
theorem Th42:
  for L being non degenerated comRing, x being Element of L, n
  being Element of NAT, p being Polynomial of L st n < len p holds eval(
  poly_shift(p,n),x) = x*eval(poly_shift(p,n+1),x) + p.n
proof
  let L be non degenerated comRing, x being Element of L, n be Element of NAT,
  p being Polynomial of L such that
A1: n < len p;
  set ps = poly_shift(p,n), ps1 = poly_shift(p,n+1);
  consider f be FinSequence of L such that
A2: eval(ps,x) = Sum f and
A3: len f = len ps and
A4: for k be Element of NAT st k in dom f holds f.k = ps.(k-'1) * (power
  L).(x,k-'1) by POLYNOM4:def 2;
  consider f1 be FinSequence of L such that
A5: eval(ps1,x) = Sum f1 and
A6: len f1 = len ps1 and
A7: for k be Element of NAT st k in dom f1 holds f1.k = ps1.(k-'1) * (
  power L).(x,k-'1) by POLYNOM4:def 2;
  rng f1 c= the carrier of L & dom (x multfield) = the carrier of L by
FUNCT_2:def 1;
  then
A8: x*f1 = (x multfield)*f1 & dom ((x multfield)*f1) = dom f1 by FVSUM_1:def 6
,RELAT_1:27;
A9: 1_L = 1.L;
  now
A10: n+1 <= len p by A1,NAT_1:13;
A11: len ps1 +1+n = len ps1 + (n+1) .= len p by A10,Th41
      .= len ps + n by A1,Th41;
    thus len f = len f;
A12: len <*p.n*> = 1 by FINSEQ_1:40;
A13: len ((x*f1)) = len f1 by A8,FINSEQ_3:29;
    hence len (<*p.n*>^(x*f1)) = len f by A3,A6,A11,A12,FINSEQ_1:22;
    let j be Nat such that
A14: j in dom f;
A15: 1 <= j by A14,FINSEQ_3:25;
A16: j <= len f by A14,FINSEQ_3:25;
    per cases by A15,XXREAL_0:1;
    suppose
A17:  j = 1;
A18:  1 in dom <*p.n*> by A12,FINSEQ_3:25;
      thus f.j = ps.(1-'1) * (power L).(x,1-'1) by A4,A14,A17
        .= ps.0 * (power L).(x,1-'1) by XREAL_1:232
        .= ps.0 * (power L).(x,0) by XREAL_1:232
        .= ps.0 * 1.L by A9,GROUP_1:def 7
        .= ps.0
        .= p.(n+(0 qua Nat)) by Def5
        .= <*p.n*>.1
        .= (<*p.n*>^(x*f1)).j by A17,A18,FINSEQ_1:def 7;
    end;
    suppose
A19:  1 < j;
      1-1 <= j-1 by A15,XREAL_1:9;
      then reconsider j1 = j-1 as Element of NAT by INT_1:3;
A20:  1+1 <= j by A19,NAT_1:13;
      then
A21:  1+1-1 <= j-1 by XREAL_1:9;
      then ex j2 being Nat st j1 = j2+1 by NAT_1:6;
      then
A22:  j1-'1+1 = j1 by NAT_D:34;
      j = j1+1;
      then
A23:  j1 = j-'1 by NAT_D:34;
      j-1 <= len f1 + 1-1 by A3,A6,A11,A16,XREAL_1:9;
      then
A24:  j1 in dom f1 by A21,FINSEQ_3:25;
      then reconsider f1j = f1.j1 as Element of L by FINSEQ_2:11;
      thus f.j = ps.(j-'1) * (power L).(x,j-'1) by A4,A14
        .= p.(n+j1) * (power L).(x,j1) by A23,Def5
        .= p.((n+1)+(j1-'1)) * (((power L).(x,j1-'1)) * x) by A22,GROUP_1:def 7
        .= x*(p.((n+1)+(j1-'1)) * (power L).(x,j1-'1)) by GROUP_1:def 3
        .= x*(ps1.(j1-'1) * (power L).(x,j1-'1)) by Def5
        .= x*f1j by A7,A24
        .= (x*f1).j1 by A8,A24,FVSUM_1:50
        .= (<*p.n*>^(x*f1)).j by A3,A6,A11,A12,A13,A16,A20,FINSEQ_1:23;
    end;
  end;
  then x*(Sum f1) = Sum (x*f1) & f = <*p.n*>^(x*f1) by FINSEQ_2:9,FVSUM_1:73;
  hence thesis by A2,A5,FVSUM_1:72;
end;
