reserve X for BCI-algebra;
reserve I for Ideal of X;
reserve a,x,y,z,u for Element of X;
reserve f,f9,g for sequence of  the carrier of X;
reserve j,i,k,n,m for Nat;

theorem
  X is BCK-algebra iff for x holds ord x=1 & x is nilpotent
proof
  thus X is BCK-algebra implies for x being Element of X holds ord x=1&x is
  nilpotent
  proof
    set k=1;
    assume
A1: X is BCK-algebra;
    let x be Element of X;
A2: x`=0.X by A1,BCIALG_1:def 8;
    then (0.X,x)to_power 1=0.X by Th2;
    then
A3: x is nilpotent;
    reconsider k as non zero Nat;
A4: for m being Nat st (0.X,x)to_power m=0.X&m <> 0 holds k<= m
    by NAT_1:14;
    (0.X,x) to_power k=0.X by A2,Th2;
    hence thesis by A3,A4,Def8;
  end;
  assume
A5: for x holds ord x = 1&x is nilpotent;
  now
    let x;
    ord x =1 & x is nilpotent by A5;
    then (0.X,x) to_power 1 = 0.X by Def8;
    hence x` =0.X by Th2;
  end;
  hence thesis by BCIALG_1:def 8;
end;
