theorem
  for X,E holds Class(E,0.X) is closed Ideal of X
proof
  let X,E;
A1: now
    let x,y be Element of X;
    assume that
A2: x\y in Class(E,0.X) and
A3: y in Class(E,0.X);
A4: [x,x] in E by EQREL_1:5;
    [0.X,y] in E by A3,EQREL_1:18;
    then [x\0.X,x\y] in E by A4,Def9;
    then [x,x\y] in E by BCIALG_1:2;
    then
A5: [x\y,x] in E by EQREL_1:6;
    [0.X,x\y] in E by A2,EQREL_1:18;
    then [0.X,x] in E by A5,EQREL_1:7;
    hence x in Class(E,0.X)by EQREL_1:18;
  end;
A6: [0.X,0.X] in E by EQREL_1:5;
  then 0.X in Class(E,0.X) by EQREL_1:18;
  then reconsider Rx=Class(E,0.X) as Ideal of X by A1,BCIALG_1:def 18;
  now
    let x be Element of Rx;
    [0.X,x] in E by EQREL_1:18;
    then [(0.X)`,x`] in E by A6,Def9;
    then [0.X,x`] in E by BCIALG_1:def 5;
    hence x` in Class(E,0.X) by EQREL_1:18;
  end;
  hence thesis by BCIALG_1:def 19;
end;
