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

<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">E</font> be   <a href="eqrel_1.html#NM1">Tolerance</a> of <font color="Maroon">X</font>;<br/>



<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>thus </b><a NAME="E1:16_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="coh_sp.html#K1" target="_self">Web</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K2" target="_self">CohSp</a> <font color="Maroon">E</font></span>)</span> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:16_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="coh_sp.html#K1" target="_self">Web</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K2" target="_self">CohSp</a> <font color="Maroon">E</font></span>)</span>
 ;<br/>

<a NAME="E2:16_1_1"/><b>then </b><i><font color="Green">E17</font></i>: 
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R2">in</a>  <a href="coh_sp.html#K2" target="_self">CohSp</a> <font color="Maroon">E</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:16_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K2">}</a></span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K2">}</a></span> )
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<b>hence </b><a NAME="E4:16_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div><b>assume </b><a NAME="E2:16_1"/><i><font color="Green">E18</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font>
 ;<br/><a NAME="E3:16_1"/><b>then </b><i><font color="Green">E25</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E4:16_1"/>
for <font color="Olive">u</font>, <font color="Olive">v</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K2">}</a></span> &amp; <font color="Olive">v</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K2">}</a></span> holds <br/><span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">u</font>,<font color="Olive">v</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:16_1_2"/>
( <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K2">}</a></span> &amp; <font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K2">}</a></span> )
 ;<br/>

<a NAME="E2:16_1_2"/><b>then </b>
( ( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> or <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> ) &amp; ( <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> or <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> ) )
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<b>hence </b><a NAME="E3:16_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">u</font>,<font color="Maroon">v</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">E</font>
 <b>by </b><i>, , <a class="ref" href="eqrel_1.html#T12">EQREL_1:12</a>, <a class="ref" href="toler_1.html#T27">TOLER_1:27</a></i>;<br/>


</div><b>end;</b></div><a NAME="E5:16_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K2">}</a></span> <a href="hidden.html#R2">in</a>  <a href="coh_sp.html#K2" target="_self">CohSp</a> <font color="Maroon">E</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E6:16_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="coh_sp.html#K1" target="_self">Web</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K2" target="_self">CohSp</a> <font color="Maroon">E</font></span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:16"/>
 <a href="coh_sp.html#K1" target="_self">Web</a> <span class="p1">(<span class="default"><a href="coh_sp.html#K2" target="_self">CohSp</a> <font color="Maroon">E</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">E</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D2">RELAT_1:def 2</a></i>;<br/>


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