theorem
  (the carrier of G)/\K is closed Ideal of G
proof
  set GK = (the carrier of G)/\K;
A1: for x,y being Element of G st x\y in GK & y in GK holds x in GK
  proof
    let x,y be Element of G;
    assume that
A2: x\y in GK and
A3: y in GK;
    the carrier of G c= the carrier of X by BCIALG_1:def 10;
    then reconsider x1=x,y1=y as Element of X;
    x\y in K by A2,XBOOLE_0:def 4;
    then
A4: x1\y1 in K by Th34;
    y1 in K by A3,XBOOLE_0:def 4;
    then x in K by A4,BCIALG_1:def 18;
    hence thesis by XBOOLE_0:def 4;
  end;
A5: 0.G = 0.X by BCIALG_1:def 10;
  then
A6: 0.G in K by BCIALG_1:def 18;
  then
A7: 0.G in GK by XBOOLE_0:def 4;
  for x being object st x in GK holds x in the carrier of G by XBOOLE_0:def 4;
  then GK is non empty Subset of G by A6,TARSKI:def 3,XBOOLE_0:def 4;
  then reconsider GK as Ideal of G by A7,A1,BCIALG_1:def 18;
  for x being Element of GK holds x` in GK
  proof
    let x be Element of GK;
A8: x in K by XBOOLE_0:def 4;
    then reconsider y=x as Element of X;
    y` in K by A8,BCIALG_1:def 19;
    then x` in K by A5,Th34;
    hence thesis by XBOOLE_0:def 4;
  end;
  hence thesis by BCIALG_1:def 19;
end;
