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 Th42:
  for A being with_empty-instruction non-empty UAStr
  holds dom Den(In(1, dom the charact of A), A) = {{}}
proof
  let A be with_empty-instruction non-empty UAStr;
  reconsider e = (the charact of A).1 as 0-ary non empty homogeneous
  quasi_total PartFunc of (the carrier of A)*, the carrier of A by Def10;

  1 in dom the charact of A by Def10;
  then
A1: Den(In(1, dom the charact of A), A) = e by SUBSET_1:def 8;
  arity e = 0 by COMPUT_1:def 21;
  then dom e = 0-tuples_on the carrier of A by COMPUT_1:22;
  hence thesis by A1,COMPUT_1:5;
end;
