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 y,z,u being
  Element of X holds (x\u)\(z\y)<=(y\u)\(z\x)
proof
  let x be Element of X;
  thus x in AtomSet(X) implies for y,z,u being Element of X holds (x\u)\(z\y)
  <=(y\u)\(z\x)
  proof
    assume x in AtomSet(X);
    then
A1: ex x1 being Element of X st x=x1 & x1 is atom;
    let y,z,u be Element of X;
    (z\(z\x))\x=0.X by Th1;
    then
    ((x\u)\(z\y))\((y\u)\(z\x))=(((z\(z\x))\u)\(z\y))\((y\u)\(z\x)) by A1;
    then ((x\u)\(z\y))\((y\u)\(z\x))=(((z\u)\(z\x))\(z\y))\((y\u)\(z\x))by Th7;
    then ((x\u)\(z\y))\((y\u)\(z\x))=(((z\u)\(z\y))\(z\x))\((y\u)\(z\x))by Th7
      .=((((z\u)\(z\y))\(z\x))\((y\u)\(z\x)))\0.X by Th2;
    then
    ((x\u)\(z\y))\((y\u)\(z\x)) =((((z\u)\(z\y))\(z\x))\((y\u)\(z\x)))\(((
    z\u)\(z\y))\(y\u)) by Th1;
    then ((x\u)\(z\y))\((y\u)\(z\x))=0.X by Def3;
    hence thesis;
  end;
  assume
A2: for y,z,u being Element of X holds (x\u)\(z\y)<=(y\u)\(z\x);
  now
    let z be Element of X;
    assume
A3: z\x=0.X;
    ((x\0.X)\(z\0.X)) <= ((0.X)`\(z\x)) by A2;
    then ((x\0.X)\(z\0.X)) \(((0.X)`\0.X))=0.X by A3;
    then ((x\0.X)\(z\0.X)) \(0.X)`=0.X by Th2;
    then ((x\0.X)\(z\0.X)) \0.X=0.X by Th2;
    then (x\0.X)\(z\0.X)=0.X by Th2;
    then (x\0.X)\z=0.X by Th2;
    then x\z = 0.X by Th2;
    hence z=x by A3,Def7;
  end;
  then x is atom;
  hence thesis;
end;
