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,y being Element of BCK-part(X) holds x\y in BCK-part(X)
proof
  let x,y be Element of BCK-part(X);
  x in {x1 where x1 is Element of X:0.X<=x1};
  then
A1: ex x1 being Element of X st x=x1 & 0.X<= x1;
  y in {y1 where y1 is Element of X:0.X<=y1};
  then
A2: ex y1 being Element of X st y=y1 & 0.X<=y1;
  (x\y)`=x`\y` by Th9;
  then (x\y)`=(y`)` by A1;
  then (x\y)`=(0.X)` by A2;
  then (x\y)`=0.X by Def5;
  then 0.X <= x\y;
  hence thesis;
end;
