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 P being set for I,J being Element of A
  st I is_terminating_wrt f,P & J is_terminating_wrt f,P &
  P is_invariant_wrt I,f holds I\;J is_terminating_wrt f,P
proof
  let P be set;
  let I,J be Element of A such that
A1: for s being Element of S st s in P
  holds [s,I] in TerminatingPrograms(A, S, T, f) and
A2: for s being Element of S st s in P
  holds [s,J] in TerminatingPrograms(A, S, T, f) and
A3: for s being Element of S st s in P holds f.(s, I) in P;
  let s be Element of S;
  assume
A4: s in P;
  then
A5: [s,I] in TerminatingPrograms(A,S,T,f) by A1;
  [f.(s,I),J] in TerminatingPrograms(A,S,T,f) by A2,A3,A4;
  hence thesis by A5,Def35;
end;
