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

theorem
  for a being Function of Y,BOOLEAN holds B_SUP(a,%I(Y))=a
proof
  let a be Function of Y,BOOLEAN;
    let y be Element of Y;
A1: now
      EqClass(y,%I(Y)) in %I(Y);
      then EqClass(y,%I(Y)) in {B:ex z being set st B={z} & z in Y} by
PARTIT1:31;
      then ex B st EqClass(y,%I(Y))=B & ex z being set st B={z} & z in Y;
      then consider z being set such that
A2:   EqClass(y,%I(Y))={z} and
      z in Y;
A3:   y in {z} by A2,EQREL_1:def 6;
      assume that
A4:   ex x being Element of Y st x in EqClass(y,%I(Y)) & a.x=TRUE and
A5:   a.y<>TRUE;
      consider x1 being Element of Y such that
A6:  x1 in EqClass(y,%I(Y)) and
A7:  a.x1=TRUE by A4;
      x1=z by A6,A2,TARSKI:def 1;
      hence contradiction by A5,A7,A3,TARSKI:def 1;
    end;
A8: now
      assume that
A9:  not(ex x being Element of Y st x in EqClass(y,%I(Y)) & a.x=TRUE ) and
A10:  a.y<>TRUE;
      a.y = FALSE by A10,XBOOLEAN:def 3;
      hence thesis by A9,Def17;
    end;
    y in EqClass(y,%I(Y)) by EQREL_1:def 6;
    hence thesis by A1,A8,Def17;
end;
