 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 Th37:
  for g be non zero Polynomial of INT.Ring st deg g = m holds
  for x be non zero Element of F_Real holds
  Sum delta_2(m,p,g,x)
    = (g.0)*(('F'(f_0(m,p))).0) - (Ext_eval(g,x))*(('F'(f_0(m,p))).0)
    proof
      let g be non zero Polynomial of INT.Ring;
      assume deg g = m; then
A2:   len g = m+1;
      for x be non zero Element of F_Real holds
      Sum delta_2(m,p,g,x)
      = (g.0)*(('F'(f_0(m,p))).0) -(Ext_eval(g,x))*(('F'(f_0(m,p))).0)
      proof
        let x be non zero Element of F_Real;
A3:     (power F_Real).(x,0) = x|^0 by BINOM:def 2 .= 1_F_Real by BINOM:8;
reconsider r=-(g.0)*(('F'(f_0(m,p))).0) as Element of F_Real by XREAL_0:def 1;
A4:     Sum (<* r *>^(delta_2(m,p,g,x))) = r + Sum (delta_2(m,p,g,x))
        by FVSUM_1:72;
        set F = <* r *>^(delta_2(m,p,g,x));
        consider u be Element of INT.Ring such that
A5:     ('F'(f_0(m,p))).0
        = ((p-'1)!)*(In((((-1)|^m)*(m!))|^p,INT.Ring)) + (In(p!,INT.Ring))*u
         by Th33;
        reconsider s = ('F'(f_0(m,p))).0 as Element of INT.Ring by A5;
A6:     len <* r *> = 1 & len delta_2(m,p,g,x) = m by Def6,FINSEQ_1:39;
        len F = 1 + m by A6,FINSEQ_1:22; then
A8:     dom F = Seg (m+1) by FINSEQ_1:def 3;
A9:     dom (delta_2(m,p,g,x)) = Seg m by A6,FINSEQ_1:def 3;
A10:    for n be Element of NAT st n in dom F holds
        F.n = -(s*g).(n-'1)*(power F_Real).(x,n-'1)
        proof
          let n be Element of NAT;
          assume
A11:      n in dom F;
          per cases;
            suppose
A12:          n = 1; then
              n-'1 = 0 by XREAL_1:232; then
              F.n = -(s*(g.(n-'1)))*(power F_Real).(x,n-'1) by A3,A12;
              hence thesis by POLYNOM5:def 4;
            end;
            suppose
A14:          n <> 1;
A15:          1 <= n <= m+1 by A8,A11,FINSEQ_1:1; then
A16:          len <*r*> + 1 <= n <= len<*r*> + len(delta_2(m,p,g,x))
                by A6,NAT_1:23,A14;
              set n1 = n-'1;
A17:          2-1 <= n-1 by NAT_1:23,A14,A15,XREAL_1:9;
              n-1 <= m+1 -1 by A15,XREAL_1:9; then
              1 <= n-'1 <= m by XREAL_1:233,A15,A17; then
A19:          n1 in dom delta_2(m,p,g,x) by A9;
              F.n = (delta_2(m,p,g,x)).(n - 1) by A16,A6,FINSEQ_1:23
              .= (delta_2(m,p,g,x)).n1 by XREAL_1:233,A15
              .= -(g.n1)*(((power F_Real).(x,n1))*s) by A19,Def6
              .= -((s*(g.n1))*((power F_Real).(x,n1)));
              hence thesis by POLYNOM5:def 4;
            end;
          end;
          per cases;
            suppose
A20:          s = 0;
A21:          for i being Nat st i in dom delta_2(m,p,g,x) holds
              (delta_2(m,p,g,x)).i = ((Seg m) --> 0).i
              proof
                let i be Nat;
                assume i in dom delta_2(m,p,g,x); then
                (delta_2(m,p,g,x)).i
                = -(g.i)*(((power F_Real).(x,i))*0.INT.Ring) by A20,Def6;
                hence thesis;
              end;
              delta_2(m,p,g,x) = m |-> 0.F_Real by A21,A6,FINSEQ_1:def 3;
              hence thesis by MATRIX_3:11,A20;
            end;
            suppose
              s <> 0; then
reconsider s0 = -s as non zero Element of INT.Ring by STRUCT_0:def 12;
              (-s)*g is Polynomial of INT.Ring; then
reconsider h = -(s*g) as Polynomial of INT.Ring by HURWITZ:15;
A25:          deg (s0*g) = deg g by RING_5:4;
reconsider h0 = (s0*g) as Polynomial of F_Real by FIELD_4:9,LIOUVIL2:5;
              deg ~@(s0*g) = deg ~@h0 by FIELD_4:20; then
A28:          len F = len h0 by A25,A2,A6,FINSEQ_1:22;
A29:          for n be Element of NAT st n in dom F holds
              F.n = h0.(n-'1)*(power F_Real).(x,n-'1)
              proof
                let n be Element of NAT;
                assume n in dom F; then
                F.n = -((s*g).(n-'1)*(power F_Real).(x,n-'1)) by A10
                .= -(s*(g.(n-'1)))*(power F_Real).(x,n-'1) by POLYNOM5:def 4
                .= (s0*(g.(n-'1)))*(power F_Real).(x,n-'1);
                hence thesis by POLYNOM5:def 4;
              end;
              reconsider s1 = s0 as Element of F_Real by XREAL_0:def 1;
              eval(h0,x) = eval(~@h0,x)
              .= Ext_eval(~@(s0*g),x) by FIELD_4:26
              .= s1*Ext_eval(g,x) by E_TRANS1:36; then
              Sum F = -(('F'(f_0(m,p))).0)*Ext_eval(g,x)
                by A29,A28,POLYNOM4:def 2;
              hence thesis by A4;
            end;
          end;
          hence thesis;
        end;
