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