
theorem ass1:
for R being domRing,
    a,b being Element of R
st a is irreducible & b is_associated_to a holds b is irreducible
proof
let R be domRing, a,b be Element of R;
assume AS: a is irreducible & b is_associated_to a;
then consider x being Element of R such that
H: x is unital & b * x = a by GCD_1:18;
now let c being Element of R;
  assume c divides b;
  then c is Unit of R or c is_associated_to a by GCD_1:2,AS;
  hence c is Unit of R or c is_associated_to b by AS,GCD_1:4;
  end;
hence thesis by AS,H;
end;
