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<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> 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>let </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> be    <a href="subset_1.html#M2">Element</a> of  <a href="euclidlp.html#K1" target="_self">line_of_REAL</a> <font color="Maroon">c<sub>1</sub></font>;<br/>













<b>assume </b><a NAME="E1:83"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="euclidlp.html#R6" target="_self">_|_</a> <font color="Maroon">c<sub>7</sub></font> )
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font> being    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:83"/><i><font color="Green">E75</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="euclid_4.html#K2">Line</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="euclid_4.html#K2">Line</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font> &amp; <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#R4" target="_self">_|_</a> <font color="Maroon">c<sub>11</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="ref" href="euclidlp.html#D8" target="_self">Def8</a></i>;<br/>
<a NAME="E4:83"/>
<font color="Maroon">c<sub>9</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>10</sub></font> <a href="euclid_4.html#R1">are_orthogonal</a> 
 
<b>by </b><i><a class="txt" href="euclidlp.html#E3:83"><i><font color="Green">E75</font></i></a>, <a class="ref" href="euclidlp.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E5:83"/><b>then </b><i><font color="Green">E76</font></i>: 
<span class="p1"><a href="euclid_4.html#K5">|(</a><span class="default"><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 class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span><a href="euclid_4.html#K5">)|</a></span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="euclid_4.html#D6">EUCLID_4:def 6</a></i>;<br/>
<a NAME="E6:83"/><i><font color="Green">E77</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K5">0*</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K5">0*</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="euclidlp.html#E1:83"><i><font color="Green">E74</font></i></a>, <a class="ref" href="euclidlp.html#T14" target="_self">Th14</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E8:83"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="euclid.html#K9">*</a> <span class="p1">(<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>
 <b>by </b><i><a class="txt" href="euclidlp.html#E1:83"><i><font color="Green">E74</font></i></a>, <a class="txt" href="euclidlp.html#E3:83"><i><font color="Green">E75</font></i></a>, <a class="ref" href="euclidlp.html#T36" target="_self">Th36</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E10:83"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="euclid.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="euclidlp.html#E1:83"><i><font color="Green">E74</font></i></a>, <a class="txt" href="euclidlp.html#E3:83"><i><font color="Green">E75</font></i></a>, <a class="ref" href="euclidlp.html#T36" target="_self">Th36</a></i>;<br/>
<span class="p1"><a href="euclid_4.html#K5">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="euclid_4.html#K5">)|</a></span> = 
<font color="Maroon">c<sub>12</sub></font> <a href="real_1.html#K4">*</a> <span class="p1"><a href="euclid_4.html#K5">|(</a><span class="default"><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 class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="euclid_4.html#K5">)|</a></span>
<b>by </b><i><a class="ref" href="euclid_4.html#T26">EUCLID_4:26</a>, <a class="txt" href="euclidlp.html#E8:83"><i><font color="Green">E78</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>12</sub></font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="real_1.html#K4">*</a> <span class="p2"><a href="euclid_4.html#K5">|(</a><span class="default"><span class="p3">(<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 class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span><a href="euclid_4.html#K5">)|</a></span></span>)</span>
<b>by </b><i><a class="ref" href="euclid_4.html#T27">EUCLID_4:27</a>, <a class="txt" href="euclidlp.html#E10:83"><i><font color="Green">E79</font></i></a></i>
<br/>.= 
0
<b>by </b><i><a class="txt" href="euclidlp.html#E5:83"><i><font color="Green">E76</font></i></a></i>
;<br/>
<a NAME="E12:83"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>4</sub></font> <a href="euclid_4.html#R1">are_orthogonal</a> 
 
<b>by </b><i><a class="ref" href="euclid_4.html#D6">EUCLID_4:def 6</a></i>;<br/>
<b>hence </b><a NAME="E13:83"/>
<font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>2</sub></font> <a href="euclidlp.html#R4" target="_self">_|_</a> <font color="Maroon">c<sub>5</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="euclidlp.html#E6:83"><i><font color="Green">E77</font></i></a>, <a class="ref" href="euclidlp.html#D3" target="_self">Def3</a></i>;<br/>


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