reserve X for BCI-algebra;
reserve n for Nat;
reserve x,y for Element of X;
reserve a,b for Element of AtomSet(X);
reserve m,n for Nat;
reserve i,j for Integer;

theorem
  X is BCK-algebra iff for x holds x is finite-period & ord x = 1
proof
  thus X is BCK-algebra implies for x holds x is finite-period & ord x = 1
  proof
    assume
A1: X is BCK-algebra;
    let x;
A2: for m being Element of NAT st x|^m in BCK-part(X) & m <> 0 holds 1<=m
    proof
      let m be Element of NAT;
      assume that
      x|^m in BCK-part(X) and
A3:   m <> 0;
      0+1<=m by A3,INT_1:7;
      hence thesis;
    end;
    x`=0.X by A1,BCIALG_1:def 8;
    then 0.X<=x;
    then x in BCK-part(X);
    then
A4: x|^1 in BCK-part(X) by Th4;
    then x is finite-period;
    hence thesis by A4,A2,Def5;
  end;
  assume
A5: for x holds x is finite-period & ord x = 1;
  for y,z being Element of X holds (y\z)\y=0.X
  proof
    let y,z be Element of X;
    z is finite-period & ord z = 1 by A5;
    then z|^1 in BCK-part(X) by Def5;
    then z in BCK-part(X) by Th4;
    then
A6: ex z1 being Element of X st z=z1 & 0.X<=z1;
    (y\z)\y = (y\y)\z by BCIALG_1:7;
    then (y\z)\y = z` by BCIALG_1:def 5;
    hence thesis by A6;
  end;
  then X is being_K;
  hence thesis by BCIALG_1:18;
end;
