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;
reserve op1,op2 for OperSymbol of S;

theorem
  for X being set holds X is OperName of S iff ex op1 st X = Name op1
proof
  let X be set;
  hereby
    assume X is OperName of S;
    then consider x being object such that
A1: x in the carrier' of S and
A2: X = Class(the Overloading of S,x) by EQREL_1:def 3;
    reconsider x1 = x as OperSymbol of S by A1;
    take x1;
    thus X = Name x1 by A2;
  end;
  given op1 such that
A3: X = Name op1;
  thus thesis by A3;
end;
