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

theorem
  for a being Function of Y,BOOLEAN,x being Element of Y holds a.x
  <= (B_SUP a).x
proof
  let a be Function of Y,BOOLEAN;
  let x be Element of Y;
A1: now
    assume not (for x being Element of Y holds a.x=FALSE);
    then B_SUP(a) = I_el(Y) by Def14;
    then (B_SUP a).x = TRUE by Def11;
    then a.x => (B_SUP a).x = TRUE;
    hence thesis;
  end;
  now
    assume for x being Element of Y holds a.x=FALSE;
    then a.x = FALSE;
    then a.x => (B_SUP a).x = TRUE;
    hence thesis;
  end;
  hence thesis by A1;
end;
