<?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/>



<a NAME="E1:26"/><i><font color="Green">E32</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K1" target="_self">max+</a> <font color="Maroon">f</font></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K2" target="_self">max-</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E2:26"/><i><font color="Green">E34</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1"><a href="mesfunc1.html#K12">|.</a><span class="default"><font color="Maroon">f</font></span><a href="mesfunc1.html#K12">.|</a></span>
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D10">MESFUNC1:def 10</a></i>;<br/>
<a NAME="E3:26"/>
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> <font color="Maroon">f</font> holds <br/><span class="p1"><a href="mesfunc1.html#K12">|.</a><span class="default"><font color="Maroon">f</font></span><a href="mesfunc1.html#K12">.|</a></span> <a href="mesfunc1.html#K4">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K1" target="_self">max+</a> <font color="Maroon">f</font></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K2" target="_self">max-</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="26_1"><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="E1:26_1"/><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> <font color="Maroon">f</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:26_1_1_1"/>
( <span class="p1">(<span class="default"><a href="mesfunc2.html#K1" 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> <font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> or <span class="p1">(<span class="default"><a href="mesfunc2.html#K1" 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="supinf_2.html#K1">0.</a>  )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_1_1"><b>suppose </b></a><a NAME="E1:26_1_1_1_1"/><i><font color="Green">E36</font></i>: 
<span class="p1">(<span class="default"><a href="mesfunc2.html#K1" 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> <font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
 ;<br/><div class="add"><b>then </b><i><font color="Green">E37</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K1" target="_self">max+</a> <font color="Maroon">f</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<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></span>)</span> = 
<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="supinf_2.html#K2">+</a> <a href="supinf_2.html#K1">0.</a> 
<b>by </b><i/>
<br/>.= 
<font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="supinf_2.html#T18">SUPINF_2:18</a></i>
;<br/><a NAME="E3:26_1_1_1_1"/>
<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#K1" target="_self">max+</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E4:26_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="mesfunc2.html#K1" 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"><font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span>,<a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i/>;<br/><a NAME="E5:26_1_1_1_1"/><b>then </b>
 <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#D9">XXREAL_0:def 9</a></i>;<br/><b>then </b><font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> = 
<span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><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><a href="extreal1.html#K4">.|</a></span>
<b>by </b><i><a class="ref" href="extreal1.html#D3">EXTREAL1:def 3</a></i>
<br/>.= 
<span class="p1"><a href="mesfunc1.html#K12">|.</a><span class="default"><font color="Maroon">f</font></span><a href="mesfunc1.html#K12">.|</a></span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
<b>by </b><i>, , <a class="ref" href="mesfunc1.html#D10">MESFUNC1:def 10</a></i>
;<br/><b>hence </b><a NAME="E7:26_1_1_1_1"/>
<span class="p1"><a href="mesfunc1.html#K12">|.</a><span class="default"><font color="Maroon">f</font></span><a href="mesfunc1.html#K12">.|</a></span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K1" target="_self">max+</a> <font color="Maroon">f</font></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K2" target="_self">max-</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>, , , <a class="ref" href="mesfunc1.html#D3">MESFUNC1:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_1_1_1_2"><b>suppose </b></a><a NAME="E1:26_1_1_1_2"/><i><font color="Green">E38</font></i>: 
<span class="p1">(<span class="default"><a href="mesfunc2.html#K1" 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="supinf_2.html#K1">0.</a> 
 ;<br/><div class="add"><b>then </b><i><font color="Green">E39</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K1" target="_self">max+</a> <font color="Maroon">f</font></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<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></span>)</span> = 
<a href="supinf_2.html#K1">0.</a>  <a href="supinf_2.html#K2">+</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>
<b>by </b><i>, </i>
<br/>.= 
 <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>
<b>by </b><i><a class="ref" href="supinf_2.html#T18">SUPINF_2:18</a></i>
;<br/><a NAME="E3:26_1_1_1_2"/>
<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#K1" target="_self">max+</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E4:26_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="mesfunc2.html#K1" 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"><font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font></span>)</span>,<a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i/>;<br/><a NAME="E5:26_1_1_1_2"/><b>then </b>
<font color="Maroon">f</font> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_2.html#K1">0.</a> 
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#D9">XXREAL_0:def 9</a></i>;<br/><b>then </b> <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> = 
<span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><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><a href="extreal1.html#K4">.|</a></span>
<b>by </b><i><a class="ref" href="extreal2.html#T55">EXTREAL2:55</a></i>
<br/>.= 
<span class="p1"><a href="mesfunc1.html#K12">|.</a><span class="default"><font color="Maroon">f</font></span><a href="mesfunc1.html#K12">.|</a></span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
<b>by </b><i>, , <a class="ref" href="mesfunc1.html#D10">MESFUNC1:def 10</a></i>
;<br/><b>hence </b><a NAME="E7:26_1_1_1_2"/>
<span class="p1"><a href="mesfunc1.html#K12">|.</a><span class="default"><font color="Maroon">f</font></span><a href="mesfunc1.html#K12">.|</a></span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K1" target="_self">max+</a> <font color="Maroon">f</font></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K2" target="_self">max-</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>, , , <a class="ref" href="mesfunc1.html#D3">MESFUNC1:def 3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:26_1"/>
<span class="p1"><a href="mesfunc1.html#K12">|.</a><span class="default"><font color="Maroon">f</font></span><a href="mesfunc1.html#K12">.|</a></span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mesfunc2.html#K1" target="_self">max+</a> <font color="Maroon">f</font></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p2">(<span class="default"><a href="mesfunc2.html#K2" target="_self">max-</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="mesfunc1.html#K4">.</a> <font color="Maroon">x</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:26"/>
<span class="p1"><a href="mesfunc1.html#K12">|.</a><span class="default"><font color="Maroon">f</font></span><a href="mesfunc1.html#K12">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K1" target="_self">max+</a> <font color="Maroon">f</font></span>)</span> <a href="mesfunc1.html#K13">+</a> <span class="p1">(<span class="default"><a href="mesfunc2.html#K2" target="_self">max-</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>
