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;
reserve f for ExecutionFunction of A,S,T;

theorem
  for C,I being Element of A
  st C is_terminating_wrt f & I is_terminating_wrt f &
  for s holds f iteration_terminates_for I\;C, s
  holds while(C,I) is_terminating_wrt f
proof
  let C,I be Element of A such that
A1: C is_terminating_wrt f and
A2: I is_terminating_wrt f and
A3: for s holds f iteration_terminates_for I\;C, s;
  let s;
  f iteration_terminates_for I\;C, f.(s,C) by A3;
  hence thesis by A1,A2,Th114;
end;
