<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">C</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">f</font> be   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">C</font>, <a href="numbers.html#K6">ExtREAL</a> ;<br/>



<i><font color="Green">E32</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K2" target="_self">max-</a> <font color="Maroon">f</font></span>)</span> = 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
<b>by </b><i/>
<br/>.= 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K9">-</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i><a class="ref" href="mesfunc1.html#D7">MESFUNC1:def 7</a></i>
;<br/>
<a NAME="E2:16"/><b>then </b><i><font color="Green">E34</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K2" target="_self">max-</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K1" target="_self">max+</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K9">-</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E3:16"/>
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K2" target="_self">max-</a> <font color="Maroon">f</font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="mesfunc2.html#K2" target="_self">max-</a> <font color="Maroon">f</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K1" target="_self">max+</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K9">-</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C</font>;<br/>

<b>assume </b><a NAME="E1:16_2"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K2" target="_self">max-</a> <font color="Maroon">f</font></span>)</span>
 ;<br/>

<a NAME="E2:16_2"/><b>then </b><i><font color="Green">E36</font></i>: 
<span class="p1">(<span class="default"><a href="mesfunc2.html#K2" target="_self">max-</a> <font color="Maroon">f</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="extreal2.html#K1">max</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K3">-</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span></span>)</span>,<a href="supinf_2.html#K1">0.</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E3:16_2"/>
 <a href="supinf_2.html#K3">-</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K9">-</a> <font color="Maroon">f</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
 
<b>by </b><i>, , <a class="ref" href="mesfunc1.html#D7">MESFUNC1:def 7</a></i>;<br/>
<b>hence </b><a NAME="E4:16_2"/>
<span class="p1">(<span class="default"><a href="mesfunc2.html#K2" target="_self">max-</a> <font color="Maroon">f</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K1" target="_self">max+</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K9">-</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, , , </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:16"/>
 <a href="mesfunc2.html#K2" target="_self">max-</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="mesfunc2.html#K1" target="_self">max+</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K9">-</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="partfun1.html#T34">PARTFUN1:34</a></i>;<br/>


</div>
