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 Th29:
  for x being Element of X holds x in AtomSet(X) iff x``=x
proof
  let x be Element of X;
  thus x in AtomSet(X) implies (x`)`=x
  proof
    assume x in AtomSet(X);
    then (0.X)`\x`= x \ 0.X by Th28;
    then (0.X)`\(x`)=x by Th2;
    hence thesis by Def5;
  end;
  assume
A1: x``=x;
  now
    let z be Element of X;
    assume
A2: z\x=0.X;
    then ((z\x)\(x`))\(z\0.X)=x\z by A1,Th2;
    then 0.X=x\z by Def3;
    hence z=x by A2,Def7;
  end;
  then x is atom;
  hence thesis;
end;
