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 (
  B_INF a).x <= 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=TRUE);
    then B_INF(a) = O_el(Y) by Def13;
    then (B_INF a).x = FALSE by Def10;
    then (B_INF a).x => a.x = TRUE;
    hence thesis;
  end;
  now
    assume for x being Element of Y holds a.x=TRUE;
    then a.x = TRUE;
    then (B_INF a).x => a.x = TRUE;
    hence thesis;
  end;
  hence thesis by A1;
end;
