reserve Y for non empty set;
reserve B for Subset of Y;

theorem
  for a being Function of Y,BOOLEAN holds O_el(Y) '<' a & a '<' I_el(Y)
proof
  let a be Function of Y,BOOLEAN;
A1: (O_el Y) 'imp' a = I_el Y
  proof
    let x be Element of Y;
    ((O_el Y) 'imp' a).x=('not' (O_el Y).x) 'or' a.x by Def8;
    then ((O_el Y) 'imp' a).x=TRUE 'or' a.x by Def10;
    hence thesis by Def11;
  end;
  a 'imp' I_el(Y)=I_el(Y)
  proof
    let x be Element of Y;
    (a 'imp' I_el Y).x='not' a.x 'or' (I_el Y).x by Def8;
    then (a 'imp' I_el Y).x='not' a.x 'or' TRUE by Def11;
    hence thesis by Def11;
  end;
  hence thesis by A1,Th15;
end;
