# Sum Less Maximum is Minimum

## Theorem

For all numbers $a, b$ where $a, b$ in $\N, \Z, \Q$ or $\R$:

$a + b - \max \left({a, b}\right) = \min \left({a, b}\right)$.

## Proof

From Sum of Maximum and Minimum we have that $a + b = \max \left({a, b}\right) + \min \left({a, b}\right)$.

Thus $a + b - \max \left({a, b}\right) = \min \left({a, b}\right)$ follows by subtracting $\max \left({a, b}\right)$ from both sides.

It is clear that this result applies when $a, b$ in $\Z, \Q$ or $\R$, as subtraction is well-defined throughout those number sets.

However, it can be seen to apply in $\N$ as well, despite the fact that $n - m$ is defined on $\N$ only when $n \ge m$.

This is because the fact that $a + b \ge \max \left({a, b}\right)$ follows immediately again from $a + b = \max \left({a, b}\right) + \min \left({a, b}\right)$.

$\blacksquare$
{"ft_lang_label":"__label__en","ft_lang_prob":0.7633198,"math_prob":1.0000086,"size":870,"snap":"2020-45-2020-50","text_gpt3_token_len":297,"char_repetition_ratio":0.23094688,"word_repetition_ratio":0.16774194,"special_character_ratio":0.36551723,"punctuation_ratio":0.13612565,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":1.0000075,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-10-23T08:35:44Z\",\"WARC-Record-ID\":\"<urn:uuid:d044cb1a-27d5-4eee-a35f-e9ec927ab7a7>\",\"Content-Length\":\"29779\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:6ec5f8cc-befd-419b-a592-34b43fdf269b>\",\"WARC-Concurrent-To\":\"<urn:uuid:aa501005-80d0-4ec7-895e-ee1904796142>\",\"WARC-IP-Address\":\"172.67.198.93\",\"WARC-Target-URI\":\"https://proofwiki.org/wiki/Sum_Less_Maximum_is_Minimum\",\"WARC-Payload-Digest\":\"sha1:NO4VV5JXVLSLNKUO5DZANRNKNU6FWOUO\",\"WARC-Block-Digest\":\"sha1:XZIXDKWSQ22F6JJHQAVPQHNS4OJZNO5G\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-45/CC-MAIN-2020-45_segments_1603107880878.30_warc_CC-MAIN-20201023073305-20201023103305-00576.warc.gz\"}"}