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 Th44:
  for A being with_catenation non-empty UAStr
  holds dom Den(In(2, dom the charact of A), A) = 2-tuples_on the carrier of A
proof
  let A be with_catenation non-empty UAStr;
  reconsider f = (the charact of A).2 as 2-ary non empty homogeneous
  quasi_total PartFunc of (the carrier of A)*, the carrier of A by Def11;
  2 in dom the charact of A by Def11;
  then
A1: Den(In(2, dom the charact of A), A) = f by SUBSET_1:def 8;
  arity f = 2 by COMPUT_1:def 21;
  hence thesis by A1,COMPUT_1:22;
end;
