reserve A,O for non empty set,
  R for Order of A,
  Ol for Equivalence_Relation of O,
  f for Function of O,A*,
  g for Function of O,A;
reserve S for OverloadedRSSign;
reserve S0 for non empty non void ManySortedSign;
reserve S for non empty Poset;
reserve s1,s2 for Element of S;
reserve w1,w2 for Element of (the carrier of S)*;
reserve S for OrderSortedSign;
reserve o,o1,o2 for OperSymbol of S;
reserve w1 for Element of (the carrier of S)*;
reserve SM for monotone OrderSortedSign,
  o,o1,o2 for OperSymbol of SM,
  w1 for Element of (the carrier of SM)*;
reserve SR for regular monotone OrderSortedSign,
  o,o1,o3,o4 for OperSymbol of SR,
  w1 for Element of (the carrier of SR)*;
reserve R for non empty Poset;
reserve z for non empty set;

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;
