core[patch]: fix loss of partially initialized variables during prompt composition (#30096)

**Description:**
This PR addresses the loss of partially initialised variables when
composing different prompts. I.e. it allows the following snippet to
run:

```python
from langchain_core.prompts import ChatPromptTemplate

prompt = ChatPromptTemplate.from_messages([('system', 'Prompt {x} {y}')]).partial(x='1')
appendix = ChatPromptTemplate.from_messages([('system', 'Appendix {z}')])

(prompt + appendix).invoke({'y': '2', 'z': '3'})
```

Previously, this would have raised a `KeyError`, stating that variable
`x` remains undefined.

**Issue**
References issue #30049

**Todo**
- [x] **Add tests and docs**: If you're adding a new integration, please
include
1. a test for the integration, preferably unit tests that do not rely on
network access,
2. an example notebook showing its use. It lives in
`docs/docs/integrations` directory.


- [x] **Lint and test**: Run `make format`, `make lint` and `make test`
from the root of the package(s) you've modified. See contribution
guidelines for more: https://python.langchain.com/docs/contributing/

---------

Co-authored-by: Eugene Yurtsev <eyurtsev@gmail.com>

libs/core/langchain_core/prompts/chat.py

@@ -1040,19 +1040,34 @@ class ChatPromptTemplate(BaseChatPromptTemplate):
         Returns:
             Combined prompt template.
         """
+        partials = {**self.partial_variables}
+
+        # Need to check that other has partial variables since it may not be
+        # a ChatPromptTemplate.
+        if hasattr(other, "partial_variables") and other.partial_variables:
+            partials.update(other.partial_variables)
+
         # Allow for easy combining
         if isinstance(other, ChatPromptTemplate):
-            return ChatPromptTemplate(messages=self.messages + other.messages)  # type: ignore[call-arg]
+            return ChatPromptTemplate(messages=self.messages + other.messages).partial(
+                **partials
+            )  # type: ignore[call-arg]
         elif isinstance(
             other, (BaseMessagePromptTemplate, BaseMessage, BaseChatPromptTemplate)
         ):
-            return ChatPromptTemplate(messages=self.messages + [other])  # type: ignore[call-arg]
+            return ChatPromptTemplate(messages=self.messages + [other]).partial(
+                **partials
+            )  # type: ignore[call-arg]
         elif isinstance(other, (list, tuple)):
             _other = ChatPromptTemplate.from_messages(other)
-            return ChatPromptTemplate(messages=self.messages + _other.messages)  # type: ignore[call-arg]
+            return ChatPromptTemplate(messages=self.messages + _other.messages).partial(
+                **partials
+            )  # type: ignore[call-arg]
         elif isinstance(other, str):
             prompt = HumanMessagePromptTemplate.from_template(other)
-            return ChatPromptTemplate(messages=self.messages + [prompt])  # type: ignore[call-arg]
+            return ChatPromptTemplate(messages=self.messages + [prompt]).partial(
+                **partials
+            )  # type: ignore[call-arg]
         else:
             msg = f"Unsupported operand type for +: {type(other)}"
             raise NotImplementedError(msg)