
theorem thEucl1:
for L being Field,
    p being Element of Polynom-Ring L,
    q being non zero Element of Polynom-Ring L
ex u,r being Element of Polynom-Ring L
st p = u * q + r &
   (r = 0.(Polynom-Ring L) or (deg* L).r < (deg* L).q)
proof
let L be Field,
    p be Element of Polynom-Ring L,
    q be non zero Element of Polynom-Ring L;
q <> 0.(Polynom-Ring L);
then AS: q <> 0_.(L) by POLYNOM3:def 10;
then reconsider q1 = q as non zero Polynomial of L
     by UPROOTS:def 5,POLYNOM3:def 10;
reconsider p1 = p as Polynomial of L by POLYNOM3:def 10;
set u = p1 div q1, r = p1 mod q1;
reconsider u,r as Element of Polynom-Ring L by POLYNOM3:def 10;
take u,r;
A: (p1 div q1) *' q1 + (p1 mod q1)
      = (p1 div q1) *' q1 + (p1 - (p1 div q1) *' q1) by HURWITZ:def 6
     .= (p1 div q1) *' q1 + (p1 + -((p1 div q1) *' q1)) by POLYNOM3:def 6
     .= ((p1 div q1) *' q1  + -((p1 div q1) *' q1)) + p1 by POLYNOM3:26
     .= ((p1 div q1) *' q1 -((p1 div q1) *' q1)) + p1 by POLYNOM3:def 6
     .= 0_.(L) + p1 by POLYNOM3:29
     .= p1 by POLYNOM3:28;
(p1 div q1) *' q1 = u * q by POLYNOM3:def 10;
hence p = u * q + r by A,POLYNOM3:def 10;
now assume r <>  0.(Polynom-Ring L);
  then C: p1 mod q1 <> 0_.(L) by POLYNOM3:def 10;
  B: (deg* L).r = deg*(p1 mod q1) by T .= deg(p1 mod q1) by C,S;
  (deg* L).q = deg* q1 by T .= deg q1 by AS,S;
  hence (deg* L).r < (deg* L).q by B,degA;
  end;
hence thesis;
end;
