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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> 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>3</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="E1:60"/><i><font color="Green">E58</font></i>: 
( not <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="euclid_4.html#NR1" target="_self">is_line</a>  )
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</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="E3:60"/><i><font color="Green">E59</font></i>: 
( <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>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="euclid_4.html#K2">Line</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="euclid_4.html#D2">EUCLID_4:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>6</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="E5:60"/><i><font color="Green">E60</font></i>: 
( <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>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>6</sub></font> <a href="euclid_4.html#R1">are_orthogonal</a>  )
 <b>by </b><i><a class="ref" href="euclidlp.html#T48" target="_self">Th48</a></i>;<br/>
<a NAME="E6:60"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>5</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>2</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>6</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#E3:60"><i><font color="Green">E59</font></i></a>, <a class="txt" href="euclidlp.html#E5:60"><i><font color="Green">E60</font></i></a>, <a class="txt" href="euclidlp.html#E1:60"><i><font color="Green">E58</font></i></a>, <a class="ref" href="euclidlp.html#T14" target="_self">Th14</a></i>;<br/>
<a NAME="E7:60"/><b>then </b><i><font color="Green">E62</font></i>: 
<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#R4" target="_self">_|_</a> <font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="txt" href="euclidlp.html#E5:60"><i><font color="Green">E60</font></i></a>, <a class="ref" href="euclidlp.html#D3" target="_self">Def3</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>7</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="E9:60"/><i><font color="Green">E63</font></i>: 
( <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>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="euclidlp.html#E1:60"><i><font color="Green">E58</font></i></a>, <a class="ref" href="euclid_4.html#T15">EUCLID_4:15</a></i>;<br/>
<a NAME="E10:60"/><i><font color="Green">E64</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#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#E9:60"><i><font color="Green">E63</font></i></a>, <a class="ref" href="euclidlp.html#T14" target="_self">Th14</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E12:60"/><i><font color="Green">E65</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>8</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#E5:60"><i><font color="Green">E60</font></i></a>, <a class="txt" href="euclidlp.html#E3:60"><i><font color="Green">E59</font></i></a>, <a class="txt" href="euclidlp.html#E9:60"><i><font color="Green">E63</font></i></a>, <a class="ref" href="euclidlp.html#T36" target="_self">Th36</a></i>;<br/>
<a NAME="E13:60"/><i><font color="Green">E66</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="euclidlp.html#R2" target="_self">//</a> <font color="Maroon">c<sub>4</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="euclidlp.html#E10:60"><i><font color="Green">E64</font></i></a>, <a class="txt" href="euclidlp.html#E6:60"><i><font color="Green">E61</font></i></a>, <a class="txt" href="euclidlp.html#E12:60"><i><font color="Green">E65</font></i></a>, <a class="ref" href="euclidlp.html#D1" target="_self">Def1</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>6</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>7</sub></font>
;<br/>

<b>thus </b><a NAME="E14:60"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="euclid_4.html#K2">Line</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>6</sub></font> <a href="euclidlp.html#R4" target="_self">_|_</a> <font color="Maroon">c<sub>7</sub></font> <a href="euclid.html#K8">-</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="euclidlp.html#E1:60"><i><font color="Green">E58</font></i></a>, <a class="txt" href="euclidlp.html#E3:60"><i><font color="Green">E59</font></i></a>, <a class="txt" href="euclidlp.html#E5:60"><i><font color="Green">E60</font></i></a>, <a class="txt" href="euclidlp.html#E9:60"><i><font color="Green">E63</font></i></a>, <a class="ref" href="euclidlp.html#T35" target="_self">Th35</a>, <a class="txt" href="euclidlp.html#E13:60"><i><font color="Green">E66</font></i></a>, <a class="txt" href="euclidlp.html#E7:60"><i><font color="Green">E62</font></i></a>, <a class="ref" href="euclidlp.html#T49" target="_self">Th49</a></i>;<br/>


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