
theorem
for F being strict Field
for p being linear Element of the carrier of Polynom-Ring F
holds embField(emb p) = F
proof
let F be strict Field;
let p be linear Element of the carrier of Polynom-Ring F;
set FP = embField(emb p), f = emb p, K = (Polynom-Ring F)/({p}-Ideal);
X: [#] K = the carrier of K & [#] F = the carrier of F;
H: the carrier of F = the carrier of embField f by polyd1;
A: 1.FP = 1.F & 0.FP = 0.F by FIELD_2:def 7;
B: dom(the addF of FP)
       = [:the carrier of FP, the carrier of FP:] by FUNCT_2:def 1
      .= dom(the addF of F) by H,FUNCT_2:def 1;
now let x be Element of dom(the addF of F);
  consider o being Element of [:the carrier of F, the carrier of F:]
  such that B1: x = o;
  consider a,b being object such that
  B2: a in the carrier of F & b in the carrier of F & o = [a,b]
      by ZFMISC_1:def 2;
  a in ((the carrier of K) \ (rng f)) \/ (the carrier of F) &
  b in ((the carrier of K) \ (rng f)) \/ (the carrier of F)
    by B2,XBOOLE_0:def 3;
  then reconsider a,b as Element of (carr f) by X,FIELD_2:def 2;
  thus (the addF of FP).x
     = (addemb f).(a,b) by B1,B2,FIELD_2:def 7
    .= addemb(f,a,b) by FIELD_2:def 4
    .= (the addF of F).(a,b) by B2,X,FIELD_2:def 3
    .= (the addF of F).x by B1,B2;
  end;
then C: the addF of FP = the addF of F by B;
B: dom(the multF of FP)
       = [:the carrier of FP, the carrier of FP:] by FUNCT_2:def 1
      .= dom(the multF of F) by H,FUNCT_2:def 1;
now let x be Element of dom(the multF of F);
  consider o being Element of [:the carrier of F, the carrier of F:]
  such that B1: x = o;
  consider a,b being object such that
  B2: a in the carrier of F & b in the carrier of F & o = [a,b]
      by ZFMISC_1:def 2;
  a in ((the carrier of K) \ (rng f)) \/ (the carrier of F) &
  b in ((the carrier of K) \ (rng f)) \/ (the carrier of F)
    by B2,XBOOLE_0:def 3;
  then reconsider a,b as Element of (carr f) by X,FIELD_2:def 2;
  thus (the multF of FP).x
     = (multemb f).(a,b) by B1,B2,FIELD_2:def 7
    .= multemb(f,a,b) by FIELD_2:def 6
    .= (the multF of F).(a,b) by B2,X,FIELD_2:def 5
    .= (the multF of F).x by B1,B2;
  end;
then the multF of FP = the multF of F by B;
hence thesis by A,C,polyd1;
end;
