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