 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 Th40:
  for x be Element of F_Real holds
  eval(~(^((tau(0))|^(p-'1))),x) = x|^(p-'1)
    proof
      let x be Element of F_Real;
      set t0 = <% In(-0,F_Real), 1.F_Real %>;
      reconsider u0 = tau(0) as Polynomial of F_Real by FIELD_4:8;
A1:   eval(u0,x) = eval(<% In(-0,F_Real),1.F_Real %>,x)
      .= 0.F_Real + (1.F_Real)*x by POLYNOM5:44
      .= x;
      set p1 = p-'1;
      eval(~(^((tau(0))|^p1)),x)
      = eval(~((^(tau(0)))|^p1),x) by Th4
      .= eval((~(^(tau(0))))`^p1,x) by Th5
      .= (power F_Real).(x,p1) by POLYNOM5:22,A1;
      hence thesis by BINOM:def 2;
    end;
