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 z,u being
  Element of X holds z\(z\(x\u))=x\u
proof
  let x be Element of X;
  thus x in AtomSet(X) implies for z,u being Element of X holds z\(z\(x\u))=x\
  u
  proof
    assume
A1: x in AtomSet(X);
    let z,u be Element of X;
    x\u =((x\u)`)` by A1,Th31
      .= ((z\z)\(x\u))` by Def5
      .= ((z\(x\u))\z)` by Th7
      .= (z\(x\u))`\z` by Th9;
    then
A2: (x\u)\( z\(z\(x\u)))=0.X by Th1;
    ( z\(z\(x\u)))\(x\u)=0.X by Th1;
    hence thesis by A2,Def7;
  end;
  assume for z,u being Element of X holds z\(z\(x\u))=x\u;
  then (x\u)``=x\u;
  hence thesis by Th31;
end;
