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 Th28:
  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;
  thus x in AtomSet(X) implies for z being Element of X holds z`\x`=x\z
  proof
    assume x in AtomSet(X);
    then
A1: ex x1 being Element of X st x=x1 & x1 is atom;
    let z be Element of X;
    (z\(z\x))\x=0.X by Th1;
    then (z\(z\x))=x by A1;
    then x\z=(z\z)\(z\x) by Th7;
    then x\z=(z\x)` by Def5;
    hence thesis by Th9;
  end;
  assume
A2: for z being Element of X holds z`\x`=x\z;
  now
    let z be Element of X;
    assume
A3: z\x=0.X;
    then (z\x)`=0.X by Def5;
    then z`\x`=0.X by Th9;
    then x\z = 0.X by A2;
    hence z=x by A3,Def7;
  end;
  then x is atom;
  hence thesis;
end;
