reserve X for BCK-algebra;
reserve x,y for Element of X;
reserve IT for non empty Subset of X;

theorem
  for X being BCI-algebra holds ((ex a be Element of X st a is
  being_greatest) implies X is BCK-algebra)
proof
  let X be BCI-algebra;
  given a being Element of X such that
A1: a is being_greatest;
  for x being Element of X holds x`=0.X
  proof
    let x be Element of X;
    x\a=0.X by A1;
    then x` = (x\x)\a by BCIALG_1:7
      .= 0.X by A1;
    hence thesis;
  end;
  hence thesis by BCIALG_1:def 8;
end;
