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
  for x being Element of X,y being Element of BCK-part(X) holds x\y <= x
proof
  let x be Element of X,y be Element of BCK-part(X);
  y in {y1 where y1 is Element of X:0.X<=y1};
  then ex y1 being Element of X st y=y1 & 0.X<=y1;
  then y`=0.X;
  then (x\x)\y=0.X by Def5;
  then (x\y)\x=0.X by Th7;
  hence thesis;
end;
