reserve X for BCI-algebra;
reserve X1 for non empty Subset of X;
reserve A,I for Ideal of X;
reserve x,y,z for Element of X;
reserve a for Element of A;

theorem
  AtomSet(X) is Ideal of X implies for x being Element of BCK-part(X),a
  being Element of AtomSet(X) st x\a in AtomSet(X) holds x=0.X
proof
  set B = BCK-part(X);
  set P = AtomSet(X);
A1: x in {0.X} iff x in B/\P
  proof
    0.X in B & 0.X in P by BCIALG_1:19;
    then 0.X in B/\P by XBOOLE_0:def 4;
    hence x in {0.X} implies x in B/\P by TARSKI:def 1;
    thus x in B/\P implies x in {0.X}
    proof
      assume
A2:   x in B/\P;
      then x in B by XBOOLE_0:def 4;
      then ex x1 being Element of X st x=x1 & 0.X<=x1;
      then
A3:   0.X\x=0.X;
      x in {x2 where x2 is Element of X:x2 is minimal} by A2,XBOOLE_0:def 4;
      then ex x2 being Element of X st x=x2 & x2 is minimal;
      then 0.X=x by A3;
      hence thesis by TARSKI:def 1;
    end;
  end;
  assume
A4: P is Ideal of X;
  for x being Element of B,a being Element of P st x\a in P holds x=0.X
  proof
    let x be Element of B;
    let a be Element of P;
    assume x\a in P;
    then x in P by A4,BCIALG_1:def 18;
    then x in B /\ P by XBOOLE_0:def 4;
    then x in {0.X} by A1;
    hence thesis by TARSKI:def 1;
  end;
  hence thesis;
end;
