x|^(d+1-'1) by A44,NAT_D:34
        .= f1.(d+1) by A6,A47;
    end;
    f1 = <*f1.1*>^f6^<*f1.(n+2)*>
    proof
      set K = <*f1.1*>^f6^<*f1.(n+2)*>;
A49:  for d being Nat st d in dom f1 holds f1.d = K.d
      proof
        let d be Nat;
        assume
A50:    d in dom f1;
        then
A51:    d>=1 by FINSEQ_3:25;
A52:    d<=n+2 by A1,A5,A50,FINSEQ_3:25;
        per cases by A51,A52,XXREAL_0:1;
        suppose
A53:      d=1;
          hence K.d = (<*f1.1*>^(f6^<*f1.(n+2)*>)).1 by FINSEQ_1:32
            .= f1.d by A53,FINSEQ_1:41;
        end;
        suppose
A54:      d>1 & d<n+2;
          then reconsider w=d-1 as Element of NAT by INT_1:3;
          d-1<n+2-1 by A54,XREAL_1:9;
          then
A55:      d-1<=n+1-1 by INT_1:7;
          d-1>=0+1 by A54,INT_1:7,XREAL_1:50;
          then w in Seg n by A55,FINSEQ_1:1;
          then
A56:      w in dom f6 by A4,A18,A22,A32,FINSEQ_1:def 3;
          then
A57:      w in dom (f6^<*f1.(n+2)*>) by FINSEQ_2:15;
          thus K.d = (<*f1.1*>^(f6^<*f1.(n+2)*>)).(w+1) by FINSEQ_1:32
            .= (f6^<*f1.(n+2)*>).w by A57,FINSEQ_3:103
            .= f6.w by A56,FINSEQ_1:def 7
            .= f1.(w+1) by A37,A56
            .= f1.d;
        end;
        suppose
A58:      d=n+2;
          set K1 = <*f1.1*>^f6;
          thus K.d = K.((n+1)+1) by A58
            .= K.(len K1 +1) by A4,A18,A22,A32,FINSEQ_5:8
            .= f1.d by A58,FINSEQ_1:42;
        end;
      end;
      len K = len (<*f1.1*>^(f6^<*f1.(n+2)*>)) by FINSEQ_1:32
        .= 1+len (f6^<*f1.(n+2)*>) by FINSEQ_5:8
        .= 1+len f6 +1 by FINSEQ_2:16
        .= len f1 by A1,A4,A5,A18,A22,A32;
      hence thesis by A49,FINSEQ_3:29;
    end;
    then Sum f1 = Sum (<*f1.1*>^(f6^<*f1.(n+2)*>)) by FINSEQ_1:32
      .= f1.1 + Sum (f6^<*f1.(n+2)*>) by RVSUM_1:76
      .= f1.1 + (Sum f6 + f1.(n+2)) by RVSUM_1:74
      .= Sum ((x-a)*f2) + r by A10,A9,A13,A16,A34
      .= (x-a) * (Poly-INT fr).x + r by A20,RVSUM_1:87;
    hence thesis by A7;
  end;
  hence thesis by A3,FINSEQ_5:62,def 3;
end;
