 reserve n for Nat;
 reserve F for Field,
         p for irreducible Element of the carrier of Polynom-Ring F,
         f for Element of the carrier of Polynom-Ring F,
         a for Element of F;

theorem Th42:
   eval(emb(f,p),KrRoot p) = Class(EqRel(Polynom-Ring F,{p}-Ideal),f)
   proof
     set z = KrRoot p, E = KroneckerField(F,p), h = canHom (emb p);
     defpred P[Nat] means
     for f st len f = $1 holds eval(emb(f,p),z)
     = Class(EqRel(Polynom-Ring F,{p}-Ideal),f);
A1:  now let f be Element of the carrier of Polynom-Ring F;
       assume len f = 0; then
       f = 0_.F by POLYNOM4:5; then
A2:    f = 0.(Polynom-Ring F) by POLYNOM3:def 10;
       0_.E = 0.(Polynom-Ring E) by POLYNOM3:def 10
          .= emb(f,p) by A2,RING_2:6;
       hence eval(emb(f,p),z) = 0.E by POLYNOM4:17
       .= Class(EqRel(Polynom-Ring F,{p}-Ideal),f) by A2,RING_1:def 6;
     end;
A3:  now let k be Nat;
       assume A4: for m being Nat st m < k holds P[m];
       now let f be Element of the carrier of Polynom-Ring F;
         assume
A5:      len f = k;
         per cases;
            suppose k = 0;
            hence eval(emb(f,p),z) = Class(EqRel(Polynom-Ring F,{p}-Ideal),f)
            by A5,A1;
         end;
         suppose k > 0; then
         consider q being Polynomial of F such that
A6:       len q < len f & f = q + Leading-Monomial(f) &
         for n be Element of NAT st n < len f-1 holds q.n = f.n
         by A5,POLYNOM4:16;
         reconsider g = q as Element of the carrier of Polynom-Ring F
         by POLYNOM3:def 10;
         reconsider LMf = LM f as Element of the carrier of Polynom-Ring F
         by POLYNOM3:def 10;
         reconsider r1 = emb(LMf,p), r2 = emb(g,p) as Polynomial of E;
A7:       emb(LMf+g,p) = emb(LMf,p) + emb(g,p) by VECTSP_1:def 20
                     .= r1 + r2 by POLYNOM3:def 10;
         Leading-Monomial(emb(f,p)) = emb(LMf,p) by Th33; then
A8:       eval(emb(LMf,p),z)=Class(EqRel(Polynom-Ring F,{p}-Ideal),LMf)
         by Lm5;
A9:       eval(emb(g,p),z)=Class(EqRel(Polynom-Ring F,{p}-Ideal),g)
          by A6,A5,A4;
A10:       eval(emb(LMf,p),z) + eval(emb(g,p),z)
         = Class(EqRel(Polynom-Ring F,{p}-Ideal),LMf+g) by A9,A8,RING_1:13
         .= Class(EqRel(Polynom-Ring F,{p}-Ideal),f) by A6,POLYNOM3:def 10;
         thus eval(emb(f,p),z) = eval(r1+r2,z) by A7,A6,POLYNOM3:def 10
         .= Class(EqRel(Polynom-Ring F,{p}-Ideal),f) by A10,POLYNOM4:19;
       end;
     end;
     hence P[k];
   end;
A11: for k being Nat holds P[k] from NAT_1:sch 4(A3);
   ex n being Nat st len f = n;
   hence thesis by A11;
end;
