reserve A for preIfWhileAlgebra,
  C,I,J for Element of A;
reserve S for non empty set,
  T for Subset of S,
  s for Element of S;

theorem
  for A being with_empty-instruction Universal_Algebra
  for o being Element of Operations A
  st o = Den(In(1, dom the charact of A), A)
  holds arity o = 0 & EmptyIns A in rng o
proof
  let A be with_empty-instruction Universal_Algebra;
  let o be Element of Operations A such that
A1: o = Den(In(1, dom the charact of A), A);
A2: dom Den(In(1, dom the charact of A), A) = {{}} by Th42;
A3: <*>the carrier of A in {{}} by TARSKI:def 1;
  hence arity o = len (<*>the carrier of A) by A1,A2,MARGREL1:def 25
    .= 0;
  thus thesis by A1,A2,A3,FUNCT_1:def 3;
end;
