Induction for Predicative Formulas, and the Order of ℕ below 1 and below n+1

In the setting of Routine Facts Used Without Citation: Arithmetic, Maps, Predicativity and Indices, let $\mathbb{N}$, $1$, the sum $n+1$ and the order $\le$ with strict order $<$ be as in The Natural Numbers and the Natural Numbers with Zero: Arithmetic, Order, Induction and Recursion (ii), The Natural Numbers and the Natural Numbers with Zero: Arithmetic, Order, Induction and Recursion (iv) and The Natural Numbers and the Natural Numbers with Zero: Arithmetic, Order, Induction and Recursion (v).

(i) Let $x$ be a set variable, and let $\varphi$ be a condition on $x$ that is expressed by a predicative formula, its parameters having their fixed values as in Conditions Expressed by a Predicative Formula, and Guards (i). For a set $a$, $\varphi(a)$ means that $\varphi$ holds when $a$ is assigned to $x$. If $\varphi(1)$ holds, and $\varphi(n)$ implies $\varphi(n+1)$ for every $n\in\mathbb{N}$, then $\varphi(n)$ holds for every $n\in\mathbb{N}$.

(ii) There is no $k\in\mathbb{N}$ with $k<1$.

(iii) For all $k,n\in\mathbb{N}$: $k<n+1$ if and only if $k<n$ or $k=n$; and $k\le n+1$ if and only if $k\le n$ or $k=n+1$.