= Solution
In Siegel's original convention, an <E-function> is an entire series $F(z)=\sum_{n\ge0}a_nz^n/n!$ with coefficients in a fixed <number field>, satisfying a nonzero <linear differential equation> over $\overline{\mathbb Q}(z)$, together with these arithmetic bounds: for every $\epsilon>0$, all conjugates of $a_0,\ldots,a_n$ have size $O_\epsilon(n^{\epsilon n})$, and some positive <integer> $d_n=O_\epsilon(n^{\epsilon n})$ makes every $d_na_j$, $j\le n$, an <algebraic integer>. A frequently used stricter convention replaces both bounds by $C^{n+1}$. The original and strict growth definitions should not simply be declared equivalent; the exponential examples below satisfy the strict one.
For sums, work in the <compositum> of the coefficient fields. Coefficient conjugate bounds add. The product of the two common denominators clears both lists; using $\epsilon/2$ in each original denominator bound gives the required $O_\epsilon(n^{\epsilon n})$ product bound. Finally all derivatives of $F+G$ lie in the sum of the two finite-dimensional derivative spaces over that <rational function> <field>, so they satisfy a nontrivial <linear dependence>, giving a differential equation for the sum. Thus \b[sums are again E-functions], the <addition closure of Siegel E-functions>.
The <Siegel–Shidlovsky theorem> states that if a vector of E-functions satisfies $F'=A(z)F$ with rational-function matrix over $\overline{\mathbb Q}$, then at every nonzero algebraic $\xi$ which is not a pole of $A$,
$$
\operatorname{trdeg}_{\overline{\mathbb Q}}\overline{\mathbb Q}(F_1(\xi),\ldots,F_s(\xi))=\operatorname{trdeg}_{\overline{\mathbb Q}(z)}\overline{\mathbb Q}(z)(F_1(z),\ldots,F_s(z)).
$$
In particular algebraically independent functions give algebraically independent values.
Take $F_j(z)=e^{\gamma_jz}$ for <algebraic numbers> $\gamma_j$ linearly independent over $\mathbb Q$. Their coefficients are $\gamma_j^n$ with exponential conjugate and denominator bounds, and $F_j'=\gamma_jF_j$. A <polynomial> relation among the functions expands into a relation among exponentials with distinct exponents $\sum m_j\gamma_j$ and <polynomial> coefficients after clearing rational-function denominators; exponential-polynomial independence excludes it. At $\xi=1$ the theorem gives <algebraic independence> of the $e^{\gamma_j}$. For arbitrary distinct algebraic $\alpha_i$, choose a rational basis of their span and divide that basis by a common denominator. The $e^{\alpha_i}$ are distinct Laurent monomials in the corresponding algebraically independent exponential values, so are linearly independent over $\overline{\mathbb Q}$. This recovers the general <Lindemann–Weierstrass theorem>. Its simplest instance uses just $F(z)=e^z$ at nonzero algebraic $\xi$.
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