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

theorem
  for a being Function of Y,BOOLEAN holds B_INF(a,%O(Y))=B_INF(a)
proof
  let a be Function of Y,BOOLEAN;
    let y be Element of Y;
A1: now
      EqClass(y,%O(Y)) in %O(Y);
      then EqClass(y,%O(Y)) in {Y} by PARTIT1:def 8;
      then
A2:   EqClass(y,%O(Y))=Y by TARSKI:def 1;
      assume ( not(for x being Element of Y holds a.x=TRUE))& for x being
      Element of Y st x in EqClass(y,%O(Y)) holds a.x=TRUE;
      hence contradiction by A2;
    end;
A3: now
      assume that
A4:   not(for x being Element of Y holds a.x=TRUE) and
A5:   not(for x being Element of Y st x in EqClass(y,%O(Y)) holds a.x = TRUE);
      B_INF(a) = O_el(Y) by A4,Def13;
      then (B_INF a).y = FALSE by Def10;
      hence thesis by A5,Def16;
    end;
    now
      assume that
A6:  for x being Element of Y holds a.x=TRUE and
A7:  for x being Element of Y st x in EqClass(y,%O(Y)) holds a.x=TRUE;
      B_INF(a) = I_el(Y) by A6,Def13;
      then (B_INF a).y = TRUE by Def11;
      hence thesis by A7,Def16;
    end;
    hence thesis by A1,A3;
end;
