<?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">p</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>;<br/>



<a NAME="E1:9"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K1" target="_self">LSeg</a> <font color="Maroon">p</font>,<font color="Maroon">p</font>
 
<b>by </b><i/>;<br/>
<a NAME="E2:9"/><b>then </b><i><font color="Green">E37</font></i>: 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">p</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K1" target="_self">LSeg</a> <font color="Maroon">p</font>,<font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/>
<a NAME="E3:9"/>
 <a href="topreal1.html#K1" target="_self">LSeg</a> <font color="Maroon">p</font>,<font color="Maroon">p</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">p</font></span><a href="tarski.html#K1">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_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="E1:9_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K1" target="_self">LSeg</a> <font color="Maroon">p</font>,<font color="Maroon">p</font>
 ;<br/>

<a NAME="E2:9_1"/><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">p</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">p</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="E4:9_1"/><i><font color="Green">E38</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">p</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">p</font></span>)</span>
 <b>and </b><br/><a NAME="E5:9_1"/>
( 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/>
<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">lambda</font></span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">lambda</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p</font>
<b>by </b><i>, <a class="ref" href="euclid.html#T37">EUCLID:37</a></i>
<br/>.= 
<font color="Maroon">p</font>
<b>by </b><i><a class="ref" href="euclid.html#T33">EUCLID:33</a></i>
;<br/>
<b>hence </b><a NAME="E7:9_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">p</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:9"/>
 <a href="topreal1.html#K1" target="_self">LSeg</a> <font color="Maroon">p</font>,<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">p</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div>
