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 Th30:
  for x being Element of X holds x in AtomSet(X) iff for z being
  Element of X holds (z\x)`=x\z
proof
  let x be Element of X;
A1: (for z being Element of X holds z`\x`=x\z) implies for z being Element
  of X holds (z\x)`=x\z
  proof
    assume
A2: for z being Element of X holds z`\x`=x\z;
    let z be Element of X;
    z`\x`=x\z by A2;
    hence thesis by Th9;
  end;
  (for z being Element of X holds (z\x)`=x\z) implies for z being Element
  of X holds z`\x`=x\z
  proof
    assume
A3: for z being Element of X holds (z\x)`=x\z;
    let z be Element of X;
    (z\x)`=x\z by A3;
    hence thesis by Th9;
  end;
  hence thesis by A1,Th28;
end;
