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
  for x being Element of X holds x in AtomSet(X) iff for u,z being
  Element of X holds (z\u)\(z\x)=x\u
proof
  let x be Element of X;
  thus x in AtomSet(X) implies for u,z being Element of X holds (z\u)\(z\x)=x\
  u
  proof
    assume x in AtomSet(X);
    then
A1: ex x1 being Element of X st x=x1 & x1 is atom;
    let u,z be Element of X;
    (z\(z\x))\x=0.X by Th1;
    then (z\(z\x))=x by A1;
    hence thesis by Th7;
  end;
  assume
A2: for u,z being Element of X holds (z\u)\(z\x)=x\u;
  now
    let z be Element of X;
    assume z\x=0.X;
    then (z\0.X)\0.X=x\0.X by A2;
    then (z\0.X)\0.X=x by Th2;
    then z\0.X=x by Th2;
    hence z=x by Th2;
  end;
  then x is atom;
  hence thesis;
end;
