 reserve n,k for Nat;
 reserve L for comRing;
 reserve R for domRing;
 reserve x0 for positive Real;

theorem Th31:
::: adjust POLYDIFF:61 to derivation
   for p be Element of the carrier of Polynom-Ring F_Real holds
   (Eval(~p))`| = Eval(~((Der1(F_Real)).p))
   proof
     let p be Element of the carrier of Polynom-Ring F_Real;
     set DR = Der1 F_Real;
     set f = (Eval(~p))`|;
     set g = Eval(~((Der1(F_Real)).p));
     defpred P[Nat] means
     for p be Element of the carrier of Polynom-Ring F_Real
     st len ~p <= $1 holds (Eval(~p))`| = Eval(~(DR.p));
A1:  P[0]
     proof
       let p be Element of the carrier of Polynom-Ring F_Real;
       assume len ~p <= 0; then
A3:    len (~p) = 0; then
A4:    p = 0_.F_Real by POLYNOM4:5;
A5:    ~p = 0_.F_Real by A3,POLYNOM4:5;
       ~(DR.p) = 0_.F_Real by A4,FIELD_14:58;
       hence thesis by POLYDIFF:52,A5,POLYDIFF:54;
     end;
A7:  P[n] implies P[n+1]
     proof
       assume
A8:    P[n];
       let p be Element of the carrier of Polynom-Ring F_Real such that
A9:    len ~p <= n+1;
       set m = len (~p)-'1;
       set q = @((~p)||m);
A11:   now
        per cases;
          suppose
            ~p <> 0_.F_Real;
            hence len (~q) < n+1 by A9,XXREAL_0:2,POLYDIFF:36;
          end;
          suppose ~p = 0_.F_Real;
            hence len (~q) < n+1 by POLYNOM4:3;
          end;
        end;
        set l = Leading-Monomial(~p);
A13:    DR is additive;
A14:    ~(DR.(@(l+ (~q)))) = DR.(@(~q) + @l) by POLYNOM3:def 10
       .= ~(DR.(@(~q)) + DR.(@l)) by A13
       .= ~(DR.(@LM(~p))) + ~(DR.q) by POLYNOM3:def 10;
A15:   @(~q + l) = @(~p) = p by POLYDIFF:37;
A16:   Eval(LM(~p))`| = Eval(~(DR.(@LM(~p)))) by Lm22;
       (Eval(~p))`| = (Eval(~q+LM(~p)))`| by POLYDIFF:37
       .= (Eval(~q)+Eval(LM(~p)))`| by POLYDIFF:55
       .= (Eval(~q))`| + (Eval(LM(~p)))`| by POLYDIFF:14
       .= Eval(~(DR.(@LM(~p)))) + Eval(~(DR.q)) by A8,A11,NAT_1:13, A16
       .= Eval(~(DR.p)) by A14,A15,POLYDIFF:55;
       hence thesis;
     end;
A17: P[n] from NAT_1:sch 2(A1,A7);
     len ~p = len ~p;
     hence thesis by A17;
   end;
