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 Th40:
  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 for i being
Integer holds (s.x > i iff g.(s, x gt i) in Funcs(X,INT)\(b,0)) & (s.x <
  i iff g.(s, x lt i) 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 be Variable of g;
  let i be Integer;
  g.(s, x gt i) in Funcs(X,INT)\(b,0) iff g.(s, x gt i).b <> 0 by Th2;
  hence s.x > i iff g.(s, x gt i) in Funcs(X,INT)\(b,0) by Th38;
  g.(s, x lt i) in Funcs(X,INT)\(b,0) iff g.(s, x lt i).b <> 0 by Th2;
  hence thesis by Th38;
end;
