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 I,J being Element of A st I is_terminating_wrt f & J is_terminating_wrt f
  holds I\;J is_terminating_wrt f
proof
  let I,J be Element of A such that
A1: for s being Element of S holds [s,I] in TerminatingPrograms(A, S, T, f) and
A2: for s being Element of S holds [s,J] in TerminatingPrograms(A, S, T, f);
  let s be Element of S;
A3: [s,I] in TerminatingPrograms(A,S,T,f) by A1;
  [f.(s,I),J] in TerminatingPrograms(A,S,T,f) by A2;
  hence thesis by A3,Def35;
end;
