
theorem Th90:
for R being domRing,
    a being Element of R
holds a|R is Unit of Polynom-Ring R iff a is Unit of R
proof
let L be domRing, a be Element of L;
set R = Polynom-Ring L;
H1: 1.R = 1_.(L) by POLYNOM3:def 10;
reconsider p = a|L as Element of Polynom-Ring L
  by POLYNOM3:def 10;
A: now assume AS: p is Unit of R;
   then (a|L) divides 1_.(L) by H1,GCD_1:def 20;
   then consider q being Polynomial of L such that
   H3: (a|L) *' q = 1_.(L) by T2;
   reconsider qq = q as Element of R by POLYNOM3:def 10;
   p <> 0.R by AS; then
   H4: a|L <> 0_.(L) by POLYNOM3:def 10;
   q <> 0_.(L) by H3,POLYNOM3:34; then
   H6: deg(a|L) + deg(q) = deg(1_.(L)) by H3,H4,HURWITZ:23
                        .= 1 - 1 by POLYNOM4:4;
   deg(a|L) is Element of NAT by H4,T8b;
   then qq is constant by H6;
   then consider b being Element of L such that H5: qq = b|L by T11;
   (a*b)|L = (1.L)|L by H3,H5,T4;
   then a divides 1.L by T9;
   hence a is Unit of L by GCD_1:def 20;
   end;
now assume a is Unit of L;
   then a divides 1.L by GCD_1:def 20;
   then consider b being Element of L such that
   H3: a * b = 1.L;
   (a|L) *' (b|L) = (1.L)|L by T4,H3 .= 1_.(L);
   then (a|L) divides 1_.(L) by T2;
   hence p is Unit of R by H1,GCD_1:def 20;
   end;
hence thesis by A;
end;
