
theorem
ex p being non monic Element of the carrier of Polynom-Ring(INT.Ring)
st deg p = 1 & p is reducible
proof
set R = INT.Ring; set K = Polynom-Ring R;
set a = 1.R + 1.R; set p = (a|R) *' rpoly(1,0.R);
a <> 0.R; then
reconsider a as non zero Element of R by STRUCT_0:def 12;
reconsider q = a|R as Element of the carrier of K by POLYNOM3:def 10;
q is constant by RATFUNC1:def 2; then
reconsider q as non zero constant Element of the carrier of K;
reconsider r = rpoly(1,0.R) as Element of the carrier of K
  by POLYNOM3:def 10;
A1: p = (a * 1_.(R)) *' rpoly(1,0.R) by LX1
     .= a * ((1_.(R)) *' rpoly(1,0.R)) by poly2
     .= a * rpoly(1,0.R) by poly1;
A2: deg p = deg q + deg rpoly(1,0.R) by HURWITZ:23
         .= 0 + deg rpoly(1,0.R) by LX
         .= 1 by HURWITZ:27;
then p.(len p -' 1) = (a * rpoly(1,0.R)).(2-1) by A1,XREAL_1:233
                   .= a * (rpoly(1,0.R)).1 by POLYNOM5:def 4
                   .= a * 1.R by HURWITZ:25
                   .= a; then
LC p = a; then
reconsider pp = p as non monic Element of the carrier of K
    by RATFUNC1:def 7,POLYNOM3:def 10;
take pp;
thus deg pp = 1 by A2;
B0: p = q * r by POLYNOM3:def 10;
B1: q divides pp by B0,GCD_1:def 1;
B4: now assume q is Unit of K;
    then a is Unit of R by Th90;
    then a divides 1.R by GCD_1:def 20;
    then consider c being Element of the carrier of R such that
    C1: a * c = 1.R;
    thus contradiction by C1,INT_1:9;
    end;
now assume q is_associated_to pp;
  then consider c being Element of the carrier of K such that
  C1: c is unital & q * c = p by GCD_1:18;
  C2: (a|R)  *' c = p by C1,POLYNOM3:def 10;
  deg c = 0 by C1,T88; then
  c <> 0_.(R) by HURWITZ:20; then
  1 = deg q + deg c by HURWITZ:23,C2,A2 .= 0 + deg c by LX;
  hence contradiction by C1,T88;
  end;
hence pp is reducible by B1,B4,RING_2:def 9;
end;
