theorem
  for b being Element of X for g being Euclidean ExecutionFunction of A,
  Funcs(X,INT), Funcs(X,INT)\(b,0) for x being Variable of g holds (s.x is odd
iff g.(s, x is_odd) in Funcs(X,INT)\(b,0)) & (s.x is even iff g.(s, x is_even)
  in Funcs(X,INT)\(b,0))
proof
  let b be Element of X;
  let f be Euclidean ExecutionFunction of A,Funcs(X,INT), Funcs(X,INT)\(b,0);
  let x be Variable of f;
  (.x).s = s.x by Th22;
  hence thesis by Th50;
end;
