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 Th52:
  for A being preIfWhileAlgebra for C,I being Element of A
  holds while(C,I) nin ElementaryInstructions A
proof
  let A be preIfWhileAlgebra;
  let C,I1 be Element of A;
  set I = while(C,I1);
  reconsider f = (the charact of A).4 as 2-ary non empty homogeneous
  quasi_total PartFunc of (the carrier of A)*, the carrier of A by Def13;
  4 in dom the charact of A by Def13;
  then In(4, dom the charact of A) = 4 by SUBSET_1:def 8;
  then dom Den(In(4, dom the charact of A), A)
  = (arity f)-tuples_on the carrier of A by COMPUT_1:22
    .= 2-tuples_on the carrier of A by COMPUT_1:def 21;
  then <*C,I1*> in dom Den(In(4, dom the charact of A), A) by FINSEQ_2:137;
  then I in rng Den(In(4, dom the charact of A), A) by FUNCT_1:def 3;
  then I nin (the carrier of A) \ {EmptyIns A}
  \ rng Den(In(3, dom the charact of A), A)
  \ rng Den(In(4, dom the charact of A), A) by XBOOLE_0:def 5;
  hence thesis by XBOOLE_0:def 5;
end;
