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
  for f being Function of [:S, the carrier of A:], S
  st f is complying_with_empty-instruction & f complies_with_if_wrt T
  for s being Element of S holds
  f.(s,C) nin T implies f.(s,if-then(C,I)) = f.(s,C)
proof
  let f be Function of [:S, the carrier of A:], S such that
A1: f is complying_with_empty-instruction and
A2: f complies_with_if_wrt T;
  let s be Element of S;
  assume f.(s,C) nin T;
  hence f.(s,if-then(C,I)) = f.(f.(s,C),EmptyIns A) by A2
    .= f.(s,C) by A1;
end;
