reserve X for BCI-algebra;
reserve x,y,z,u,a,b for Element of X;
reserve IT for non empty Subset of X;

theorem
  IT is Ideal of X implies for x,y being Element of X st x in IT & y<=x
  holds y in IT
proof
  assume
A1: IT is Ideal of X;
  let x,y be Element of X;
  assume that
A2: x in IT and
A3: y<=x;
  y\0.X <= x by A3,Th2;
  then
A4: y in {z where z is Element of X : z\0.X<=x};
  0.X is Element of IT by A1,Def18;
  then {z where z is Element of X : z\0.X<=x} c= IT by A1,A2,Lm5;
  hence thesis by A4;
end;
