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