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 P is_invariant_wrt I, f & P is_invariant_wrt J, f
  holds P is_invariant_wrt I\;J, f
proof
  let P be set;
  let I,J be Element of A;
  assume that
A1: for s being Element of S st s in P holds f.(s, I) in P and
A2: for s being Element of S st s in P holds f.(s, J) in P;
  let s be Element of S;
  assume s in P;
  then
A3: f.(s, I) in P by A1;
  f.(s, I\;J) = f.(f.(s, I), J) by Def29;
  hence thesis by A2,A3;
end;
