<?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">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>, <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>assume </b><a NAME="E1:19"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">p1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p2</font>
 ;<br/>

<b>set </b><font color="Maroon">P</font> =  <a href="topreal1.html#K3" target="_self">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>;<br/>
<a NAME="E2:19"/><i><font color="Green">E38</font></i>: 
( the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font> )
 
<b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">p1'</font> = <font color="Maroon">p1</font>, <font color="Maroon">p2'</font> = <font color="Maroon">p2</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font> <b>by </b><i/>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>for <font color="Olive">x</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Olive">x</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Olive">x</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span>;<br/>
<a NAME="E4:19"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span> holds <br/> ex <font color="Olive">u</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">a</font>,<font color="Olive">u</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:19_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">a</font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<b>take </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span>
;<br/>

<b>thus </b><a NAME="E3:19_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">a</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">x</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span>]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E6:19"/><i><font color="Green">E50</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 <b>and </b><br/><a NAME="E7:19"/><i><font color="Green">E54</font></i>: 
for <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">a</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">a</font>]
 <b>from </b><i><a class="ref" href="zfrefle1.html#S1">ZFREFLE1:sch 1</a>();<br/></i>
<a NAME="E8:19"/><i><font color="Green">E56</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" target="_self">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:19_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 ;<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="E3:19_2"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">y</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#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:19_2"/>
<font color="Maroon">x</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> : ( 0 <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> 1 ) </span>}</span> 
 
<b>by </b><i>, , <a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/>
<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="E6:19_2"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; 0 <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> 1 )
 ;<br/>
<a NAME="E7:19_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span>
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E8:19_2"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Olive">lambda</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><font color="Olive">lambda</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span> where <font color="Olive">lambda</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">lambda</font> &amp; <font color="Olive">lambda</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E9:19_2"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3" target="_self">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 
;<br/>
<a NAME="E10:19_2"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" target="_self">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<b>hence </b><a NAME="E11:19_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" target="_self">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font></span>)</span></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as   <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" target="_self">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font></span>)</span></span>)</span> <b>by </b><i>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<a NAME="E10:19"/><i><font color="Green">E67</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" target="_self">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>;<br/>
<a NAME="E11:19"/><i><font color="Green">E68</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" target="_self">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:19_3"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" target="_self">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E2:19_3"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3" target="_self">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<a NAME="E3:19_3"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Olive">lambda</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><font color="Olive">lambda</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span> where <font color="Olive">lambda</font> is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">lambda</font> &amp; <font color="Olive">lambda</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 
;<br/>
<b>then consider </b><font color="Maroon">lambda</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E5:19_3"/><i><font color="Green">E69</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">lambda</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">lambda</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">lambda</font> &amp; <font color="Maroon">lambda</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<br/>
<a NAME="E6:19_3"/>
<font color="Maroon">lambda</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> : ( 0 <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> 1 ) </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E7:19_3"/><b>then </b><i><font color="Green">E70</font></i>: 
<font color="Maroon">lambda</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/>
<a NAME="E8:19_3"/><b>then </b>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">lambda</font>
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E9:19_3"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:19"/><i><font color="Green">E71</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>;<br/>
<a NAME="E13:19"/><i><font color="Green">E74</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" target="_self">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font></span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:19_4"/><i><font color="Green">E75</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> )
 ;<br/><a NAME="E2:19_4"/><b>then </b>
<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> : ( 0 <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> 1 ) </span>}</span> 
 <b>by </b><i>, <a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">r1</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:19_4"/><i><font color="Green">E76</font></i>: 
( <font color="Maroon">r1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r1</font> &amp; <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<br/><a NAME="E5:19_4"/>
<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> : ( 0 <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> 1 ) </span>}</span> 
 <b>by </b><i>, , <a class="ref" href="rcomp_1.html#D1">RCOMP_1:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">r2</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E7:19_4"/><i><font color="Green">E77</font></i>: 
( <font color="Maroon">r2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r2</font> &amp; <font color="Maroon">r2</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<br/><a NAME="E8:19_4"/>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">r1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span> )
 <b>by </b><i>, , , , <a class="txt" href="topreal1.html#E1"><i><font color="Green">Lemma25</font></i></a></i>;<br/><a NAME="E9:19_4"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="euclid.html#T30">EUCLID:30</a></i>;<br/><a NAME="E10:19_4"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r1</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span>
 <b>by </b><i><a class="ref" href="euclid.html#T37">EUCLID:37</a></i>;<br/><a NAME="E11:19_4"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span>
 <b>by </b><i><a class="ref" href="euclid.html#T30">EUCLID:30</a></i>;<br/><a NAME="E12:19_4"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r1</font> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">r1</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span>
 <b>by </b><i><a class="ref" href="euclid.html#T37">EUCLID:37</a></i>;<br/><a NAME="E13:19_4"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K16">0.REAL</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span>
 <b>by </b><i><a class="ref" href="euclid.html#T33">EUCLID:33</a></i>;<br/><a NAME="E14:19_4"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="euclid.html#T31">EUCLID:31</a></i>;<br/><a NAME="E15:19_4"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span>
 <b>by </b><i><a class="ref" href="euclid.html#T30">EUCLID:30</a></i>;<br/><a NAME="E16:19_4"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span>
 <b>by </b><i><a class="ref" href="euclid.html#T37">EUCLID:37</a></i>;<br/><a NAME="E17:19_4"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="euclid.html#K16">0.REAL</a> <font color="Maroon">n</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span>
 <b>by </b><i><a class="ref" href="euclid.html#T33">EUCLID:33</a></i>;<br/><a NAME="E18:19_4"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r2</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font>
 <b>by </b><i><a class="ref" href="euclid.html#T31">EUCLID:31</a></i>;<br/><a NAME="E19:19_4"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">r1</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">r2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font>
 <b>by </b><i><a class="ref" href="euclid.html#T37">EUCLID:37</a></i>;<br/><a NAME="E20:19_4"/><b>then </b>
<font color="Maroon">r2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">r1</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="ref" href="euclid.html#T38">EUCLID:38</a></i>;<br/><b>hence </b><a NAME="E21:19_4"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 <b>by </b><i>, <a class="txt" href="topreal1.html#E1"><i><font color="Green">Lemma25</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E15:19"/><b>then </b><i><font color="Green">E78</font></i>: 
<font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>
<b>reconsider </b><font color="Maroon">PP</font> =  <a href="topreal1.html#K3" target="_self">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span> <b>by </b><i/>;<br/>
<a NAME="E17:19"/>
not <font color="Maroon">PP</font> is <a href="xboole_0.html#V1">empty</a>
 
;<br/>
<b>then reconsider </b><font color="Maroon">PP</font> =  <a href="topreal1.html#K3" target="_self">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span> ;<br/>
<a NAME="E19:19"/><i><font color="Green">E79</font></i>: 
<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" target="_self">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pcomps_1.html#K4">TopSpaceMetr</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span> <a href="topmetr.html#K1">|</a> <font color="Maroon">PP</font></span>)</span>
 
<b>by </b><i><a class="ref" href="topmetr.html#T20">TOPMETR:20</a></i>;<br/>
<a NAME="E20:19"/>
for <font color="Olive">W</font> being  <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> <br/> for <font color="Olive">G</font> being   <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">W</font>  ex <font color="Olive">H</font> being   <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Olive">W</font> st <font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <font color="Olive">H</font> <a href="tarski.html#R1">c=</a> <font color="Olive">G</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">W</font> be   <a href="pre_topc.html#NM2">Point</a> of <a href="topmetr.html#K5">I[01]</a> ;<br/><b>let </b><font color="Maroon">G</font> be    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">W</font>;<br/>



<a NAME="E1:19_5"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">W</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="connsp_2.html#D1">CONNSP_2:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">Q</font> being   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" target="_self">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E3:19_5"/><i><font color="Green">E80</font></i>: 
( <font color="Maroon">Q</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">Q</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font> &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> )
 <b>by </b><i><a class="ref" href="tops_1.html#T54">TOPS_1:54</a></i>;<br/>
<a NAME="E4:19_5"/>
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" target="_self">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3" target="_self">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<a NAME="E5:19_5"/><b>then </b>
( the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" target="_self">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3" target="_self">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span> <a href="topmetr.html#K1">|</a> <font color="Maroon">PP</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3" target="_self">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font> )
 
<b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a>, <a class="ref" href="topmetr.html#D2">TOPMETR:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">Y</font> = <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">W</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span> <a href="topmetr.html#K1">|</a> <font color="Maroon">PP</font></span>)</span> ;<br/>
<b>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="E8:19_5"/><i><font color="Green">E81</font></i>: 
( <font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">Y</font>,<font color="Maroon">r</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q</font> )
 <b>by </b><i>, <a class="txt" href="topreal1.html#E1"><i><font color="Green">Lemma25</font></i></a>, <a class="ref" href="topmetr.html#T22">TOPMETR:22</a></i>;<br/>
<b>reconsider </b><font color="Maroon">r</font> = <font color="Maroon">r</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<b>set </b><font color="Maroon">delta</font> = <font color="Maroon">r</font> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K13">Pitag_dist</a> <font color="Maroon">n</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">p1'</font>,<font color="Maroon">p2'</font></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon">W'</font> = <font color="Maroon">W</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K2">Closed-Interval-MSpace</a> 0,1</span>)</span> <b>by </b><i><a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="topmetr.html#T14">TOPMETR:14</a></i>;<br/>
<b>reconsider </b><font color="Maroon">H</font> =  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">W'</font>,<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K13">Pitag_dist</a> <font color="Maroon">n</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">p1'</font>,<font color="Maroon">p2'</font></span>)</span></span>)</span> as   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K5">I[01]</a>  <b>by </b><i><a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="topmetr.html#T14">TOPMETR:14</a></i>;<br/>
<a NAME="E12:19_5"/>
(  <a href="topmetr.html#K5">I[01]</a>  <a href="hidden.html#R1">=</a>  <a href="topmetr.html#K4">Closed-Interval-TSpace</a> 0,1 &amp;  <a href="topmetr.html#K4">Closed-Interval-TSpace</a> 0,1 <a href="hidden.html#R1">=</a>  <a href="pcomps_1.html#K4">TopSpaceMetr</a> <span class="p1">(<span class="default"><a href="topmetr.html#K2">Closed-Interval-MSpace</a> 0,1</span>)</span> )
 
<b>by </b><i><a class="ref" href="topmetr.html#T27">TOPMETR:27</a>, <a class="ref" href="topmetr.html#D8">TOPMETR:def 8</a></i>;<br/>
<a NAME="E13:19_5"/><b>then </b><i><font color="Green">E82</font></i>: 
<font color="Maroon">H</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="topmetr.html#T21">TOPMETR:21</a></i>;<br/>
<b>reconsider </b><font color="Maroon">p11</font> = <font color="Maroon">p1</font>, <font color="Maroon">p22</font> = <font color="Maroon">p2</font> as   <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> <b>by </b><i/>;<br/>
<a NAME="E15:19_5"/><i><font color="Green">E83</font></i>: 
<span class="p1">(<span class="default"><a href="euclid.html#K13">Pitag_dist</a> <font color="Maroon">n</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">p1'</font>,<font color="Maroon">p2'</font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K4">dist</a> <font color="Maroon">p11</font>,<font color="Maroon">p22</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#D1">METRIC_1:def 1</a></i>;<br/>
<a NAME="E16:19_5"/><i><font color="Green">E104</font></i>: 
(  <a href="metric_1.html#K4">dist</a> <font color="Maroon">p11</font>,<font color="Maroon">p22</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 &amp;  <a href="metric_1.html#K4">dist</a> <font color="Maroon">p11</font>,<font color="Maroon">p22</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 
<b>by </b><i>, <a class="ref" href="metric_1.html#T2">METRIC_1:2</a>, <a class="ref" href="metric_1.html#T5">METRIC_1:5</a></i>;<br/>
<a NAME="E17:19_5"/><b>then </b><i><font color="Green">E105</font></i>: 
<font color="Maroon">r</font> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K13">Pitag_dist</a> <font color="Maroon">n</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">p1'</font>,<font color="Maroon">p2'</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i>, , <a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/>
<a NAME="E18:19_5"/><b>then </b>
<font color="Maroon">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font>
 
<b>by </b><i><a class="ref" href="tbsp_1.html#T16">TBSP_1:16</a></i>;<br/>
<a NAME="E19:19_5"/><b>then </b>
<font color="Maroon">W</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <font color="Maroon">H</font>
 
<b>by </b><i>, <a class="ref" href="tops_1.html#T55">TOPS_1:55</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">H</font> = <font color="Maroon">H</font> as    <a href="connsp_2.html#M1">a_neighborhood</a> of <font color="Maroon">W</font> <b>by </b><i><a class="ref" href="connsp_2.html#D1">CONNSP_2:def 1</a></i>;<br/>
<b>take </b>
<font color="Maroon">H</font>
;<br/>

<b>reconsider </b><font color="Maroon">W1</font> = <font color="Maroon">W</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E22:19_5"/>
(  <a href="topreal1.html#K3" target="_self">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span> <a href="topmetr.html#K1">|</a> <font color="Maroon">PP</font></span>)</span> &amp;  <a href="topreal1.html#K3" target="_self">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> &amp; the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</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="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="topmetr.html#D2">TOPMETR:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">WW'</font> = <font color="Maroon">Y</font> as   <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> <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E24:19_5"/>
<font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">H</font> <a href="tarski.html#R1">c=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">Y</font>,<font color="Maroon">r</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_5_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:19_5_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">H</font>
 ;<br/>

<b>then consider </b><font color="Maroon">u</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:19_5_1"/><i><font color="Green">E108</font></i>: 
( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <font color="Maroon">H</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">u1</font> = <font color="Maroon">u</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">u'</font> = <font color="Maroon">u</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K2">Closed-Interval-MSpace</a> 0,1</span>)</span> <b>by </b><i/>;<br/>
<a NAME="E6:19_5_1"/>
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" target="_self">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font></span>)</span></span>)</span> &amp;  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" target="_self">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3" target="_self">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font> )
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a>, <a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a>, <a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<a NAME="E7:19_5_1"/><b>then </b>
( the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" target="_self">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3" target="_self">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp; the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span> <a href="topmetr.html#K1">|</a> <font color="Maroon">PP</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3" target="_self">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" target="_self">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a>, <a class="ref" href="topmetr.html#D2">TOPMETR:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">aa</font> = <font color="Maroon">a</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span> <a href="topmetr.html#K1">|</a> <font color="Maroon">PP</font></span>)</span> <b>by </b><i/>;<br/>
<a NAME="E9:19_5_1"/>
(  <a href="topreal1.html#K3" target="_self">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span> <a href="topmetr.html#K1">|</a> <font color="Maroon">PP</font></span>)</span> &amp;  <a href="topreal1.html#K3" target="_self">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> &amp; the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</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="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="topmetr.html#D2">TOPMETR:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">aa'</font> = <font color="Maroon">aa</font> as   <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> <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">aa1</font> = <font color="Maroon">aa'</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font> ;<br/>
<b>reconsider </b><font color="Maroon">WW1</font> = <font color="Maroon">WW'</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font> ;<br/>
<a NAME="E13:19_5_1"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">W</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">W1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a></i>;<br/>
<a NAME="E14:19_5_1"/>
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" target="_self">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D9">PRE_TOPC:def 9</a></i>;<br/>
<a NAME="E15:19_5_1"/><b>then </b>
(  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" target="_self">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" target="_self">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font></span>)</span></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a>, <a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a>, <a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<a NAME="E16:19_5_1"/><b>then </b>
( the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" target="_self">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3" target="_self">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">a</font>, <font color="Maroon">fW</font> = <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">W</font> as   <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> <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">p12</font> = <font color="Maroon">p1</font> <a href="euclid.html#K20">-</a> <font color="Maroon">p2</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font> <b>by </b><i/>;<br/>
<font color="Maroon">WW1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">aa1</font> = 
<font color="Maroon">fW</font> <a href="euclid.html#K20">-</a> <font color="Maroon">a'</font>
<b>by </b><i><a class="ref" href="euclid.html#D13">EUCLID:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">W1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><font color="Maroon">W1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span> <a href="euclid.html#K20">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">u1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><font color="Maroon">u1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span>
<b>by </b><i>, , , </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">W1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">W1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span></span>)</span> <a href="euclid.html#K20">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">u1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span></span>)</span> <a href="euclid.html#K20">-</a> <span class="p1">(<span class="default"><font color="Maroon">u1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T50">EUCLID:50</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">W1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">W1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">u1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K20">-</a> <span class="p1">(<span class="default"><font color="Maroon">u1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T49">EUCLID:49</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">W1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">W1</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">u1</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span></span>)</span> <a href="euclid.html#K20">-</a> <span class="p1">(<span class="default"><font color="Maroon">u1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T54">EUCLID:54</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">u1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">W1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">W1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p2">(<span class="default"><font color="Maroon">u1</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T49">EUCLID:49</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">u1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">W1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">W1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">u1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T54">EUCLID:54</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">u1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">W1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><font color="Maroon">u1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">W1</font></span>)</span></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">u1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">W1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K19">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">u1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">W1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T44">EUCLID:44</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">u1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">W1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">u1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">W1</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T45">EUCLID:45</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">u1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">W1</font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="euclid.html#K20">-</a> <font color="Maroon">p2</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T53">EUCLID:53</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">u1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">W1</font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">p12</font>
<b>by </b><i><a class="ref" href="euclid.html#D11">EUCLID:def 11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">u1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">W1</font></span>)</span> <a href="euclid.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">p1'</font> <a href="euclid.html#K8">-</a> <font color="Maroon">p2'</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#D13">EUCLID:def 13</a></i>
;<br/>
<a NAME="E20:19_5_1"/><b>then </b><i><font color="Green">E119</font></i>: 
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">WW1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">aa1</font></span>)</span></span><a href="euclid.html#K12">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">u1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">W1</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1'</font> <a href="euclid.html#K8">-</a> <font color="Maroon">p2'</font></span>)</span></span><a href="euclid.html#K12">.|</a></span>
 
<b>by </b><i><a class="ref" href="euclid.html#T14">EUCLID:14</a></i>;<br/>
<a NAME="E21:19_5_1"/><i><font color="Green">E120</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">W'</font>,<font color="Maroon">u'</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K13">Pitag_dist</a> <font color="Maroon">n</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">p1'</font>,<font color="Maroon">p2'</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">W''</font> = <font color="Maroon">W'</font>, <font color="Maroon">u''</font> = <font color="Maroon">u'</font> as   <a href="pre_topc.html#NM2">Point</a> of <a href="metric_1.html#K8">RealSpace</a>  <b>by </b><i><a class="ref" href="topmetr.html#T12">TOPMETR:12</a></i>;<br/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">W'</font>,<font color="Maroon">u'</font> = 
the <a href="metric_1.html#U1">distance</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K2">Closed-Interval-MSpace</a> 0,1</span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">W'</font>,<font color="Maroon">u'</font>
<b>by </b><i><a class="ref" href="metric_1.html#D1">METRIC_1:def 1</a></i>
<br/>.= 
the <a href="metric_1.html#U1">distance</a> of <a href="metric_1.html#K8">RealSpace</a>  <a href="metric_1.html#K1">.</a> <font color="Maroon">W''</font>,<font color="Maroon">u''</font>
<b>by </b><i><a class="ref" href="topmetr.html#D1">TOPMETR:def 1</a></i>
<br/>.= 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">W''</font>,<font color="Maroon">u''</font>
<b>by </b><i><a class="ref" href="metric_1.html#D1">METRIC_1:def 1</a></i>
;<br/>
<a NAME="E24:19_5_1"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">W1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">u1</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K13">Pitag_dist</a> <font color="Maroon">n</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">p1'</font>,<font color="Maroon">p2'</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="topmetr.html#T15">TOPMETR:15</a></i>;<br/>
<a NAME="E25:19_5_1"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><font color="Maroon">u1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">W1</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K13">Pitag_dist</a> <font color="Maroon">n</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">p1'</font>,<font color="Maroon">p2'</font></span>)</span>
 
;<br/>
<a NAME="E26:19_5_1"/><b>then </b><i><font color="Green">E123</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">u1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">W1</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K13">Pitag_dist</a> <font color="Maroon">n</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">p1'</font>,<font color="Maroon">p2'</font></span>)</span>
 
<b>by </b><i><a class="ref" href="complex1.html#T138">COMPLEX1:138</a></i>;<br/>
<a NAME="E27:19_5_1"/><i><font color="Green">E124</font></i>: 
( <font color="Maroon">r</font> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K13">Pitag_dist</a> <font color="Maroon">n</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">p1'</font>,<font color="Maroon">p2'</font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp; <span class="p1">(<span class="default"><a href="euclid.html#K13">Pitag_dist</a> <font color="Maroon">n</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">p1'</font>,<font color="Maroon">p2'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 
<b>by </b><i>, , , <a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/>
<a NAME="E28:19_5_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="euclid.html#K13">Pitag_dist</a> <font color="Maroon">n</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">p1'</font>,<font color="Maroon">p2'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K13">Pitag_dist</a> <font color="Maroon">n</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">p1'</font>,<font color="Maroon">p2'</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xcmplx_1.html#T54">XCMPLX_1:54</a></i>;<br/>
<a NAME="E29:19_5_1"/><b>then </b><i><font color="Green">E125</font></i>: 
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1'</font> <a href="euclid.html#K8">-</a> <font color="Maroon">p2'</font></span>)</span></span><a href="euclid.html#K12">.|</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K13">Pitag_dist</a> <font color="Maroon">n</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">p1'</font>,<font color="Maroon">p2'</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="euclid.html#D6">EUCLID:def 6</a></i>;<br/>
<a NAME="E30:19_5_1"/>
<font color="Maroon">r</font> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K13">Pitag_dist</a> <font color="Maroon">n</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">p1'</font>,<font color="Maroon">p2'</font></span>)</span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i>, , <a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/>
<a NAME="E31:19_5_1"/><b>then </b>
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">WW1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">aa1</font></span>)</span></span><a href="euclid.html#K12">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K13">Pitag_dist</a> <font color="Maroon">n</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">p1'</font>,<font color="Maroon">p2'</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="euclid.html#K13">Pitag_dist</a> <font color="Maroon">n</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">p1'</font>,<font color="Maroon">p2'</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="txt" href="topreal1.html#E1:19"><i><font color="Green">E37</font></i></a>, <a class="ref" href="xreal_1.html#T70">XREAL_1:70</a></i>;<br/>
<a NAME="E32:19_5_1"/><b>then </b>
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">WW1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">aa1</font></span>)</span></span><a href="euclid.html#K12">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>;<br/>
<a NAME="E33:19_5_1"/><b>then </b>
the <a href="metric_1.html#U1">distance</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">WW'</font>,<font color="Maroon">aa'</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="euclid.html#D6">EUCLID:def 6</a></i>;<br/>
<a NAME="E34:19_5_1"/><b>then </b>
the <a href="metric_1.html#U1">distance</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span> <a href="topmetr.html#K1">|</a> <font color="Maroon">PP</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">Y</font>,<font color="Maroon">aa</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="topmetr.html#D1">TOPMETR:def 1</a></i>;<br/>
<a NAME="E35:19_5_1"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">Y</font>,<font color="Maroon">aa</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#D1">METRIC_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E36:19_5_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">Y</font>,<font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E25:19_5"/><b>then </b>
<font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">H</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Q</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E26:19_5"/>
<font color="Maroon">f</font> <a href="relat_1.html#K9">.:</a> <font color="Maroon">H</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">G</font>
 <b>by </b><i><a class="txt" href="topreal1.html#E1"><i><font color="Green">Lemma25</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E21:19"/><b>then </b><i><font color="Green">E126</font></i>: 
<font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D3">BORSUK_1:def 3</a></i>;<br/>
<b>take </b>
<font color="Maroon">f</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="topreal1.html#D2" target="_self">TOPREAL1:def 2</a><br/>

<a NAME="E22:19"/><i><font color="Green">E127</font></i>: 
 <a href="topmetr.html#K5">I[01]</a>  is <a href="compts_1.html#V2">compact</a>
 
<b>by </b><i><a class="ref" href="heine.html#T11">HEINE:11</a>, <a class="ref" href="topmetr.html#T27">TOPMETR:27</a></i>;<br/>
<a NAME="E23:19"/>
 <a href="pcomps_1.html#K4">TopSpaceMetr</a> <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span> <a href="compts_1.html#NR2" target="_self">is_T2</a> 
 
<b>by </b><i><a class="ref" href="pcomps_1.html#T38">PCOMPS_1:38</a></i>;<br/>
<a NAME="E24:19"/><b>then </b>
 <a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font> <a href="compts_1.html#NR2" target="_self">is_T2</a> 
 
;<br/>
<a NAME="E25:19"/><b>then </b>
<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" target="_self">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font></span>)</span> <a href="compts_1.html#NR2" target="_self">is_T2</a> 
 
<b>by </b><i><a class="ref" href="topmetr.html#T3">TOPMETR:3</a></i>;<br/>
<b>hence </b><a NAME="E26:19"/>
<font color="Maroon">f</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 <b>by </b><i>, , , , <a class="txt" href="topreal1.html#E1"><i><font color="Green">Lemma25</font></i></a>, , <a class="ref" href="compts_1.html#T26">COMPTS_1:26</a></i>;<br/>

<a NAME="E27:19"/>
0 <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</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/>
<b>hence </b><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> 0</span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default">0 <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">p1</font> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default">0 <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T33">EUCLID:33</a></i>
<br/>.= 
<font color="Maroon">p1</font> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K16">0.REAL</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T33">EUCLID:33</a></i>
<br/>.= 
<font color="Maroon">p1</font>
<b>by </b><i><a class="ref" href="euclid.html#T31">EUCLID:31</a></i>
;<br/><br/>

<a NAME="E29:19"/>
1 <a href="hidden.html#R2">in</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</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/>
<b>hence </b><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default">1 <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="euclid.html#K16">0.REAL</a> <font color="Maroon">n</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default">1 <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T33">EUCLID:33</a></i>
<br/>.= 
1 <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font>
<b>by </b><i><a class="ref" href="euclid.html#T31">EUCLID:31</a></i>
<br/>.= 
<font color="Maroon">p2</font>
<b>by </b><i><a class="ref" href="euclid.html#T33">EUCLID:33</a></i>
;<br/><br/>


</div>
