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 Th25:
  the_arity_of o1 <= the_arity_of o2 implies Args(o1,A) c= Args(o2 ,A)
proof
  reconsider M = the Sorts of A as OrderSortedSet of S by Th17;
A1: M#.(the_arity_of o1) = M#.((the Arity of S).o1) by MSUALG_1:def 1
    .= (M# * (the Arity of S)).o1 by FUNCT_2:15
    .= Args(o1,A) by MSUALG_1:def 4;
A2: M#.(the_arity_of o2) = M#.((the Arity of S).o2) by MSUALG_1:def 1
    .= (M# * (the Arity of S)).o2 by FUNCT_2:15
    .= Args(o2,A) by MSUALG_1:def 4;
  assume the_arity_of o1 <= the_arity_of o2;
  hence thesis by A1,A2,Th20;
end;
