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

<b>let </b><font color="Maroon">cn</font> be   <a href="real_1.html#NM1">Real</a>;<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> 2</span>)</span>;<br/>



<b>assume </b><a NAME="E1:155"/><i><font color="Green">E34</font></i>: 
(  <a href="real_1.html#K1">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">cn</font> &amp; <font color="Maroon">cn</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 )
 ;<br/>

<b>reconsider </b><font color="Maroon">O</font> =  <a href="euclid.html#K16">0.REAL</a> 2 as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">V0</font> =  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">O</font>,<span class="p1">(<span class="default"><span class="p2"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">p2</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="real_1.html#K3">+</a> 1</span>)</span> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="ref" href="euclid.html#T25">EUCLID:25</a></i>;<br/>
<a NAME="E4:155"/><i><font color="Green">E36</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">p2</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">p2</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="real_1.html#K3">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/>
<a NAME="E5:155"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">p2</font></span><a href="toprns_1.html#K5">.|</a></span>
 
<b>by </b><i><a class="ref" href="toprns_1.html#T26">TOPRNS_1:26</a></i>;<br/>
<a NAME="E6:155"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">O</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V0</font>
 
<b>by </b><i>, <a class="ref" href="goboard6.html#T4">GOBOARD6:4</a></i>;<br/>
<a NAME="E7:155"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">V0</font> <a href="tarski.html#R1">c=</a>  <a href="metric_1.html#K10">cl_Ball</a> <font color="Maroon">O</font>,<span class="p1">(<span class="default"><span class="p2"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">p2</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="real_1.html#K3">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="metric_1.html#T15">METRIC_1:15</a></i>;<br/>
<b>reconsider </b><font color="Maroon">K01</font> =  <a href="metric_1.html#K10">cl_Ball</a> <font color="Maroon">O</font>,<span class="p1">(<span class="default"><span class="p2"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">p2</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="real_1.html#K3">+</a> 1</span>)</span> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="ref" href="euclid.html#T25">EUCLID:25</a></i>;<br/>
<a NAME="E9:155"/>
<font color="Maroon">K01</font> is <a href="compts_1.html#V6">compact</a>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">K0</font> =  <a href="metric_1.html#K10">cl_Ball</a> <font color="Maroon">O</font>,<span class="p1">(<span class="default"><span class="p2"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">p2</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="real_1.html#K3">+</a> 1</span>)</span> as  non <a href="xboole_0.html#V1">empty</a> <a href="compts_1.html#V6">compact</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">u2</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> 2</span>)</span> <b>by </b><i><a class="ref" href="euclid.html#T25">EUCLID:25</a></i>;<br/>
<a NAME="E12:155"/><i><font color="Green">E40</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">cn</font> <a href="jgraph_4.html#K9" target="_self">-FanMorphS</a> </span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">K0</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">K0</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="155_1"><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="E1:155_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">cn</font> <a href="jgraph_4.html#K9" target="_self">-FanMorphS</a> </span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">K0</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="E3:155_1"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">cn</font> <a href="jgraph_4.html#K9" target="_self">-FanMorphS</a> </span>)</span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K0</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">cn</font> <a href="jgraph_4.html#K9" target="_self">-FanMorphS</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">q</font> = <font color="Maroon">x</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> 2</span>)</span> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">uq</font> = <font color="Maroon">q</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span> <b>by </b><i><a class="ref" href="euclid.html#T25">EUCLID:25</a></i>;<br/>
<a NAME="E6:155_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">cn</font> <a href="jgraph_4.html#K9" target="_self">-FanMorphS</a> </span>)</span>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">u</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> 2</span>)</span> <b>by </b><i><a class="ref" href="euclid.html#T25">EUCLID:25</a></i>;<br/>
<b>reconsider </b><font color="Maroon">q4</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#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i/>;<br/>
<a NAME="E9:155_1"/><i><font color="Green">E43</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q4</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q</font></span><a href="toprns_1.html#K5">.|</a></span>
 
<b>by </b><i>, <a class="txt" href="jgraph_4.html#E34"><i><font color="Green">Lemma141</font></i></a></i>;<br/>
<a NAME="E10:155_1"/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">O</font>,<font color="Maroon">uq</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">p2</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="real_1.html#K3">+</a> 1
 
<b>by </b><i>, <a class="ref" href="metric_1.html#T13">METRIC_1:13</a></i>;<br/>
<a NAME="E11:155_1"/><b>then </b>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K16">0.REAL</a> 2</span>)</span> <a href="euclid.html#K20">-</a> <font color="Maroon">q</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">p2</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="real_1.html#K3">+</a> 1
 
<b>by </b><i><a class="ref" href="jgraph_1.html#T45">JGRAPH_1:45</a></i>;<br/>
<a NAME="E12:155_1"/><b>then </b>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K16">0.REAL</a> 2</span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><a href="euclid.html#K19">-</a> <font color="Maroon">q</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">p2</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="real_1.html#K3">+</a> 1
 
<b>by </b><i><a class="ref" href="euclid.html#T45">EUCLID:45</a></i>;<br/>
<a NAME="E13:155_1"/><b>then </b>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K19">-</a> <font color="Maroon">q</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">p2</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="real_1.html#K3">+</a> 1
 
<b>by </b><i><a class="ref" href="euclid.html#T31">EUCLID:31</a></i>;<br/>
<a NAME="E14:155_1"/><b>then </b>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">p2</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="real_1.html#K3">+</a> 1
 
<b>by </b><i><a class="ref" href="toprns_1.html#T27">TOPRNS_1:27</a></i>;<br/>
<a NAME="E15:155_1"/><b>then </b>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K19">-</a> <font color="Maroon">q4</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">p2</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="real_1.html#K3">+</a> 1
 
<b>by </b><i>, <a class="ref" href="toprns_1.html#T27">TOPRNS_1:27</a></i>;<br/>
<a NAME="E16:155_1"/><b>then </b>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K16">0.REAL</a> 2</span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><a href="euclid.html#K19">-</a> <font color="Maroon">q4</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">p2</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="real_1.html#K3">+</a> 1
 
<b>by </b><i><a class="ref" href="euclid.html#T31">EUCLID:31</a></i>;<br/>
<a NAME="E17:155_1"/><b>then </b>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K16">0.REAL</a> 2</span>)</span> <a href="euclid.html#K20">-</a> <font color="Maroon">q4</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">p2</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="real_1.html#K3">+</a> 1
 
<b>by </b><i><a class="ref" href="euclid.html#T45">EUCLID:45</a></i>;<br/>
<a NAME="E18:155_1"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">O</font>,<font color="Maroon">u</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">p2</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="real_1.html#K3">+</a> 1
 
<b>by </b><i><a class="ref" href="jgraph_1.html#T45">JGRAPH_1:45</a></i>;<br/>
<b>hence </b><a NAME="E19:155_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K0</font>
 <b>by </b><i><a class="ref" href="metric_1.html#T13">METRIC_1:13</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:155"/>
<font color="Maroon">K0</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><font color="Maroon">cn</font> <a href="jgraph_4.html#K9" target="_self">-FanMorphS</a> </span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">K0</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="155_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="E1:155_2"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K0</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">u</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> 2</span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">q4</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#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i/>;<br/>
<a NAME="E4:155_2"/>
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">cn</font> <a href="jgraph_4.html#K9" target="_self">-FanMorphS</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="jgraph_4.html#T30" target="_self">Th30</a>, <a class="ref" href="jgraph_4.html#T28" target="_self">Th28</a></i>;<br/>
<a NAME="E5:155_2"/><b>then </b>
<font color="Maroon">q4</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">cn</font> <a href="jgraph_4.html#K9" target="_self">-FanMorphS</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<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="E7:155_2"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">cn</font> <a href="jgraph_4.html#K9" target="_self">-FanMorphS</a> </span>)</span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">cn</font> <a href="jgraph_4.html#K9" target="_self">-FanMorphS</a> </span>)</span> <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/>
<b>reconsider </b><font color="Maroon">q</font> = <font color="Maroon">x</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> 2</span>)</span> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">uq</font> = <font color="Maroon">q</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span> <b>by </b><i><a class="ref" href="euclid.html#T25">EUCLID:25</a></i>;<br/>
<a NAME="E10:155_2"/><i><font color="Green">E55</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q4</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">q</font></span><a href="toprns_1.html#K5">.|</a></span>
 
<b>by </b><i>, <a class="txt" href="jgraph_4.html#E34"><i><font color="Green">Lemma141</font></i></a></i>;<br/>
<a NAME="E11:155_2"/>
( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">uq</font> &amp; <font color="Maroon">q4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> &amp; <font color="Maroon">O</font> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K16">0.REAL</a> 2 )
 
;<br/>
<a NAME="E12:155_2"/><b>then </b>
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K0</font>
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E13:155_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">cn</font> <a href="jgraph_4.html#K9" target="_self">-FanMorphS</a> </span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">K0</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E14:155"/><b>then </b><i><font color="Green">E56</font></i>: 
<font color="Maroon">K0</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">cn</font> <a href="jgraph_4.html#K9" target="_self">-FanMorphS</a> </span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">K0</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E15:155"/>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K19">-</a> <font color="Maroon">p2</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">p2</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="real_1.html#K3">+</a> 1
 
<b>by </b><i>, <a class="ref" href="toprns_1.html#T27">TOPRNS_1:27</a></i>;<br/>
<a NAME="E16:155"/><b>then </b>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K16">0.REAL</a> 2</span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><a href="euclid.html#K19">-</a> <font color="Maroon">p2</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">p2</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="real_1.html#K3">+</a> 1
 
<b>by </b><i><a class="ref" href="euclid.html#T31">EUCLID:31</a></i>;<br/>
<a NAME="E17:155"/><b>then </b>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K16">0.REAL</a> 2</span>)</span> <a href="euclid.html#K20">-</a> <font color="Maroon">p2</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">p2</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="real_1.html#K3">+</a> 1
 
<b>by </b><i><a class="ref" href="euclid.html#T45">EUCLID:45</a></i>;<br/>
<a NAME="E18:155"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">O</font>,<font color="Maroon">u2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">p2</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="real_1.html#K3">+</a> 1
 
<b>by </b><i><a class="ref" href="jgraph_1.html#T45">JGRAPH_1:45</a></i>;<br/>
<a NAME="E19:155"/><b>then </b><i><font color="Green">E58</font></i>: 
<font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V0</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/>
<a NAME="E20:155"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">V0</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>set </b><font color="Maroon">q3</font> = <span class="p1">(<span class="default"><font color="Maroon">cn</font> <a href="jgraph_4.html#K9" target="_self">-FanMorphS</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font>;<br/>
<b>reconsider </b><font color="Maroon">u3</font> = <span class="p1">(<span class="default"><font color="Maroon">cn</font> <a href="jgraph_4.html#K9" target="_self">-FanMorphS</a> </span>)</span> <a href="funct_2.html#K3">.</a> <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> 2</span>)</span> <b>by </b><i><a class="ref" href="euclid.html#T25">EUCLID:25</a></i>;<br/>
<a NAME="E22:155"/>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">cn</font> <a href="jgraph_4.html#K9" target="_self">-FanMorphS</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">p2</font></span><a href="toprns_1.html#K5">.|</a></span>
 
<b>by </b><i><a class="txt" href="jgraph_4.html#E34"><i><font color="Green">Lemma141</font></i></a></i>;<br/>
<a NAME="E23:155"/><b>then </b>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K19">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">cn</font> <a href="jgraph_4.html#K9" target="_self">-FanMorphS</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">p2</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="real_1.html#K3">+</a> 1
 
<b>by </b><i>, <a class="ref" href="toprns_1.html#T27">TOPRNS_1:27</a></i>;<br/>
<a NAME="E24:155"/><b>then </b>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K16">0.REAL</a> 2</span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><a href="euclid.html#K19">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">cn</font> <a href="jgraph_4.html#K9" target="_self">-FanMorphS</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">p2</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="real_1.html#K3">+</a> 1
 
<b>by </b><i><a class="ref" href="euclid.html#T31">EUCLID:31</a></i>;<br/>
<a NAME="E25:155"/><b>then </b>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K16">0.REAL</a> 2</span>)</span> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">cn</font> <a href="jgraph_4.html#K9" target="_self">-FanMorphS</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">p2</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="real_1.html#K3">+</a> 1
 
<b>by </b><i><a class="ref" href="euclid.html#T45">EUCLID:45</a></i>;<br/>
<a NAME="E26:155"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">O</font>,<font color="Maroon">u3</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">p2</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="real_1.html#K3">+</a> 1
 
<b>by </b><i><a class="ref" href="jgraph_1.html#T45">JGRAPH_1:45</a></i>;<br/>
<a NAME="E27:155"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">cn</font> <a href="jgraph_4.html#K9" target="_self">-FanMorphS</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V0</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/>
<b>hence </b><a NAME="E28:155"/>
 ex <font color="Olive">K</font> being non <a href="xboole_0.html#V1">empty</a> <a href="compts_1.html#V6">compact</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive">K</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">cn</font> <a href="jgraph_4.html#K9" target="_self">-FanMorphS</a> </span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Olive">K</font> &amp;  ex <font color="Olive">V2</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Olive">V2</font> &amp; <font color="Olive">V2</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">V2</font> <a href="tarski.html#R1">c=</a> <font color="Olive">K</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">cn</font> <a href="jgraph_4.html#K9" target="_self">-FanMorphS</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Olive">V2</font> ) )
 <b>by </b><i>, , , </i>;<br/>


</div>
