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 Th31:
  for x being Element of X holds x in AtomSet(X) iff for z being
  Element of X holds ((x\z)`)`=x\z
proof
  let x be Element of X;
  thus x in AtomSet(X) implies for z being Element of X holds ((x\z)`)`=x\z
  proof
    assume
A1: x in AtomSet(X);
    let z be Element of X;
A2: (z\(z\x))\x=0.X by Th1;
    ex x1 being Element of X st x=x1 & x1 is atom by A1;
    then (z\(z\x))=x by A2;
    then ((x\z)`)`=(((z\z)\(z\x))`)` by Th7;
    then ((x\z)`)`=(((z\x)`)`)` by Def5;
    then ((x\z)`)`=(z\x)` by Th8;
    hence thesis by A1,Th30;
  end;
  assume for z being Element of X holds (x\z)``=x\z;
  then ((x\0.X)`)`=x\0.X;
  then (x`)`=x\0.X by Th2;
  then (x`)`=x by Th2;
  hence thesis by Th29;
end;
