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

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">p1</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>;<br/><b>let </b><font color="Maroon">p2</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>;<br/><b>let </b><font color="Maroon">p</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>;<br/><b>let </b><font color="Maroon">u</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span>;<br/><b>let </b><font color="Maroon">P</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>;<br/><b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>;<br/>















<b>assume </b><b>that </b><br/><a NAME="E1:87"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>and </b><br/><a NAME="E2:87"/><i><font color="Green">E56</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font>
 <b>and </b><br/><a NAME="E3:87"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> )
 <b>and </b><br/><a NAME="E4:87"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">u</font>,<font color="Maroon">r</font>
 <b>and </b><br/><a NAME="E5:87"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">p2</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">u</font>,<font color="Maroon">r</font>
 <b>and </b><br/><a NAME="E6:87"/><i><font color="Green">E60</font></i>: 
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">u</font>,<font color="Maroon">r</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font>
 ;<br/>

<b>consider </b><font color="Maroon">f3</font> being   <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span><b> such that </b><br/><a NAME="E8:87"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">f3</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">f3</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">f3</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f3</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Maroon">u</font>,<font color="Maroon">r</font></span>)</span> )
 <b>by </b><i><a class="ref" href="jordan2c.html#T9" target="_self">Th9</a>, <a class="ref" href="jordan2c.html#T12" target="_self">Th12</a>, , <a class="ref" href="jordan2c.html#T13" target="_self">Th13</a>, </i>;<br/>
<a NAME="E9:87"/>
<span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Maroon">u</font>,<font color="Maroon">r</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T11" target="_self">Th11</a>, , <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/>
<a NAME="E10:87"/><b>then </b>
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f3</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font> &amp; <font color="Maroon">f3</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">f3</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> )
 
<b>by </b><i><a class="ref" href="jordan2c.html#T15" target="_self">Th15</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E11:87"/>
 ex <font color="Olive">f5</font> being  <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> st <br/>( <font color="Olive">f5</font> is <a href="pre_topc.html#V5">continuous</a> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Olive">f5</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font> &amp; <font color="Olive">f5</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> &amp; <font color="Olive">f5</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> )
 <b>by </b><i><a class="ref" href="jordan2c.html#T15" target="_self">Th15</a></i>;<br/>


</div>
