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 BCI-algebra,x,y being Element of X holds (x\y)\(y\x)=x\y)
  implies X is BCK-algebra
proof
  assume
A1: for X being BCI-algebra,x,y being Element of X holds (x\y)\(y\x)=x\y;
  for s being Element of X holds s` = 0.X
  proof
    let s be Element of X;
    s` \ (s \ 0.X ) = s` by A1;
    then s`\(s` \ s) =s`\s` by Th2;
    then s`\(s` \ s) =0.X by Def5;
    hence thesis by Th1;
  end;
  hence thesis by Def8;
end;
