
theorem lemgcdi:
for R being EuclidianRing
for a,b being Element of R
for g being a_gcd of a,b holds {g}-Ideal = {a,b}-Ideal
proof
let R be EuclidianRing, a,b be Element of R, g be a_gcd of a,b;
HX: {a,b}-Ideal = the set of all a*r + b*s where r,s is Element of R &
    {g}-Ideal = the set of all g*r where r is Element of R
    by IDEAL_1:64,IDEAL_1:65;
consider r,s being Element of R such that
H1: g = a * r + b * s by lemgcdii;
H2: g in {a,b}-Ideal by H1,HX;
now let o be object;
  assume o in {a,b}-Ideal; then
  consider r,s being Element of R such that
  H1: o = a * r + b * s by HX;
  g divides a by RING_4:def 10; then
  consider c being Element of R such that
  H2: a = g * c by GCD_1:def 1;
  g divides b by RING_4:def 10; then
  consider e being Element of R such that
  H3: b = g * e by GCD_1:def 1;
  o = g * (c * r) + (g * e) * s by H1,H2,H3,GROUP_1:def 3
   .= g * (c * r) + g * (e * s) by GROUP_1:def 3
   .= g * ((c * r) + (e * s)) by VECTSP_1:def 2;
  hence o in {g}-Ideal by HX;
  end;then
{a,b}-Ideal c= {g}-Ideal;
hence thesis by H2,IDEAL_1:67;
end;
