[proofplan] Fix an infinite cardinal $\kappa \geq |L| + \aleph_0$ and an arbitrary parameter set $A$ of cardinality $\kappa$. The hypothesis assigns to each complete $1$-type over $A$ an invariant datum in a set $D_A$ of size at most $|A|$, and injectivity ensures that distinct types give distinct data. Therefore the number of complete $1$-types over $A$ is bounded by $|D_A|$, hence by $\kappa$. Since $A$ was arbitrary, this is exactly $\kappa$-stability in the stated sense. [/proofplan] [step:Apply the type-counting hypothesis to an arbitrary parameter set of size $\kappa$] Let $\kappa$ be an infinite cardinal such that $\kappa \geq |L| + \aleph_0$. Let $A \subseteq \mathfrak{C}$ be an arbitrary parameter set with $|A| = \kappa$. Since $A$ is infinite and $|A| = \kappa \geq |L| + \aleph_0$, the hypothesis applies to $A$. Hence there exist a set $D_A$ and an injective map \begin{align*} \Phi_A: S_1(A) &\to D_A \end{align*} such that \begin{align*} |D_A| \leq |A|. \end{align*} [guided] We fix the cardinal $\kappa$ because $\kappa$-stability is a statement about all parameter sets of that fixed size. Thus let $\kappa$ be an infinite cardinal satisfying \begin{align*} \kappa \geq |L| + \aleph_0. \end{align*} To prove $\kappa$-stability, we must consider an arbitrary parameter set of size $\kappa$. Let $A \subseteq \mathfrak{C}$ be such a parameter set, so \begin{align*} |A| = \kappa. \end{align*} The hypothesis of the theorem applies exactly to infinite parameter sets whose size is at least $|L| + \aleph_0$. Our chosen set $A$ satisfies both conditions: it is infinite because $\kappa$ is infinite, and it satisfies \begin{align*} |A| = \kappa \geq |L| + \aleph_0. \end{align*} Therefore the hypothesis supplies a set $D_A$ and an injective map \begin{align*} \Phi_A: S_1(A) &\to D_A \end{align*} with \begin{align*} |D_A| \leq |A|. \end{align*} The role of $\Phi_A$ is to encode each complete $1$-type over $A$ by one datum in $D_A$, with injectivity guaranteeing that no two different types are assigned the same datum. [/guided] [/step] [step:Bound the number of complete $1$-types over $A$ by $\kappa$] Because $\Phi_A: S_1(A) \to D_A$ is injective, cardinal monotonicity under injections gives \begin{align*} |S_1(A)| \leq |D_A|. \end{align*} Combining this with the bound on $D_A$ and the equality $|A| = \kappa$, we obtain \begin{align*} |S_1(A)| \leq |D_A| \leq |A| = \kappa. \end{align*} Thus every parameter set $A \subseteq \mathfrak{C}$ of cardinality $\kappa$ satisfies $|S_1(A)| \leq \kappa$. [/step] [step:Conclude $\kappa$-stability for every allowed cardinal] The parameter set $A \subseteq \mathfrak{C}$ with $|A| = \kappa$ was arbitrary. Therefore, for this fixed infinite cardinal $\kappa \geq |L| + \aleph_0$, every parameter set of cardinality $\kappa$ has at most $\kappa$ complete $1$-types over it. Hence $T$ is $\kappa$-stable in the stated sense. Since the argument used only the assumptions that $\kappa$ is infinite and $\kappa \geq |L| + \aleph_0$, it applies to every such cardinal $\kappa$. This proves the theorem. [/step]