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

theorem
  for a being Function of Y,BOOLEAN holds B_SUP(a,%O(Y))=B_SUP(a)
proof
  let a be Function of Y,BOOLEAN;
    let y be Element of Y;
    EqClass(y,%O(Y)) in %O(Y);
    then EqClass(y,%O(Y)) in {Y} by PARTIT1:def 8;
    then
A1: EqClass(y,%O(Y))=Y by TARSKI:def 1;
A2: now
      assume that
A3:  not(ex x being Element of Y st x in EqClass(y,%O(Y)) & a.x=TRUE ) and
A4:  for x being Element of Y holds a.x=FALSE;
      B_SUP(a) = O_el(Y) by A4,Def14;
      then (B_SUP a).y = FALSE by Def10;
      hence thesis by A3,Def17;
    end;
    now
      assume that
A5:  ex x being Element of Y st x in EqClass(y,%O(Y)) & a.x=TRUE and
      not(for x being Element of Y holds a.x=FALSE);
      B_SUP(a) = I_el(Y) by A5,Def14;
      then (B_SUP a).y = TRUE by Def11;
      hence thesis by A5,Def17;
    end;
    hence thesis by A2,A1,XBOOLEAN:def 3;
end;
