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Now since (f_k^{-1} in G_k), it follows from the induction statement (4.2) that for every ((z,w) in B(0 M)) begin{aligned} |pi _1 circ f_k^{-1} z,w)| le theta ^{-k}M text quad { and } quad |pi _2 circ f_k^{-1} z,w)| le theta ^{-k}M + kC texta ^{-(k-1)} M^d d. end{aligned}This implies that ((z_k,w_k)rightarrow 0) as (krightarrow infty).

By induction assumption, the statement of the theorem holds for dimension (n-1), and so sum_{i=1}^{m} lambda_{i} X_{k}(t_{i}) =0,quad k=1,ldots, n-1, (10) with some (t_{1},ldots, t_{m} in S^{ast}subset S) (here we use the fact that (X^{ast}_{i}(s) = X_{i}(s)) for (sin S^{ast})).

This theory is obtained from Q by adding Σ0-induction and the statement asserting that exponentiation is total.

We prove this statement by induction.

(8) We prove this statement by induction.

Proof: We prove this statement through induction.

Proof (a) We prove the statement by induction on n.

Let us prove the statement by induction, using the matrix recursion from Example 5.8.

Now, Γ ′(B k′)=k for every k, because one can show easily by induction the following stronger statement.

The above two equalities hold if and only if G ≅ C 2 k + 1 n − 2 k − 1. Proof We prove the statement by induction on n − 2 k − 1. Obviously, the result holds for n − 2 k − 1 = 0, 1.

We shall prove statements by induction on N. If ({N = 0}), then (4.10) holds trivially.