
theorem copr1:
for F being Field
for p,q being Element of the carrier of Polynom-Ring F
holds p,q are_coprime iff p gcd q = 1_.(F)
proof
let F be Field, p,q be Element of the carrier of Polynom-Ring F;
set P = Polynom-Ring F;
reconsider o = 0_.(F), e = 1_.(F) as Element of the carrier of P
   by POLYNOM3:def 10;
consider a,b being Element of Polynom-Ring F such that
H: a = p & b = q & p gcd q = a gcd b by RING_4:def 12;
Y: 1.P = 1_.(F) by POLYNOM3:def 10;
now assume AS: p,q are_coprime; then
   AS1: 1.P divides p & 1.P divides q &
        for r being Element of the carrier of P st r divides p & r divides q
        holds r divides 1.P by RING_4:def 10;
   now assume p = 0_.(F) & q = 0_.(F);
      then p gcd q = o by H,RING_4:def 11;
      then o divides p & o divides q &
        for r being Element of the carrier of P
        st r divides p & r divides q holds r divides o by H,RING_4:51;
      then 0_.(F) divides 1_.(F) by AS1,Y;
      then consider r2 being Polynomial of F such that
      E: (0_.(F)) *' r2 = 1_.(F) by RING_4:1;
      thus contradiction by E;
      end;
   hence p gcd q = 1_.(F) by H,AS,Y,RING_4:def 11;
   end;
hence thesis by H,POLYNOM3:def 10;
end;
