
theorem polyd1:
for F being Field
for p being linear Element of the carrier of Polynom-Ring F
holds (Polynom-Ring F)/({p}-Ideal), F are_isomorphic &
      the carrier of embField(emb p) = the carrier of F
proof
let F be Field; let p be linear Element of the carrier of Polynom-Ring F;
set FP = (Polynom-Ring F)/({p}-Ideal), I = {p}-Ideal, f = emb p,
    FX = Polynom-Ring F;
A: FP,(Polynom-Ring p) are_isomorphic by RING_4:48;
(Polynom-Ring p), F are_isomorphic by polyd;
hence (Polynom-Ring F)/({p}-Ideal), F are_isomorphic by A,RING_3:44;
B: now let q be Element of the carrier of FX;
   B1: q = (q div p) *' p + (q mod p) by RING_4:4;
   reconsider r = q mod p, d = q div p as
                       Element of the carrier of FX by POLYNOM3:def 10;
   deg r < deg p by divmod; then
   deg r < 1 by defl; then
   deg r + 1 <= 1 by INT_1:7; then
   (deg r + 1) - 1 <= 1 - 1 by XREAL_1:9; then
   reconsider r as constant Element of the carrier of FX by RING_4:def 4;
   (q div p) *' p = d * p by POLYNOM3:def 10; then
   q = d * p + r by B1,POLYNOM3:def 10; then
   B2: q - r = d * p + (r - r) by RLVECT_1:28
            .= d * p + 0.FX by RLVECT_1:15;
   {p}-Ideal = the set of all p*c where c is Element of FX by IDEAL_1:64;
   then p * d in I;
   then d * p in I by GROUP_1:def 12;
   then Class(EqRel(FX,I),q) = Class(EqRel(FX,I),r) by B2,RING_1:6;
   hence ex r being constant Element of the carrier of FX
         st Class(EqRel(FX,I),q) = Class(EqRel(FX,I),r);
   end;
E: now let o be object;
   assume o in the carrier of FP;
   then consider q being Element of the carrier of FX such that
   E1: o = Class(EqRel(FX,I),q) by RING_1:11;
   consider r being constant Element of the carrier of FX such that
   E2: Class(EqRel(FX,I),q) = Class(EqRel(FX,I),r) by B;
   consider a being Element of F such that
   E3: a|F = r by RING_4:20;
   E4: f.a = o by E1,E2,E3,FIELD_1:39;
   dom f = the carrier of F by FUNCT_2:def 1;
   hence o in rng f by E4,FUNCT_1:def 3;
   end;
now let o be object;
  assume B: o in rng f;
  rng f c= the carrier of FP by RELAT_1:def 19;
  hence o in the carrier of FP by B;
  end;
then C: rng f = the carrier of FP by E,TARSKI:2;
X: [#] F = the carrier of F & [#] FP = the carrier of FP;
thus the carrier of embField(emb p)
   = carr f by FIELD_2:def 7
  .= ((the carrier of FP)\(rng f))\/(the carrier of F) by X,FIELD_2:def 2
  .= {} \/ (the carrier of F) by C,XBOOLE_1:37
  .= the carrier of F;
end;
