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 Th24:
  the_result_sort_of o1 <= the_result_sort_of o2 implies Result(o1
  ,A) c= Result(o2,A)
proof
  reconsider M = the Sorts of A as OrderSortedSet of S by Th17;
A1: Result(o2,A) = ((the Sorts of A) * the ResultSort of S).o2 by
MSUALG_1:def 5
    .= (the Sorts of A).((the ResultSort of S).o2) by FUNCT_2:15
    .= (the Sorts of A).(the_result_sort_of o2) by MSUALG_1:def 2;
  assume the_result_sort_of o1 <= the_result_sort_of o2;
  then
A2: M.(the_result_sort_of o1) c= M.(the_result_sort_of o2) by Def16;
  Result(o1,A) = ((the Sorts of A) * the ResultSort of S).o1 by MSUALG_1:def 5
    .= (the Sorts of A).((the ResultSort of S).o1) by FUNCT_2:15
    .= (the Sorts of A).(the_result_sort_of o1) by MSUALG_1:def 2;
  hence thesis by A2,A1;
end;
