Let M be a structure < N, <,  0,  1,  +, {[/n] | n=1,2,…}, f > of signature {<,  0,  1,  +, {[/n] | n=1,2,…}, f }.
A unbounded f is called weakly monotone in M iff f(0)=0, for any natural m there is a natural number k such that f(x)> mx for x>k, and for any x<y from M, f(x)≤f(y). For a weakly monotone f, there is a strong monotone sequence of natural numbers s⁽⁰⁾(f), s⁽¹⁾(f),…, s⁽ⁿ⁾(f),… such that

1. s⁽⁰⁾(f)=0 and f(s⁽ⁱ⁾(f))< f(s⁽ⁱ⁺¹⁾(f)) for any natural i
   
2. for any natural i and s⁽ⁱ⁾(f) ≤ x<s⁽ⁱ⁺¹⁾(f), f(x)=f(s⁽ⁱ⁾(f))

For a weakly monotone f, let f^-1(x)=s⁽ⁱ⁾(f) ⇔ f(s⁽ⁱ⁾(f))≤ x<f(s⁽ⁱ⁺¹⁾(f)). Let [u] denote f^-1(f(u)).

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