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

<b>let </b><font color="Maroon">ra</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">rb</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">a</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">b</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>







<b>assume </b><a NAME="E1:7"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">ra</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">rb</font>
 ;<br/>

<b>let </b><font color="Maroon">f</font> be  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">ra</font>,<font color="Maroon">rb</font></span>)</span>,<a href="topmetr.html#K3">R^1</a> ;<br/><b>let </b><font color="Maroon">d</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>



<b>assume </b><a NAME="E2:7"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">ra</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">rb</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> &amp; <font color="Maroon">a</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">d</font> &amp; <font color="Maroon">d</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">b</font> )
 ;<br/>

<a NAME="E3:7"/><i><font color="Green">E55</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">ra</font>,<font color="Maroon">rb</font></span>)</span> is <a href="connsp_1.html#V2">connected</a>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:7"/><i><font color="Green">E56</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">ra</font>,<font color="Maroon">rb</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">ra</font>,<font color="Maroon">rb</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>;<br/>
<a NAME="E5:7"/><i><font color="Green">E57</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">ra</font>,<font color="Maroon">rb</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">ra</font>,<font color="Maroon">rb</font></span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i>, <a class="ref" href="topmetr.html#T25">TOPMETR:25</a></i>;<br/>
<a NAME="E6:7"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">ra</font>,<font color="Maroon">rb</font></span>)</span></span>)</span> is <a href="connsp_1.html#V2">connected</a>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:7"/><i><font color="Green">E64</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">ra</font>,<font color="Maroon">rb</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:7_1"/><i><font color="Green">E65</font></i>: 
for <font color="Olive">rc</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>   st <font color="Maroon">ra</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">rc</font> &amp; <font color="Olive">rc</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">rb</font> holds <br/><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">rc</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">d</font>
 ;<br/><div><i><font color="Green">E66</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:7_1_1"/>
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">ra</font>,<font color="Maroon">rb</font></span>)</span></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:7_1_1"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">ra</font>,<font color="Maroon">rb</font></span>)</span> &amp; <font color="Maroon">d</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><a NAME="E4:7_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">ra</font>,<font color="Maroon">rb</font></span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i>, </i>;<br/><b>then reconsider </b><font color="Maroon">r</font> = <font color="Maroon">x</font> as   <a href="real_1.html#NM1">Real</a> ;<br/><a NAME="E6:7_1_1"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">ra</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">rb</font> )
 <b>by </b><i>, , </i>;<br/><a NAME="E7:7_1_1"/><b>then </b><i><font color="Green">E73</font></i>: 
<font color="Maroon">ra</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i>, , <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E8:7_1_1"/>
<font color="Maroon">rb</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">r</font>
 <b>by </b><i>, , , <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><b>hence </b><a NAME="E9:7_1_1"/>
contradiction
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><a NAME="E3:7_1"/>
<font color="Maroon">ra</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">ra</font>,<font color="Maroon">rb</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="E4:7_1"/><b>then </b><i><font color="Green">E76</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">ra</font>,<font color="Maroon">rb</font></span>)</span></span>)</span>
 <b>by </b><i>, , , , <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><a NAME="E5:7_1"/>
<font color="Maroon">rb</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">ra</font>,<font color="Maroon">rb</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="E6:7_1"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <span class="p2">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">ra</font>,<font color="Maroon">rb</font></span>)</span></span>)</span>
 <b>by </b><i>, , , , <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/><b>hence </b><a NAME="E7:7_1"/>
contradiction
 <b>by </b><i>, , , , </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E9:7"/>
 ex <font color="Olive">rc</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  st <br/>( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">rc</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> &amp; <font color="Maroon">ra</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">rc</font> &amp; <font color="Olive">rc</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">rb</font> )
 ;<br/>


</div>
