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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></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="E1:29"/><i><font color="Green">E16</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K22">`2</a>  )
 ;<br/>

<b>set </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">c<sub>3</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> =  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>
<a NAME="E2:29"/><i><font color="Green">E17</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K1">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:29_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K1">}</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></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="E1:29_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:29_1_1"/><b>then </b><i><font color="Green">E18</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:29_1_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></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">b<sub>1</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K18">*</a> <span class="p4"><a href="euclid.html#K23">|[</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span></span>)</span> where B is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 
;<br/>
<b>then consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E5:29_1_1"/><i><font color="Green">E19</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<br/>
<b>set </b><font color="Maroon">c<sub>10</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span>;<br/>
<i><font color="Green">E20</font></i>: <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">c<sub>9</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K18">*</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span></span>)</span> <a href="euclid.html#K21">`1</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">c<sub>9</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K18">*</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>
<b>by </b><i><a class="ref" href="topreal3.html#T7" target="_self">Th7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K18">*</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>
<b>by </b><i><a class="ref" href="topreal3.html#T9" target="_self">Th9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="topreal3.html#T9" target="_self">Th9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a> 

;<br/>
<a NAME="E7:29_1_1"/>
<font color="Maroon">c<sub>8</sub></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">b<sub>1</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> where B is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="topreal3.html#E2:29_1_1"><i><font color="Green">E18</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E9:29_1_1"/><i><font color="Green">E21</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<br/>
<i><font color="Green">E22</font></i>: <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">c<sub>11</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="euclid.html#K22">`2</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">c<sub>11</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>
<b>by </b><i><a class="ref" href="topreal3.html#T7" target="_self">Th7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>
<b>by </b><i><a class="ref" href="topreal3.html#T9" target="_self">Th9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="topreal3.html#T9" target="_self">Th9</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K22">`2</a> 
<b>by </b><i><a class="txt" href="topreal3.html#E1:29"><i><font color="Green">E16</font></i></a></i>
;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> = 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K18">*</a> <span class="p5"><a href="euclid.html#K23">|[</a><span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p0">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K18">*</a> <span class="p5"><a href="euclid.html#K23">|[</a><span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p0">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T57">EUCLID:57</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="topreal3.html#E5:29_1_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="topreal3.html#E6:29_1_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="topreal3.html#E9:29_1_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="topreal3.html#E10:29_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="euclid.html#T57">EUCLID:57</a></i>
;<br/>
<b>hence </b><a NAME="E12:29_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="topreal3.html#E5:29_1_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


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

<b>let </b><font color="Maroon">c<sub>8</sub></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="E2:29_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E3:29_1"/><b>then </b><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E4:29_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/>
<b>hence </b><a NAME="E5:29_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="topreal3.html#E3:29_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:29"/><i><font color="Green">E18</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:29_2"/><i><font color="Green">E19</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>8</sub></font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/>
<a NAME="E3:29_2"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="topreal3.html#E1:29_2"><i><font color="Green">E19</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E4:29_2"/><b>then </b>
<font color="Maroon">c<sub>8</sub></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">b<sub>1</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p4"><a href="euclid.html#K23">|[</a><span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> where B is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 
;<br/>
<b>then consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E6:29_2"/><i><font color="Green">E21</font></i>: 
( <font color="Maroon">c<sub>8</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="Maroon">c<sub>9</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<br/>
<i><font color="Green">E22</font></i>: <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">c<sub>9</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="euclid.html#K22">`2</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">c<sub>9</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>
<b>by </b><i><a class="ref" href="topreal3.html#T7" target="_self">Th7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>
<b>by </b><i><a class="ref" href="topreal3.html#T9" target="_self">Th9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="topreal3.html#T9" target="_self">Th9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K22">`2</a> 

;<br/>
<a NAME="E8:29_2"/>
<font color="Maroon">c<sub>8</sub></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">b<sub>1</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> where B is   <a href="real_1.html#NM1">Real</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="topreal3.html#E3:29_2"><i><font color="Green">E20</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E10:29_2"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">c<sub>8</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="Maroon">c<sub>10</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<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">c<sub>10</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="euclid.html#K22">`2</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">c<sub>10</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>
<b>by </b><i><a class="ref" href="topreal3.html#T7" target="_self">Th7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K22">`2</a> </span>)</span>
<b>by </b><i><a class="ref" href="topreal3.html#T9" target="_self">Th9</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K22">`2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="topreal3.html#T9" target="_self">Th9</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K22">`2</a> 
<b>by </b><i><a class="txt" href="topreal3.html#E1:29"><i><font color="Green">E16</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E12:29_2"/>
contradiction
 <b>by </b><i><a class="txt" href="topreal3.html#E1:29"><i><font color="Green">E16</font></i></a>, <a class="txt" href="topreal3.html#E6:29_2"><i><font color="Green">E21</font></i></a>, <a class="txt" href="topreal3.html#E7:29_2"><i><font color="Green">E22</font></i></a>, <a class="txt" href="topreal3.html#E10:29_2"><i><font color="Green">E23</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K21">`1</a> </span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K22">`2</a> </span>)</span></span><a href="euclid.html#K23">]|</a></span>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T23">XBOOLE_1:23</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="tarski.html#K1">}</a></span>
<b>by </b><i><a class="txt" href="topreal3.html#E2:29"><i><font color="Green">E17</font></i></a>, <a class="txt" href="topreal3.html#E3:29"><i><font color="Green">E18</font></i></a></i>
;<br/>


</div>
