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

<b>let </b><font color="Maroon">p</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">g</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a> ;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:17"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>and </b><br/><a NAME="E2:17"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">g</font> &amp; <font color="Maroon">f</font> <a href="fcont_1.html#R2">is_continuous_on</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">g</font></span><a href="rcomp_1.html#K1">.]</a></span> )
 <b>and </b><br/><a NAME="E3:17"/><i><font color="Green">E43</font></i>: 
( not <font color="Maroon">f</font> <a href="rfunct_2.html#R1">is_increasing_on</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">g</font></span><a href="rcomp_1.html#K1">.]</a></span> &amp; not <font color="Maroon">f</font> <a href="rfunct_2.html#R2">is_decreasing_on</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">g</font></span><a href="rcomp_1.html#K1">.]</a></span> )
 ;<br/>

<a NAME="E4:17"/><i><font color="Green">E44</font></i>: 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">g</font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="fcont_1.html#D2">FCONT_1:def 2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:17_1_1"/>
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> or <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">g</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1"><b>suppose </b></a><a NAME="E1:17_1_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font>
 ;<br/><div class="add"><a NAME="E2:17_1_1_1"/><b>then </b>
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">g</font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default"><font color="Maroon">p</font></span><a href="seq_4.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="rcomp_1.html#T14">RCOMP_1:14</a></i>;<br/><b>hence </b><a NAME="E3:17_1_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="rfunct_2.html#T75">RFUNCT_2:75</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_2"><b>suppose </b></a><a NAME="E1:17_1_1_2"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">g</font>
 ;<br/><div class="add"><a NAME="E2:17_1_1_2"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">g</font></span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">g</font></span><a href="rcomp_1.html#K1">.]</a></span> )
 <b>by </b><i>, <a class="ref" href="rcomp_1.html#T15">RCOMP_1:15</a></i>;<br/><a NAME="E3:17_1_1_2"/><b>then </b><i><font color="Green">E47</font></i>: 
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">g</font>
 <b>by </b><i>, , , <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:17_1_1_2_1_1"/>
( <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">g</font> or <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">g</font> )
 </span><b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:17_1_1_2_1_1_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">g</font>
 ;<br/><div class="add"><a NAME="E2:17_1_1_2_1_1_1"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">x1</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">x1</font> &amp; <font color="Olive">x1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">g</font> holds <br/>( <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x1</font> &amp; <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">g</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1_2_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x1</font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:17_1_1_2_1_1_1_1"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x1</font> &amp; <font color="Maroon">x1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">g</font> )
 <b>and </b><br/><a NAME="E2:17_1_1_2_1_1_1_1"/><i><font color="Green">E51</font></i>: 
(  not <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font> or  not <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">g</font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1_2_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_2_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:17_1_1_2_1_1_1_1_1_1"/>
( <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font> or <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">g</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font> )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_2_1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:17_1_1_2_1_1_1_1_1_1_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1_2_1_1_1_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_2_1_1_1_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:17_1_1_2_1_1_1_1_1_1_1_1_1"/>
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> or <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_2_1_1_1_1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:17_1_1_2_1_1_1_1_1_1_1_1_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:17_1_1_2_1_1_1_1_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_2_1_1_1_1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:17_1_1_2_1_1_1_1_1_1_1_1_1_2"/>
<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x1</font>
 ;<br/><div class="add"><a NAME="E2:17_1_1_2_1_1_1_1_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E53</font></i>: 
<font color="Maroon">x1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">p</font>
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:17_1_1_2_1_1_1_1_1_1_1_1_1_2"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">g</font> ) </span>}</span> 
 <b>by </b><i/>;<br/><a NAME="E4:17_1_1_2_1_1_1_1_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E54</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">g</font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/><a NAME="E5:17_1_1_2_1_1_1_1_1_1_1_1_1_2"/>
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">g</font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i>, <a class="ref" href="rcomp_1.html#T15">RCOMP_1:15</a></i>;<br/><a NAME="E6:17_1_1_2_1_1_1_1_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E59</font></i>: 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">g</font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">g</font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i>, <a class="ref" href="rcomp_1.html#T16">RCOMP_1:16</a></i>;<br/><a NAME="E7:17_1_1_2_1_1_1_1_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E60</font></i>: 
<font color="Maroon">f</font> <a href="fcont_1.html#R2">is_continuous_on</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">g</font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i>, <a class="ref" href="fcont_1.html#T17">FCONT_1:17</a></i>;<br/><a NAME="E8:17_1_1_2_1_1_1_1_1_1_1_1_1_2"/>
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">g</font> ) </span>}</span> 
 <b>by </b><i>, </i>;<br/><a NAME="E9:17_1_1_2_1_1_1_1_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">g</font></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/><a NAME="E10:17_1_1_2_1_1_1_1_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">g</font></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">g</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>then consider </b><font color="Maroon">s</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E12:17_1_1_2_1_1_1_1_1_1_1_1_1_2"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">g</font></span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font> )
 <b>by </b><i>, , </i>;<br/><a NAME="E13:17_1_1_2_1_1_1_1_1_1_1_1_1_2"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">g</font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i>, </i>;<br/><a NAME="E14:17_1_1_2_1_1_1_1_1_1_1_1_1_2"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">t</font> where <font color="Olive">t</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">x1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">t</font> &amp; <font color="Olive">t</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">g</font> ) </span>}</span> 
 <b>by </b><i>, <a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/><a NAME="E15:17_1_1_2_1_1_1_1_1_1_1_1_1_2"/><b>then </b>
 ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> &amp; <font color="Maroon">x1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">g</font> )
 ;<br/><b>hence </b><a NAME="E16:17_1_1_2_1_1_1_1_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i>, , , , , , <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:17_1_1_2_1_1_1_1_1_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_2_1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:17_1_1_2_1_1_1_1_1_1_2"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">g</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1_2_1_1_1_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_2_1_1_1_1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:17_1_1_2_1_1_1_1_1_1_2_1_1"/>
( <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> or <font color="Maroon">g</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_2_1_1_1_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:17_1_1_2_1_1_1_1_1_1_2_1_1_1"/>
<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:17_1_1_2_1_1_1_1_1_1_2_1_1_1"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_2_1_1_1_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:17_1_1_2_1_1_1_1_1_1_2_1_1_2"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x1</font>
 ;<br/><div class="add"><a NAME="E2:17_1_1_2_1_1_1_1_1_1_2_1_1_2"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">g</font> ) </span>}</span> 
 <b>by </b><i/>;<br/><a NAME="E3:17_1_1_2_1_1_1_1_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E65</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">g</font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/><a NAME="E4:17_1_1_2_1_1_1_1_1_1_2_1_1_2"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">g</font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i>, <a class="ref" href="rcomp_1.html#T15">RCOMP_1:15</a></i>;<br/><a NAME="E5:17_1_1_2_1_1_1_1_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E66</font></i>: 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">x1</font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">g</font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i>, <a class="ref" href="rcomp_1.html#T16">RCOMP_1:16</a></i>;<br/><a NAME="E6:17_1_1_2_1_1_1_1_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E67</font></i>: 
<font color="Maroon">f</font> <a href="fcont_1.html#R2">is_continuous_on</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">x1</font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i>, <a class="ref" href="fcont_1.html#T17">FCONT_1:17</a></i>;<br/><a NAME="E7:17_1_1_2_1_1_1_1_1_1_2_1_1_2"/>
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font> ) </span>}</span> 
 <b>by </b><i>, </i>;<br/><a NAME="E8:17_1_1_2_1_1_1_1_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/><a NAME="E9:17_1_1_2_1_1_1_1_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>then consider </b><font color="Maroon">s</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E11:17_1_1_2_1_1_1_1_1_1_2_1_1_2"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">x1</font></span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">g</font> )
 <b>by </b><i>, , </i>;<br/><a NAME="E12:17_1_1_2_1_1_1_1_1_1_2_1_1_2"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">g</font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i>, </i>;<br/><a NAME="E13:17_1_1_2_1_1_1_1_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E71</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font>
 <b>by </b><i>, , , , <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><a NAME="E14:17_1_1_2_1_1_1_1_1_1_2_1_1_2"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">t</font> where <font color="Olive">t</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">t</font> &amp; <font color="Olive">t</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x1</font> ) </span>}</span> 
 <b>by </b><i>, <a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/><a NAME="E15:17_1_1_2_1_1_1_1_1_1_2_1_1_2"/><b>then </b>
 ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> &amp; <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x1</font> )
 ;<br/><b>hence </b><a NAME="E16:17_1_1_2_1_1_1_1_1_1_2_1_1_2"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:17_1_1_2_1_1_1_1_1_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:17_1_1_2_1_1_1_1"/>
contradiction
 ;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon">x1</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">x2</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:17_1_1_2_1_1_1"/><i><font color="Green">E72</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">g</font></span><a href="rcomp_1.html#K1">.]</a></span> <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> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">g</font></span><a href="rcomp_1.html#K1">.]</a></span> <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> &amp; <font color="Maroon">x1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x2</font> &amp; <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font> )
 <b>by </b><i>, <a class="ref" href="rfunct_2.html#D2">RFUNCT_2:def 2</a></i>;<br/><a NAME="E5:17_1_1_2_1_1_1"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">g</font></span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">g</font></span><a href="rcomp_1.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> <font color="Maroon">f</font> &amp; <font color="Maroon">x2</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/><a NAME="E6:17_1_1_2_1_1_1"/><b>then </b>
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">t</font> where <font color="Olive">t</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">t</font> &amp; <font color="Olive">t</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">g</font> ) </span>}</span>  &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">g</font> ) </span>}</span>  )
 <b>by </b><i><a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/><a NAME="E7:17_1_1_2_1_1_1"/><b>then </b>
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">t</font> where <font color="Olive">t</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">t</font> &amp; <font color="Olive">t</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">g</font> ) </span>}</span>  &amp;  ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> &amp; <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">g</font> ) )
 ;<br/><a NAME="E8:17_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E74</font></i>: 
(  ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> &amp; <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">g</font> ) &amp; <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x2</font> &amp; <font color="Maroon">x2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">g</font> )
 ;<br/><a NAME="E9:17_1_1_2_1_1_1"/><b>then </b>
( <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font> &amp; <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">g</font> )
 <b>by </b><i/>;<br/><a NAME="E10:17_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font> ) </span>}</span> 
 <b>by </b><i/>;<br/><a NAME="E11:17_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/><a NAME="E12:17_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E75</font></i>: 
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E13:17_1_1_2_1_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">g</font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i>, <a class="ref" href="rcomp_1.html#T15">RCOMP_1:15</a></i>;<br/><a NAME="E14:17_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E76</font></i>: 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">x1</font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">g</font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i>, <a class="ref" href="rcomp_1.html#T16">RCOMP_1:16</a></i>;<br/><a NAME="E15:17_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">f</font> <a href="fcont_1.html#R2">is_continuous_on</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">x1</font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i>, <a class="ref" href="fcont_1.html#T17">FCONT_1:17</a></i>;<br/><b>then consider </b><font color="Maroon">s</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E17:17_1_1_2_1_1_1"/><i><font color="Green">E77</font></i>: 
( <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">x1</font></span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font> )
 <b>by </b><i>, , </i>;<br/><a NAME="E18:17_1_1_2_1_1_1"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">g</font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i>, </i>;<br/><a NAME="E19:17_1_1_2_1_1_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">t</font> where <font color="Olive">t</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">t</font> &amp; <font color="Olive">t</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x1</font> ) </span>}</span> 
 <b>by </b><i>, <a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/><a NAME="E20:17_1_1_2_1_1_1"/><b>then </b>
 ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> &amp; <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x1</font> )
 ;<br/><b>hence </b><a NAME="E21:17_1_1_2_1_1_1"/>
contradiction
 <b>by </b><i>, , , , , , <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:17_1_1_2_1_1_2"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">g</font>
 ;<br/><div class="add"><a NAME="E2:17_1_1_2_1_1_2"/><i><font color="Green">E80</font></i>: 
for <font color="Olive">x1</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">x1</font> &amp; <font color="Olive">x1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">g</font> holds <br/>( <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">g</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x1</font> &amp; <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Olive">x1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1_2_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x1</font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:17_1_1_2_1_1_2_1"/><i><font color="Green">E81</font></i>: 
( <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x1</font> &amp; <font color="Maroon">x1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">g</font> )
 <b>and </b><br/><a NAME="E2:17_1_1_2_1_1_2_1"/><i><font color="Green">E82</font></i>: 
(  not <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">g</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font> or  not <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1_2_1_1_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_2_1_1_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:17_1_1_2_1_1_2_1_1_1"/>
( <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">g</font> or <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font> )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_2_1_1_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:17_1_1_2_1_1_2_1_1_1_1"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">g</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1_2_1_1_2_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_2_1_1_2_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:17_1_1_2_1_1_2_1_1_1_1_1_1"/>
( <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> or <font color="Maroon">g</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_2_1_1_2_1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:17_1_1_2_1_1_2_1_1_1_1_1_1_1"/>
<font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:17_1_1_2_1_1_2_1_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_2_1_1_2_1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:17_1_1_2_1_1_2_1_1_1_1_1_1_2"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">g</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x1</font>
 ;<br/><div class="add"><a NAME="E2:17_1_1_2_1_1_2_1_1_1_1_1_1_2"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">g</font> ) </span>}</span> 
 <b>by </b><i/>;<br/><a NAME="E3:17_1_1_2_1_1_2_1_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E92</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">g</font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/><a NAME="E4:17_1_1_2_1_1_2_1_1_1_1_1_1_2"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">g</font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i>, <a class="ref" href="rcomp_1.html#T15">RCOMP_1:15</a></i>;<br/><a NAME="E5:17_1_1_2_1_1_2_1_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E93</font></i>: 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">x1</font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">g</font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i>, <a class="ref" href="rcomp_1.html#T16">RCOMP_1:16</a></i>;<br/><a NAME="E6:17_1_1_2_1_1_2_1_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E94</font></i>: 
<font color="Maroon">f</font> <a href="fcont_1.html#R2">is_continuous_on</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">x1</font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i>, <a class="ref" href="fcont_1.html#T17">FCONT_1:17</a></i>;<br/><a NAME="E7:17_1_1_2_1_1_2_1_1_1_1_1_1_2"/>
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font> ) </span>}</span> 
 <b>by </b><i>, </i>;<br/><a NAME="E8:17_1_1_2_1_1_2_1_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/><a NAME="E9:17_1_1_2_1_1_2_1_1_1_1_1_1_2"/><b>then </b>
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>then consider </b><font color="Maroon">s</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E11:17_1_1_2_1_1_2_1_1_1_1_1_1_2"/><i><font color="Green">E95</font></i>: 
( <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">x1</font></span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">g</font> )
 <b>by </b><i>, , </i>;<br/><a NAME="E12:17_1_1_2_1_1_2_1_1_1_1_1_1_2"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">g</font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i>, </i>;<br/><a NAME="E13:17_1_1_2_1_1_2_1_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E96</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font>
 <b>by </b><i>, , , , <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><a NAME="E14:17_1_1_2_1_1_2_1_1_1_1_1_1_2"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">t</font> where <font color="Olive">t</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">t</font> &amp; <font color="Olive">t</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x1</font> ) </span>}</span> 
 <b>by </b><i>, <a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/><a NAME="E15:17_1_1_2_1_1_2_1_1_1_1_1_1_2"/><b>then </b>
 ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> &amp; <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x1</font> )
 ;<br/><b>hence </b><a NAME="E16:17_1_1_2_1_1_2_1_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:17_1_1_2_1_1_2_1_1_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_2_1_1_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:17_1_1_2_1_1_2_1_1_1_2"/><i><font color="Green">E97</font></i>: 
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1_2_1_1_2_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_2_1_1_2_1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:17_1_1_2_1_1_2_1_1_1_2_1_1"/>
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> or <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_2_1_1_2_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:17_1_1_2_1_1_2_1_1_1_2_1_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:17_1_1_2_1_1_2_1_1_1_2_1_1_1"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_2_1_1_2_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:17_1_1_2_1_1_2_1_1_1_2_1_1_2"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x1</font>
 ;<br/><div class="add"><a NAME="E2:17_1_1_2_1_1_2_1_1_1_2_1_1_2"/>
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">g</font> ) </span>}</span> 
 <b>by </b><i/>;<br/><a NAME="E3:17_1_1_2_1_1_2_1_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E99</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">g</font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/><a NAME="E4:17_1_1_2_1_1_2_1_1_1_2_1_1_2"/>
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">g</font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i>, <a class="ref" href="rcomp_1.html#T15">RCOMP_1:15</a></i>;<br/><a NAME="E5:17_1_1_2_1_1_2_1_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E102</font></i>: 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">g</font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">g</font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i>, <a class="ref" href="rcomp_1.html#T16">RCOMP_1:16</a></i>;<br/><a NAME="E6:17_1_1_2_1_1_2_1_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E104</font></i>: 
<font color="Maroon">f</font> <a href="fcont_1.html#R2">is_continuous_on</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">g</font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i>, <a class="ref" href="fcont_1.html#T17">FCONT_1:17</a></i>;<br/><a NAME="E7:17_1_1_2_1_1_2_1_1_1_2_1_1_2"/>
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">g</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font> ) </span>}</span> 
 <b>by </b><i>, </i>;<br/><a NAME="E8:17_1_1_2_1_1_2_1_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">g</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/><a NAME="E9:17_1_1_2_1_1_2_1_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">g</font></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">g</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>then consider </b><font color="Maroon">s</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E11:17_1_1_2_1_1_2_1_1_1_2_1_1_2"/><i><font color="Green">E106</font></i>: 
( <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">g</font></span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font> )
 <b>by </b><i>, , </i>;<br/><a NAME="E12:17_1_1_2_1_1_2_1_1_1_2_1_1_2"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">g</font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i>, </i>;<br/><a NAME="E13:17_1_1_2_1_1_2_1_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E107</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font>
 <b>by </b><i>, , , , <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><a NAME="E14:17_1_1_2_1_1_2_1_1_1_2_1_1_2"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">t</font> where <font color="Olive">t</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">x1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">t</font> &amp; <font color="Olive">t</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">g</font> ) </span>}</span> 
 <b>by </b><i>, <a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/><a NAME="E15:17_1_1_2_1_1_2_1_1_1_2_1_1_2"/><b>then </b>
 ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> &amp; <font color="Maroon">x1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">g</font> )
 ;<br/><b>hence </b><a NAME="E16:17_1_1_2_1_1_2_1_1_1_2_1_1_2"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:17_1_1_2_1_1_2_1_1_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:17_1_1_2_1_1_2_1"/>
contradiction
 ;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon">x1</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">x2</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:17_1_1_2_1_1_2"/><i><font color="Green">E108</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">g</font></span><a href="rcomp_1.html#K1">.]</a></span> <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> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">g</font></span><a href="rcomp_1.html#K1">.]</a></span> <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> &amp; <font color="Maroon">x1</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">x2</font> &amp; <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font> )
 <b>by </b><i>, <a class="ref" href="rfunct_2.html#D3">RFUNCT_2:def 3</a></i>;<br/><a NAME="E5:17_1_1_2_1_1_2"/><i><font color="Green">E109</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">g</font></span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">g</font></span><a href="rcomp_1.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> <font color="Maroon">f</font> &amp; <font color="Maroon">x2</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/><a NAME="E6:17_1_1_2_1_1_2"/><b>then </b>
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">t</font> where <font color="Olive">t</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">t</font> &amp; <font color="Olive">t</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">g</font> ) </span>}</span>  &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">g</font> ) </span>}</span>  )
 <b>by </b><i><a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/><a NAME="E7:17_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E110</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">t</font> where <font color="Olive">t</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">t</font> &amp; <font color="Olive">t</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">g</font> ) </span>}</span>  &amp;  ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> &amp; <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">g</font> ) )
 ;<br/><a NAME="E8:17_1_1_2_1_1_2"/><b>then </b>
(  ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> &amp; <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">g</font> ) &amp; <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x2</font> &amp; <font color="Maroon">x2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">g</font> )
 ;<br/><a NAME="E9:17_1_1_2_1_1_2"/><b>then </b>
( <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">g</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font> &amp; <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">p</font> )
 <b>by </b><i/>;<br/><a NAME="E10:17_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">g</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font> ) </span>}</span> 
 <b>by </b><i/>;<br/><a NAME="E11:17_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">g</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/><a NAME="E12:17_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E111</font></i>: 
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">g</font></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">g</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x2</font></span>)</span></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E13:17_1_1_2_1_1_2"/>
<font color="Maroon">g</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">g</font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i>, <a class="ref" href="rcomp_1.html#T15">RCOMP_1:15</a></i>;<br/><a NAME="E14:17_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E112</font></i>: 
<span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">x2</font>,<font color="Maroon">g</font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">g</font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i>, <a class="ref" href="rcomp_1.html#T16">RCOMP_1:16</a></i>;<br/><a NAME="E15:17_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">f</font> <a href="fcont_1.html#R2">is_continuous_on</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">x2</font>,<font color="Maroon">g</font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i>, <a class="ref" href="fcont_1.html#T17">FCONT_1:17</a></i>;<br/><b>then consider </b><font color="Maroon">s</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E17:17_1_1_2_1_1_2"/><i><font color="Green">E113</font></i>: 
( <font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">x2</font>,<font color="Maroon">g</font></span><a href="rcomp_1.html#K1">.]</a></span> &amp; <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x1</font> )
 <b>by </b><i>, , </i>;<br/><a NAME="E18:17_1_1_2_1_1_2"/><i><font color="Green">E114</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">g</font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i>, </i>;<br/><a NAME="E19:17_1_1_2_1_1_2"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">t</font> where <font color="Olive">t</font> is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">x2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">t</font> &amp; <font color="Olive">t</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">g</font> ) </span>}</span> 
 <b>by </b><i>, <a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/><a NAME="E20:17_1_1_2_1_1_2"/><b>then </b>
 ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> &amp; <font color="Maroon">x2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">g</font> )
 ;<br/><b>hence </b><a NAME="E21:17_1_1_2_1_1_2"/>
contradiction
 <b>by </b><i>, , , , , , <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:17_1_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E6:17"/>
contradiction
 ;<br/>


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