New reals: Can live with them, can live without them
英文摘要：
We give a self-contained proof of the preservation theorem for proper countable support iterations known as "tools-preservation," "Case A" or "first preservation theorem" in the literature. We do not assume that the forcings add reals.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
