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

theorem Th20:
  for a being constant Function of Y,BOOLEAN holds a=O_el(Y) or a=I_el(Y)
proof
  let a be constant Function of Y,BOOLEAN;
A1: (for n1,n2 being set st n1 in Y & n2 in Y holds a.n1=a.n2) implies (for
  x being Element of Y holds a.x=TRUE ) or for x being Element of Y holds a.x=
  FALSE
  proof
    assume
A2: for n1,n2 being set st n1 in Y & n2 in Y holds a.n1=a.n2;
    now
      assume that
A3:   not( for x being Element of Y holds a.x=TRUE ) and
A4:   not(for x being Element of Y holds a.x=FALSE);
      consider x1 being Element of Y such that
A5:   a.x1<>TRUE by A3;
      a.x1 = FALSE by A5,XBOOLEAN:def 3;
      hence contradiction by A2,A4;
    end;
    hence thesis;
  end;
  dom a = Y by PARTFUN1:def 2;
hence thesis by A1,Def10,Def11,FUNCT_1:def 10;
end;
