<?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>, <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font>, <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> be    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>1</sub></font>;<br/>

















<b>assume </b><a NAME="E1:45"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>5</sub></font> <a href="euclidlp.html#R3" target="_self">are_lindependent2</a>  &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="euclid_4.html#K2">Line</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="euclid_4.html#K2">Line</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="euclid_4.html#K2">Line</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="euclid_4.html#K2">Line</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>9</sub></font> )
 ;<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="E3:45"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="euclid.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="euclidlp.html#T36" target="_self">Th36</a></i>;<br/>
<a NAME="E4:45"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="euclidlp.html#E1:45"><i><font color="Green">E45</font></i></a>, <a class="txt" href="euclidlp.html#E3:45"><i><font color="Green">E46</font></i></a>, <a class="ref" href="euclidlp.html#T15" target="_self">Th15</a></i>;<br/>
<b>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="E6:45"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="euclid.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="euclidlp.html#E1:45"><i><font color="Green">E45</font></i></a>, <a class="ref" href="euclidlp.html#T36" target="_self">Th36</a></i>;<br/>
<a NAME="E7:45"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="euclidlp.html#E1:45"><i><font color="Green">E45</font></i></a>, <a class="txt" href="euclidlp.html#E6:45"><i><font color="Green">E48</font></i></a>, <a class="ref" href="euclidlp.html#T15" target="_self">Th15</a></i>;<br/>
<a NAME="E8:45"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="real_1.html#NM1">Real</a> st <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="euclid.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="euclid.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K5">0*</a> <font color="Maroon">c<sub>1</sub></font> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>13</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/>



<b>assume </b><a NAME="E1:45_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="euclid.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="euclid.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K5">0*</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>then </b> <a href="euclid.html#K5">0*</a> <font color="Maroon">c<sub>1</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>10</sub></font></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#K8">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="euclid.html#K7">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="euclid.html#K9">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="euclid.html#K9">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></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#E6:45"><i><font color="Green">E48</font></i></a>, <a class="txt" href="euclidlp.html#E3:45"><i><font color="Green">E46</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>10</sub></font></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#K8">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></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>13</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>11</sub></font></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#K8">-</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid_4.html#T4">EUCLID_4:4</a></i>
;<br/>
<a NAME="E3:45_1"/><b>then </b><i><font color="Green">E50</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>13</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i><a class="txt" href="euclidlp.html#E1:45"><i><font color="Green">E45</font></i></a>, <a class="ref" href="euclidlp.html#D2" target="_self">Def2</a></i>;<br/>
<i><font color="Green">E51</font></i>: <font color="Maroon">c<sub>12</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>10</sub></font>
<b>by </b><i><a class="txt" href="euclidlp.html#E4:45"><i><font color="Green">E47</font></i></a>, <a class="ref" href="xcmplx_1.html#T90">XCMPLX_1:90</a></i>
<br/>.= 
0
<b>by </b><i><a class="txt" href="euclidlp.html#E3:45_1"><i><font color="Green">E50</font></i></a></i>
;<br/>
<font color="Maroon">c<sub>13</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="real_1.html#K4">*</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>11</sub></font>
<b>by </b><i><a class="txt" href="euclidlp.html#E7:45"><i><font color="Green">E49</font></i></a>, <a class="ref" href="xcmplx_1.html#T90">XCMPLX_1:90</a></i>
<br/>.= 
0
<b>by </b><i><a class="txt" href="euclidlp.html#E3:45_1"><i><font color="Green">E50</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E6:45_1"/>
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="txt" href="euclidlp.html#E4:45_1"><i><font color="Green">E51</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E9:45"/>
<font color="Maroon">c<sub>7</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font> <a href="euclidlp.html#R3" target="_self">are_lindependent2</a> 
 <b>by </b><i><a class="ref" href="euclidlp.html#D2" target="_self">Def2</a></i>;<br/>


</div>
