 reserve R for domRing;
 reserve p for odd prime Nat, m for positive Nat;
 reserve g for non zero Polynomial of INT.Ring;
 reserve f for Element of the carrier of Polynom-Ring INT.Ring;
 reserve z0 for non zero Element of F_Real;

theorem Th39:
  for x be Element of F_Real st 0 < x <= m holds
  for i be Nat st i in Seg m holds |.eval(~(^((x.(m,p))/.i)),x) .| <= m|^p
    proof
      let x be Element of F_Real;
      set PRI = Polynom-Ring INT.Ring;
      set PRF = Polynom-Ring F_Real;
      set FR = F_Real;
      assume
A1:   0 < x <= m;
      let i be Nat;
      assume
A2:   i in Seg m;
A3:   dom x.(m,p) = Seg (len x.(m,p)) by FINSEQ_1:def 3 .= Seg m by Def2;
      then
A4:   (x.(m,p))/.i = (x.(m,p)).i by A2,PARTFUN1:def 6
        .= (tau(i))|^p by A2,A3,Def2;
A5:   PRI is Subring of PRF by FIELD_4:def 1;
      p in NAT by ORDINAL1:def 12; then
A6:   ^((tau(i))|^p) = (^(tau(i))) |^ p by A5,FIELD_6:19;
      reconsider ui = tau(i) as Polynomial of F_Real by FIELD_4:8;
A7:   (^(tau(i))) |^ p = ui`^p by RINGDER1:37;
      reconsider z0 = -i as Element of FR by XREAL_0:def 1;
A8:   eval(ui,x) = eval(<% In(-i,FR),1.FR %>,x)
        .= z0+(1.FR)*x by POLYNOM5:44
        .= z0 + x;
A9:   eval(~(^((x.(m,p))/.i)),x)
        = (power F_Real).(eval(ui,x),p) by A6,A4,A7,POLYNOM5:22
        .= (z0 + x)|^p by A8,BINOM:def 2;
      |. (i -x) .| = |. -(i -x) .| by COMPLEX1:52
        .= |. z0 + x .|; then
A10:  |. z0 + x .| <= m by Lm15,A1,A2;
      |.(z0 + x)|^p.| <= m|^p
      proof
        per cases;
          suppose
A11:        |. z0 + x .| <> 0; then
A12:        z0 + x <> 0.F_Real;
            |.(z0 + x)|^p.| = |.power(z0 + x,p).| by BINOM:def 2
            .= |.(z0 + x) .| to_power p by A12,Lm14;
            hence thesis by A11,PREPOWER:9,A10;
          end;
          suppose
            |. z0 + x .| = 0; then
            z0 + x = 0.F_Real;
            hence thesis;
          end;
        end;
        hence thesis by A9;
    end;
