= Solution
An <integer> version of <Siegel lemma> is as follows. For $M<N$ homogeneous <linear equations> $Az=0$, with <integer> <matrix> entries bounded in absolute value by $A_0\geq1$, there is a nonzero <integer> solution satisfying
$$
\boxed{\|z\|_\infty\leq\left\lfloor(2NA_0)^{M/(N-M)}\right\rfloor.}
$$
Put the displayed <integer> bound equal to $Q$ and evaluate the equations at every <vector> in $\{0,\ldots,Q\}^N$. There are $(Q+1)^N$ inputs and at most $(2NA_0Q+1)^M$ outputs. But $2NA_0Q+1\leq2NA_0(Q+1)$ and $(Q+1)^{N-M}>(2NA_0)^M$, so the number of inputs is larger. Two distinct inputs have identical outputs; their difference proves the claim. This is the <pigeonhole proof of integer Siegel lemma>. Rational coefficients are handled by clearing denominators. Over a <number field> of degree $d$, expanding in a rational basis gives at most $dM$ rational equations, so the same construction applies when $N>dM$, with the height of the resulting <integer> <matrix> controlling the bound.
Siegel's original arithmetic definition starts with an entire series $E(z)=\sum_{n\geq0}a_nz^n/n!$, with all $a_n$ in a fixed <number field>. For every $\epsilon>0$ there must be positive <integers> $d_n$ such that $d_na_0,\ldots,d_na_n$ are <algebraic integers>, and both $d_n$ and the absolute values of every conjugate of $a_n$ are $O_\epsilon(n^{\epsilon n})$. The growth condition ensures convergence everywhere. The stronger, now customary strict <E-function> definition uses bounds $C^{n+1}$ for a fixed $C$ and also includes the requirement of a nonzero <linear differential equation> over $\overline{\mathbb Q}(z)$. In the original formulation, that differential condition is explicitly part of the system hypothesis in the theorem below. These definitions should be distinguished; the exponentials used here satisfy even the stronger conditions.
The <Siegel–Shidlovsky theorem> states that if a <vector> of <E-functions> satisfies $f'=A(z)f$, with $A(z)$ a <matrix> of <rational functions> over $\overline{\mathbb Q}$, then for every nonzero algebraic $\xi$ which is not a pole of $A$,
$$
\boxed{\operatorname{trdeg}_{\overline{\mathbb Q}}\overline{\mathbb Q}(f_1(\xi),\ldots,f_s(\xi))
=\operatorname{trdeg}_{\overline{\mathbb Q}(z)}\overline{\mathbb Q}(z)(f_1(z),\ldots,f_s(z)).}
$$
In particular, <algebraic independence> of the functions implies <algebraic independence> of the values. The exclusion of zero is necessary: their Taylor values there are algebraic.
Take rationally independent algebraic $\gamma_1,\ldots,\gamma_s$ and $f_j(z)=e^{\gamma_jz}$. Their coefficients are $\gamma_j^n$, their conjugates grow exponentially, and powers of a fixed common denominator clear all coefficients through degree $n$. They satisfy the nonsingular diagonal system $f_j'=\gamma_jf_j$.
To check functional independence, a <polynomial> relation among these exponentials, after clearing rational-function denominators, would be $\sum_mP_m(z)e^{\lambda_mz}=0$ with distinct $\lambda_m=\sum_jm_j\gamma_j$. Such an <exponential polynomial> cannot vanish identically. To isolate a nonzero term with exponent $\lambda_0$, apply $\prod_{\lambda\ne\lambda_0}(D-\lambda)^{\deg P_\lambda+1}$. All other terms vanish, while the remaining <polynomial> is acted on by operators $D+\lambda_0-\lambda$, each injective on <polynomials> because its nonzero constant preserves a nonzero leading coefficient. Thus the remaining term is nonzero, a contradiction.
The theorem at $\xi=1$ now makes $e^{\gamma_j}$ algebraically independent. For arbitrary distinct algebraic exponents $\alpha_i$, choose a rational basis of their span and divide its elements by a common denominator so that every $\alpha_i$ is an <integer> combination of the resulting $\gamma_j$. Then $e^{\alpha_i}$ are distinct Laurent monomials in $e^{\gamma_j}$. A linear relation among them becomes a nonzero <polynomial> relation after multiplying by a common monomial, contradicting independence. These particular <E-functions> therefore yield the full <Lindemann–Weierstrass theorem>.
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