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

<b>let </b><font color="Maroon">X</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">X</font>, <a href="numbers.html#K6">ExtREAL</a> ;<br/><b>let </b><font color="Maroon">A</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">a</font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/>







<b>assume </b><a NAME="E1:32"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 ;<br/>

<a NAME="E2:32"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<a NAME="E3:32"/><b>then </b><i><font color="Green">E27</font></i>: 
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="txt" href="mesfunc1.html#E2"><i><font color="Green">Lemma29</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E4:32"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K17" target="_self">great_dom</a> <font color="Maroon">f</font>,<font color="Maroon">a</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K16" target="_self">less_eq_dom</a> <font color="Maroon">f</font>,<font color="Maroon">a</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1"><b>proof </b></a><div class="add">

<a NAME="E1:32_1"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K17" target="_self">great_dom</a> <font color="Maroon">f</font>,<font color="Maroon">a</font></span>)</span> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K16" target="_self">less_eq_dom</a> <font color="Maroon">f</font>,<font color="Maroon">a</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:32_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K17" target="_self">great_dom</a> <font color="Maroon">f</font>,<font color="Maroon">a</font></span>)</span>
 ;<br/>

<a NAME="E2:32_1_1"/><b>then </b><i><font color="Green">E34</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="mesfunc1.html#K17" target="_self">great_dom</a> <font color="Maroon">f</font>,<font color="Maroon">a</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">y1</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E4:32_1_1"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="mesfunc1.html#K4" target="_self">.</a> <font color="Maroon">x</font> &amp; <font color="Maroon">a</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">y1</font> )
 <b>by </b><i><a class="txt" href="mesfunc1.html#E1:32"><i><font color="Green">E24</font></i></a></i>;<br/>
<a NAME="E5:32_1_1"/>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="mesfunc1.html#K16" target="_self">less_eq_dom</a> <font color="Maroon">f</font>,<font color="Maroon">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:32_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="mesfunc1.html#K16" target="_self">less_eq_dom</a> <font color="Maroon">f</font>,<font color="Maroon">a</font>
 ;<br/>

<b>then consider </b><font color="Maroon">y2</font> being   <a href="supinf_1.html#NM3">R_eal</a><b> such that </b><br/><a NAME="E3:32_1_1_1"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">y2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="mesfunc1.html#K4" target="_self">.</a> <font color="Maroon">x</font> &amp; <font color="Maroon">y2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font> )
 <b>by </b><i/>;<br/>
<b>thus </b><a NAME="E4:32_1_1_1"/>
contradiction
 <b>by </b><i>, <a class="txt" href="mesfunc1.html#E2:32_1_1"><i><font color="Green">E34</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:32_1_1"/><b>then </b>
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K16" target="_self">less_eq_dom</a> <font color="Maroon">f</font>,<font color="Maroon">a</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E7:32_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K16" target="_self">less_eq_dom</a> <font color="Maroon">f</font>,<font color="Maroon">a</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:32_1"/>
<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K17" target="_self">great_dom</a> <font color="Maroon">f</font>,<font color="Maroon">a</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K16" target="_self">less_eq_dom</a> <font color="Maroon">f</font>,<font color="Maroon">a</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:32"/>
<font color="Maroon">A</font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K16" target="_self">less_eq_dom</a> <font color="Maroon">f</font>,<font color="Maroon">a</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K17" target="_self">great_dom</a> <font color="Maroon">f</font>,<font color="Maroon">a</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2"><b>proof </b></a><div class="add">

<a NAME="E1:32_2"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K16" target="_self">less_eq_dom</a> <font color="Maroon">f</font>,<font color="Maroon">a</font></span>)</span></span>)</span> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K17" target="_self">great_dom</a> <font color="Maroon">f</font>,<font color="Maroon">a</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:32_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K16" target="_self">less_eq_dom</a> <font color="Maroon">f</font>,<font color="Maroon">a</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:32_2_1"/><b>then </b><i><font color="Green">E39</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K16" target="_self">less_eq_dom</a> <font color="Maroon">f</font>,<font color="Maroon">a</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E3:32_2_1"/><b>then </b><i><font color="Green">E40</font></i>: 
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="mesfunc1.html#K16" target="_self">less_eq_dom</a> <font color="Maroon">f</font>,<font color="Maroon">a</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="mesfunc1.html#T1" target="_self">Th1</a>, </i>;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">f</font> <a href="mesfunc1.html#K4" target="_self">.</a> <font color="Maroon">x</font> as   <a href="supinf_1.html#NM3">R_eal</a> ;<br/>
<a NAME="E6:32_2_1"/>
not <font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a</font>
 
<b>by </b><i><a class="txt" href="mesfunc1.html#E2"><i><font color="Green">Lemma29</font></i></a>, , , </i>;<br/>
<a NAME="E7:32_2_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="mesfunc1.html#K17" target="_self">great_dom</a> <font color="Maroon">f</font>,<font color="Maroon">a</font>
 
<b>by </b><i><a class="txt" href="mesfunc1.html#E2"><i><font color="Green">Lemma29</font></i></a>, , <a class="txt" href="mesfunc1.html#E1:32"><i><font color="Green">E24</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:32_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K17" target="_self">great_dom</a> <font color="Maroon">f</font>,<font color="Maroon">a</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:32_2"/>
<font color="Maroon">A</font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K16" target="_self">less_eq_dom</a> <font color="Maroon">f</font>,<font color="Maroon">a</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K17" target="_self">great_dom</a> <font color="Maroon">f</font>,<font color="Maroon">a</font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:32"/>
<font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K17" target="_self">great_dom</a> <font color="Maroon">f</font>,<font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="subset_1.html#K6">\</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="mesfunc1.html#K16" target="_self">less_eq_dom</a> <font color="Maroon">f</font>,<font color="Maroon">a</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div>
