reserve A for preIfWhileAlgebra;
reserve A for Euclidean preIfWhileAlgebra;
reserve X for non empty countable set;
reserve T for Subset of Funcs(X, INT);
reserve f for Euclidean ExecutionFunction of A, Funcs(X, INT), T;
reserve A for Euclidean preIfWhileAlgebra,
  X for non empty countable set,
   z for (Element of X),
  s,s9 for (Element of Funcs(X, INT)),
  T for Subset of Funcs(X, INT),
  f for Euclidean ExecutionFunction of A, Funcs(X, INT), T,
  v for INT-Variable of A,f,
  t for INT-Expression of A,f;
reserve i for Integer;

theorem
  for b being Element of X for g being Euclidean ExecutionFunction of A,
Funcs(X,INT), Funcs(X,INT)\(b,0) for x,y being Variable of g holds (s.x <= s.y
  iff g.(s, x leq y) in Funcs(X,INT)\(b,0)) & (s.x >= s.y iff g.(s, x geq y) in
  Funcs(X,INT)\(b,0))
proof
  let b be Element of X;
  let g be Euclidean ExecutionFunction of A,Funcs(X,INT), Funcs(X,INT)\(b,0);
  let x,y be Variable of g;
  g.(s, x leq y) in Funcs(X,INT)\(b,0) iff g.(s, x leq y).b <> 0 by Th2;
  hence s.x <= s.y iff g.(s, x leq y) in Funcs(X,INT)\(b,0) by Th35;
  g.(s, x geq y) in Funcs(X,INT)\(b,0) iff g.(s, x geq y).b <> 0 by Th2;
  hence thesis by Th35;
end;
