<?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/>



<a NAME="E1:10"/>
for <font color="Olive">a</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">a</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K1" target="_self">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font> iff <font color="Olive">a</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K1" target="_self">LSeg</a> <font color="Maroon">p2</font>,<font color="Maroon">p1</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_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>thus </b><a NAME="E1:10_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">p1</font>,<font color="Maroon">p2</font> implies <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K1" target="_self">LSeg</a> <font color="Maroon">p2</font>,<font color="Maroon">p1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:10_1_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">p1</font>,<font color="Maroon">p2</font>
 ;<br/>

<a NAME="E2:10_1_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">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="E4:10_1_1"/><i><font color="Green">E37</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>
 <b>and </b><br/><a NAME="E5:10_1_1"/><i><font color="Green">E38</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">lambda</font>
 <b>and </b><br/><a NAME="E6:10_1_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">lambda</font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 ;<br/>
<a NAME="E7:10_1_1"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K5">-</a> <font color="Maroon">lambda</font> &amp; 1 <a href="real_1.html#K5">-</a> <font color="Maroon">lambda</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K5">-</a> 0 &amp; 1 <a href="real_1.html#K5">-</a> 0 <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</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> <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">lambda</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span> &amp; 1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">lambda</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">lambda</font> )
 
<b>by </b><i>, , , <a class="ref" href="real_2.html#T106">REAL_2:106</a>, <a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/>
<a NAME="E8:10_1_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">r</font></span>)</span> <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"><font color="Olive">r</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p1</font></span>)</span></span>)</span> 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> 
 
;<br/>
<b>hence </b><a NAME="E9:10_1_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">p2</font>,<font color="Maroon">p1</font>
 ;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:10_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">p2</font>,<font color="Maroon">p1</font>
 ;<br/>

<a NAME="E3:10_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">p2</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">p1</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="E5:10_1"/><i><font color="Green">E50</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">p2</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">p1</font></span>)</span>
 <b>and </b><br/><a NAME="E6:10_1"/><i><font color="Green">E54</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">lambda</font>
 <b>and </b><br/><a NAME="E7:10_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">lambda</font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 ;<br/>
<a NAME="E8:10_1"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K5">-</a> <font color="Maroon">lambda</font> &amp; 1 <a href="real_1.html#K5">-</a> <font color="Maroon">lambda</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K5">-</a> 0 &amp; 1 <a href="real_1.html#K5">-</a> 0 <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">lambda</font> <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">lambda</font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span> &amp; 1 <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">lambda</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">lambda</font> )
 
<b>by </b><i>, , , <a class="ref" href="real_2.html#T106">REAL_2:106</a>, <a class="ref" href="xreal_1.html#T50">XREAL_1:50</a></i>;<br/>
<a NAME="E9:10_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">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="p3">(<span class="default"><font color="Olive">r</font> <a href="euclid.html#K18">*</a> <font color="Maroon">p2</font></span>)</span></span>)</span> 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> 
 
;<br/>
<b>hence </b><a NAME="E10:10_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">p1</font>,<font color="Maroon">p2</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:10"/>
 <a href="topreal1.html#K1" target="_self">LSeg</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K1" target="_self">LSeg</a> <font color="Maroon">p2</font>,<font color="Maroon">p1</font>
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


</div>
