<?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>, <font color="Maroon">c<sub>4</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font> be    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>5</sub></font>;<br/>



















<b>assume </b><a NAME="E1:42"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="euclidlp.html#R3" target="_self">are_lindependent2</a> 
 ;<br/>

<b>assume </b><a NAME="E2:42"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> )
 ;<br/>

<i><font color="Green">E45</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="euclid.html#K8">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="euclid.html#K8">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K9">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclidlp.html#T26" target="_self">Th26</a>, <a class="txt" href="euclidlp.html#E2:42"><i><font color="Green">E44</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="euclid.html#K8">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclidlp.html#T26" target="_self">Th26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="euclidlp.html#T30" target="_self">Th30</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="euclid_4.html#T3">EUCLID_4:3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="ref" href="euclid_4.html#T1">EUCLID_4:1</a></i>
;<br/>
<a NAME="E4:42"/><i><font color="Green">E46</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:42_2"/><i><font color="Green">E47</font></i>: 
not <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/>

<i><font color="Green">E48</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclidlp.html#T8" target="_self">Th8</a></i>
<br/>.= 
0 <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="ref" href="euclidlp.html#T6" target="_self">Th6</a>, <a class="txt" href="euclidlp.html#E3:42"><i><font color="Green">E45</font></i></a>, <a class="txt" href="euclidlp.html#E1:42_2"><i><font color="Green">E47</font></i></a></i>
<br/>.= 
 <a href="euclid.html#K5">0*</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="euclid_4.html#T3">EUCLID_4:3</a></i>
;<br/>
<a NAME="E3:42_2"/><b>then </b><i><font color="Green">E49</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0 &amp;  <a href="real_1.html#K1">-</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i><a class="txt" href="euclidlp.html#E1:42"><i><font color="Green">E43</font></i></a>, <a class="ref" href="euclidlp.html#D2" target="_self">Def2</a></i>;<br/>
<i><font color="Green">E50</font></i>: <font color="Maroon">c<sub>2</sub></font> = 
 <a href="real_1.html#K1">-</a> 0
<b>by </b><i><a class="txt" href="euclidlp.html#E3:42_2"><i><font color="Green">E49</font></i></a></i>
<br/>.= 
0

;<br/>
<i><font color="Green">E51</font></i>: <font color="Maroon">c<sub>10</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="euclid_4.html#T3">EUCLID_4:3</a>, <a class="txt" href="euclidlp.html#E2:42"><i><font color="Green">E44</font></i></a>, <a class="txt" href="euclidlp.html#E4:42_2"><i><font color="Green">E50</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="ref" href="euclid_4.html#T1">EUCLID_4:1</a></i>
;<br/>
<i><font color="Green">E52</font></i>: <font color="Maroon">c<sub>6</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default">0 <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="euclidlp.html#E2:42"><i><font color="Green">E44</font></i></a>, <a class="txt" href="euclidlp.html#E2:42_2"><i><font color="Green">E48</font></i></a>, <a class="txt" href="euclidlp.html#E1:42"><i><font color="Green">E43</font></i></a>, <a class="ref" href="euclidlp.html#D2" target="_self">Def2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="euclid_4.html#T3">EUCLID_4:3</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="ref" href="euclid_4.html#T1">EUCLID_4:1</a></i>
;<br/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid_4.html#T4">EUCLID_4:4</a>, <a class="txt" href="euclidlp.html#E5:42_2"><i><font color="Green">E51</font></i></a>, <a class="txt" href="euclidlp.html#E6:42_2"><i><font color="Green">E52</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="euclid_4.html#T4">EUCLID_4:4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="ref" href="euclid_4.html#T7">EUCLID_4:7</a></i>
<br/>.= 
 <a href="euclid.html#K5">0*</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="euclid_4.html#T3">EUCLID_4:3</a></i>
;<br/>
<a NAME="E8:42_2"/><b>then </b><i><font color="Green">E53</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0 &amp;  <a href="real_1.html#K1">-</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i><a class="txt" href="euclidlp.html#E1:42"><i><font color="Green">E43</font></i></a>, <a class="ref" href="euclidlp.html#D2" target="_self">Def2</a></i>;<br/>
<i><font color="Green">E54</font></i>: <font color="Maroon">c<sub>1</sub></font> = 
 <a href="real_1.html#K1">-</a> 0
<b>by </b><i><a class="txt" href="euclidlp.html#E8:42_2"><i><font color="Green">E53</font></i></a></i>
<br/>.= 
0

;<br/>
<font color="Maroon">c<sub>10</sub></font> = 
<span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default">0 <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="euclid_4.html#T3">EUCLID_4:3</a>, <a class="txt" href="euclidlp.html#E2:42"><i><font color="Green">E44</font></i></a>, <a class="txt" href="euclidlp.html#E4:42_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="euclidlp.html#E9:42_2"><i><font color="Green">E54</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="euclid_4.html#T3">EUCLID_4:3</a></i>
<br/>.= 
 <a href="euclid.html#K5">0*</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="euclid_4.html#T1">EUCLID_4:1</a></i>
;<br/>
<b>hence </b><a NAME="E11:42_2"/>
contradiction
 <b>by </b><i><a class="txt" href="euclidlp.html#E1:42"><i><font color="Green">E43</font></i></a>, <a class="txt" href="euclidlp.html#E43"><i><font color="Green">Lemma40</font></i></a></i>;<br/>


</div><b>end;</b></div>
<i><font color="Green">E47</font></i>: <font color="Maroon">c<sub>7</sub></font> = 
1 <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="ref" href="euclid_4.html#T3">EUCLID_4:3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="euclidlp.html#E4:42"><i><font color="Green">E46</font></i></a>, <a class="ref" href="xcmplx_1.html#T107">XCMPLX_1:107</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="euclid_4.html#T4">EUCLID_4:4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="euclid.html#K8">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclidlp.html#T17" target="_self">Th17</a>, <a class="txt" href="euclidlp.html#E3:42"><i><font color="Green">E45</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="euclid.html#K8">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclidlp.html#T1" target="_self">Th1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="euclid.html#K8">-</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#K6">/</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="euclid_4.html#T4">EUCLID_4:4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclidlp.html#T6" target="_self">Th6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="euclidlp.html#T8" target="_self">Th8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="euclid.html#K7">+</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#K6">/</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p5">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T100">XCMPLX_1:100</a></i>
;<br/>
<i><font color="Green">E48</font></i>: <span class="p1">(<span class="default"><a href="euclid.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K9">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclidlp.html#T8" target="_self">Th8</a>, <a class="txt" href="euclidlp.html#E2:42"><i><font color="Green">E44</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K9">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclidlp.html#T26" target="_self">Th26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclidlp.html#T26" target="_self">Th26</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>7</sub></font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="euclidlp.html#T28" target="_self">Th28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="euclid_4.html#T3">EUCLID_4:3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="ref" href="euclid_4.html#T1">EUCLID_4:1</a></i>
;<br/>
<i><font color="Green">E49</font></i>: <font color="Maroon">c<sub>8</sub></font> = 
1 <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="ref" href="euclid_4.html#T3">EUCLID_4:3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="txt" href="euclidlp.html#E4:42"><i><font color="Green">E46</font></i></a>, <a class="ref" href="xcmplx_1.html#T107">XCMPLX_1:107</a></i>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="euclid_4.html#T4">EUCLID_4:4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K9">*</a> <span class="p2">(<span class="default"><a href="euclid.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid_4.html#T6">EUCLID_4:6</a>, <a class="txt" href="euclidlp.html#E6:42"><i><font color="Green">E48</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K9">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclidlp.html#T8" target="_self">Th8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclidlp.html#T1" target="_self">Th1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="euclidlp.html#T1" target="_self">Th1</a></i>
;<br/>
<b>set </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>12</sub></font> = <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>13</sub></font> = <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>;<br/>
<b>take </b>
<font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
;<br/>

<b>take </b>
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
;<br/>

<b>take </b>
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
;<br/>

<b>thus </b><a NAME="E8:42"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> &amp; <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"><span class="p3">(<span class="default"><a href="real_1.html#K1">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="euclidlp.html#E5:42"><i><font color="Green">E47</font></i></a>, <a class="txt" href="euclidlp.html#E7:42"><i><font color="Green">E49</font></i></a></i>;<br/>


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