
theorem polyd:
for F being Field
for p being linear Element of the carrier of Polynom-Ring F
holds (Polynom-Ring p), F are_isomorphic &
      the carrier of embField(canHomP 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 p, f = canHomP p;
H: the carrier of FP = {q where q is Polynomial of F : deg q < deg p}
   by RING_4:def 8;
A: now let o be object;
   assume o in the carrier of FP;
   then consider q being Polynomial of F such that
   B: q = o & deg q < deg p by H;
   reconsider q as Element of the carrier of Polynom-Ring F by POLYNOM3:def 10;
   deg q < 1 by B,defl;
   then deg q + 1 <= 1 by INT_1:7;
   then (deg q + 1) - 1 <= 1 - 1 by XREAL_1:9;
   then consider a being Element of F such that
   C: q = a|F by RING_4:20,RING_4:def 4;
   D: f.a = o by B,C,defch;
   dom f = the carrier of F by FUNCT_2:def 1;
   hence o in rng f by D,FUNCT_1:def 3;
   end;
X: 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 B: rng f = the carrier of FP by A,TARSKI:2;
f is monomorphism onto by X,A,TARSKI:2;
then F,(Polynom-Ring p) are_isomorphic;
hence (Polynom-Ring p), F are_isomorphic;
X: [#] FP = the carrier of FP & [#] F = the carrier of F;
thus the carrier of embField(canHomP 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 B,XBOOLE_1:37
  .= the carrier of F;
end;
