theorem Th15:
  ConstOSSet(R,z) is non-empty & for s1,s2 being Element of R st
  s1 <= s2 holds ConstOSSet(R,z).s1 c= ConstOSSet(R,z).s2
proof
  set x = ConstOSSet(R,z);
  set D = (the carrier of R) --> z;
  for s being object st s in the carrier of R holds x.s is non empty
by FUNCOP_1:7;
  hence x is non-empty by PBOOLE:def 13;
  let s1,s2 being Element of R;
  D.s1 = z by FUNCOP_1:7
    .= D.s2 by FUNCOP_1:7;
  hence thesis;
end;
