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;
reserve s1,s2 for SortSymbol of S,
  o,o1,o2,o3 for OperSymbol of S,
  w1,w2 for Element of (the carrier of S)*;
reserve CH for ManySortedFunction of ConstOSSet(S,z)# * the Arity of S,
  ConstOSSet(S,z) * the ResultSort of S;
reserve A for OSAlgebra of S;
reserve M for MSAlgebra over S0;
reserve A for OSAlgebra of S;

theorem Th27:
  for A being non-empty OSAlgebra of S holds A is monotone iff for
  o1,o2 st o1 <= o2 holds Den(o1,A) c= Den(o2,A)
proof
  let A be non-empty OSAlgebra of S;
  hereby
    assume
A1: A is monotone;
    let o1,o2;
    assume o1 <= o2;
    then Den(o2,A)|Args(o1,A) = Den(o1,A) by A1;
    hence Den(o1,A) c= Den(o2,A) by RELAT_1:59;
  end;
  assume
A2: for o1,o2 st o1 <= o2 holds Den(o1,A) c= Den(o2,A);
  let o1,o2 such that
A3: o1 <= o2;
 dom Den(o1,A) = Args(o1,A) by FUNCT_2:def 1;
  hence Den(o2,A)|Args(o1,A) = Den(o1,A)|Args(o1,A) by A2,A3,GRFUNC_1:27
    .= Den(o1,A);
end;
