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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><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> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>



<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="euclid.html#T25">EUCLID:25</a></i>;<br/>
<a NAME="E2:60"/><i><font color="Green">E42</font></i>: 
 <a href="euclid_4.html#K2" target="_self">Line</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a>  <a href="euclid_4.html#K6" target="_self">Line</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</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="E1:60_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="euclid_4.html#K2" target="_self">Line</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:60_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>6</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>7</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 ;<br/>
<a NAME="E4:60_1"/>
( <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="ref" href="euclid.html#D11">EUCLID:def 11</a></i>;<br/>
<a NAME="E5:60_1"/><b>then </b>
<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>7</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <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>7</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>7</sub></font> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="euclid.html#D10">EUCLID:def 10</a></i>;<br/>
<b>hence </b><a NAME="E6:60_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="euclid_4.html#K6" target="_self">Line</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="euclid_4.html#E3:60_1"><i><font color="Green">E43</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:60"/><i><font color="Green">E43</font></i>: 
 <a href="euclid_4.html#K6" target="_self">Line</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="euclid_4.html#K2" target="_self">Line</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</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="E1:60_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="euclid_4.html#K6" target="_self">Line</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:60_2"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>6</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>7</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>7</sub></font> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 ;<br/>
<a NAME="E4:60_2"/>
( <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="ref" href="euclid.html#D11">EUCLID:def 11</a></i>;<br/>
<a NAME="E5:60_2"/><b>then </b>
<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>7</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <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>7</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>7</sub></font> <a href="euclid.html#K18">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="euclid.html#D10">EUCLID:def 10</a></i>;<br/>
<b>hence </b><a NAME="E6:60_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="euclid_4.html#K2" target="_self">Line</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="euclid_4.html#E3:60_2"><i><font color="Green">E44</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>take </b>
<font color="Maroon">c<sub>4</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/>

<b>thus </b><a NAME="E4:60"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp;  <a href="euclid_4.html#K2" target="_self">Line</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="euclid_4.html#K6" target="_self">Line</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="euclid_4.html#E2:60"><i><font color="Green">E42</font></i></a>, <a class="txt" href="euclid_4.html#E3:60"><i><font color="Green">E43</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div>
