<?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">X</font> be    <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#K1">REAL</a> ;<br/>





<b>assume </b><a NAME="E1:20"/>
<font color="Maroon">f</font> <a href="rfunct_1.html#R1">is_bounded_above_on</a> <font color="Maroon">X</font>
 ;<br/>

<b>then consider </b><font color="Maroon">b</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E3:20"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">c</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> holds <br/><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="rfunct_1.html#D9">RFUNCT_1:def 9</a></i>;<br/>
<a NAME="E4:20"/><i><font color="Green">E39</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="rfunct_3.html#K19">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><a class="ref" href="rfunct_3.html#D10">RFUNCT_3:def 10</a></i>;<br/>
<a NAME="E5:20"/>
 ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <br/>for <font color="Olive">c</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:20_1_1_1"/>
( <font color="Maroon">b</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 or <font color="Maroon">b</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_1_1"><b>suppose </b></a><a NAME="E1:20_1_1_1_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">b</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/><div class="add"><a NAME="E2:20_1_1_1_1"/>
for <font color="Olive">c</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">c</font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_1_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:20_1_1_1_1_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:20_1_1_1_1_1"/><b>then </b><i><font color="Green">E41</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rfunct_3.html#D10">RFUNCT_3:def 10</a></i>;<br/>
<a NAME="E3:20_1_1_1_1_1"/><b>then </b>
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:20_1_1_1_1_1"/><b>then </b>
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E5:20_1_1_1_1_1"/><b>then </b>
 <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font></span>)</span>,0 <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="xxreal_0.html#D9">XXREAL_0:def 9</a></i>;<br/>
<a NAME="E6:20_1_1_1_1_1"/><b>then </b><i><font color="Green">E42</font></i>: 
 <a href="rfunct_3.html#K2">max+</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="rfunct_3.html#D1">RFUNCT_3:def 1</a></i>;<br/>
<a NAME="E7:20_1_1_1_1_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E8:20_1_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 <b>by </b><i>, , <a class="ref" href="rfunct_3.html#D10">RFUNCT_3:def 10</a></i>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon">r</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:20_1_1_1_1"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> 0 &amp; ( for <font color="Olive">c</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> ) )
 ;<br/><b>thus </b><a NAME="E5:20_1_1_1_1"/>
 ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <br/>for <font color="Olive">c</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_1_2"><b>suppose </b></a><a NAME="E1:20_1_1_1_2"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">b</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/><div class="add"><a NAME="E2:20_1_1_1_2"/>
for <font color="Olive">c</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1_1_2_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:20_1_1_1_2_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:20_1_1_1_2_1"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rfunct_3.html#D10">RFUNCT_3:def 10</a></i>;<br/>
<a NAME="E3:20_1_1_1_2_1"/><b>then </b>
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:20_1_1_1_2_1"/><b>then </b>
 <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font></span>)</span>,0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font>
 
<b>by </b><i>, <a class="ref" href="square_1.html#T50">SQUARE_1:50</a></i>;<br/>
<a NAME="E5:20_1_1_1_2_1"/><b>then </b><i><font color="Green">E49</font></i>: 
 <a href="rfunct_3.html#K2">max+</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="ref" href="rfunct_3.html#D1">RFUNCT_3:def 1</a></i>;<br/>
<a NAME="E6:20_1_1_1_2_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E7:20_1_1_1_2_1"/>
<span class="p1">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">b</font>
 <b>by </b><i>, , <a class="ref" href="rfunct_3.html#D10">RFUNCT_3:def 10</a></i>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon">r</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E4:20_1_1_1_2"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; ( for <font color="Olive">c</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> ) )
 ;<br/><a NAME="E5:20_1_1_1_2"/>
<font color="Maroon">r</font> is   <a href="real_1.html#NM1">Real</a>
 <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/><b>hence </b><a NAME="E6:20_1_1_1_2"/>
 ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <br/>for <font color="Olive">c</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>then consider </b><font color="Maroon">r</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:20_1"/><i><font color="Green">E51</font></i>: 
for <font color="Olive">c</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 ;<br/>
<b>thus </b><a NAME="E4:20_1"/>
 ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <br/>for <font color="Olive">c</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Olive">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:20"/>
 <a href="rfunct_3.html#K19">max+</a> <font color="Maroon">f</font> <a href="rfunct_1.html#R1">is_bounded_above_on</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="rfunct_1.html#D9">RFUNCT_1:def 9</a></i>;<br/>


</div>
