<?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="E1:19"/><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#K1" 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> <font color="Maroon">f</font> &amp;  <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> <font color="Maroon">f</font> )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E2:19"/>
<span class="p1">(<span class="default"><a href="mesfunc2.html#K1" target="_self">max+</a> <font color="Maroon">f</font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#R1">misses</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#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K1">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:19_1"/>
not <span class="p1">(<span class="default"><a href="mesfunc2.html#K1" target="_self">max+</a> <font color="Maroon">f</font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#R1">misses</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#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">x1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:19_1"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</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="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">x1</font> <a href="hidden.html#R2">in</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#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x1</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C</font> <b>by </b><i/>;<br/>
<a NAME="E5:19_1"/>
( <font color="Maroon">x1</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> &amp; <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">x1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">x1</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> &amp; <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">x1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<a NAME="E6:19_1"/><b>then </b>
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <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">x1</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <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">x1</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  )
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E7:19_1"/>
contradiction
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E3:19"/><b>then </b><i><font color="Green">E35</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="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</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="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E4:19"/>
<span class="p1">(<span class="default"><a href="mesfunc2.html#K1" target="_self">max+</a> <font color="Maroon">f</font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#R1">misses</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#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:19_2"/>
not <span class="p1">(<span class="default"><a href="mesfunc2.html#K1" target="_self">max+</a> <font color="Maroon">f</font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#R1">misses</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#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">x1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:19_2"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</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="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">x1</font> <a href="hidden.html#R2">in</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#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x1</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C</font> <b>by </b><i/>;<br/>
<a NAME="E5:19_2"/>
( <font color="Maroon">x1</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> &amp; <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">x1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">x1</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> &amp; <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">x1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<a NAME="E6:19_2"/><b>then </b>
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <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">x1</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <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">x1</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  )
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E7:19_2"/>
contradiction
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E5:19"/><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="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</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="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E6:19"/>
<span class="p1">(<span class="default"><a href="mesfunc2.html#K1" target="_self">max+</a> <font color="Maroon">f</font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#R1">misses</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#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:19_3"/>
not <span class="p1">(<span class="default"><a href="mesfunc2.html#K1" target="_self">max+</a> <font color="Maroon">f</font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#R1">misses</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#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">x1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:19_3"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</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="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">x1</font> <a href="hidden.html#R2">in</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#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x1</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C</font> <b>by </b><i/>;<br/>
<a NAME="E5:19_3"/>
( <font color="Maroon">x1</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> &amp; <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">x1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<a NAME="E6:19_3"/><b>then </b>
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <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">x1</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  )
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E7:19_3"/>
contradiction
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E7:19"/><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="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</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="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E8:19"/>
<span class="p1">(<span class="default"><a href="mesfunc2.html#K1" target="_self">max+</a> <font color="Maroon">f</font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#R1">misses</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#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K1">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:19_4"/>
not <span class="p1">(<span class="default"><a href="mesfunc2.html#K1" target="_self">max+</a> <font color="Maroon">f</font></span>)</span> <a href="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span> <a href="xboole_0.html#R1">misses</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#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<b>then consider </b><font color="Maroon">x1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:19_4"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</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="relset_1.html#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">x1</font> <a href="hidden.html#R2">in</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#K11">"</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x1</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C</font> <b>by </b><i/>;<br/>
<a NAME="E5:19_4"/>
( <font color="Maroon">x1</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> &amp; <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">x1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<a NAME="E6:19_4"/><b>then </b>
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <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">x1</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  )
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E7:19_4"/>
contradiction
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E9:19"/><b>then </b>
<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="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="subset_1.html#K9">/\</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="relset_1.html#K11">"</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><a href="supinf_1.html#K1">+infty</a> </span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E10:19"/><b>then </b>
(  <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#K6">-</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="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="xboole_0.html#K1">{}</a>  <a href="xboole_0.html#K2">\/</a> <a href="xboole_0.html#K1">{}</a> </span>)</span> &amp;  <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> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="xboole_0.html#K1">{}</a>  <a href="xboole_0.html#K2">\/</a> <a href="xboole_0.html#K1">{}</a> </span>)</span> )
 
<b>by </b><i>, , , , <a class="ref" href="mesfunc1.html#D3">MESFUNC1:def 3</a>, <a class="ref" href="mesfunc1.html#D4">MESFUNC1:def 4</a></i>;<br/>
<b>hence </b><a NAME="E11:19"/>
(  <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#K6">-</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> &amp;  <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> )
 ;<br/>


</div>
