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<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">x0</font> be    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font>;<br/><b>let </b><font color="Maroon">x1</font> be    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font>;<br/><b>let </b><font color="Maroon">y0</font> be    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font>;<br/><b>let </b><font color="Maroon">y1</font> be    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font>;<br/><b>let </b><font color="Maroon">L1</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">n</font>;<br/><b>let </b><font color="Maroon">L2</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">n</font>;<br/>













<b>assume </b><a NAME="E1:83"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L1</font> &amp; <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L1</font> &amp; <font color="Maroon">x0</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x1</font> &amp; <font color="Maroon">y0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L2</font> &amp; <font color="Maroon">y1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L2</font> &amp; <font color="Maroon">y0</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y1</font> &amp; <font color="Maroon">L1</font> <a href="euclidlp.html#R6" target="_self">_|_</a> <font color="Maroon">L2</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">x2</font> being    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font>, <font color="Maroon">x3</font> being    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font>, <font color="Maroon">y2</font> being    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font>, <font color="Maroon">y3</font> being    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font><b> such that </b><br/><a NAME="E3:83"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">L1</font> <a href="hidden.html#R1">=</a>  <a href="euclid_4.html#K2">Line</a> <font color="Maroon">x2</font>,<font color="Maroon">x3</font> &amp; <font color="Maroon">L2</font> <a href="hidden.html#R1">=</a>  <a href="euclid_4.html#K2">Line</a> <font color="Maroon">y2</font>,<font color="Maroon">y3</font> &amp; <font color="Maroon">x3</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x2</font> <a href="euclidlp.html#R4" target="_self">_|_</a> <font color="Maroon">y3</font> <a href="euclid.html#K8">-</a> <font color="Maroon">y2</font> )
 <b>by </b><i/>;<br/>
<a NAME="E4:83"/>
<font color="Maroon">x3</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x2</font>,<font color="Maroon">y3</font> <a href="euclid.html#K8">-</a> <font color="Maroon">y2</font> <a href="euclid_4.html#R1">are_orthogonal</a> 
 
<b>by </b><i><a class="ref" href="euclidlp.html#T19" target="_self">Th19</a>, </i>;<br/>
<a NAME="E5:83"/><b>then </b><i><font color="Green">E88</font></i>: 
<span class="p1"><a href="euclid_4.html#K5">|(</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x3</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">y3</font> <a href="euclid.html#K8">-</a> <font color="Maroon">y2</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">E89</font></i>: 
( <font color="Maroon">x1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x0</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K5">0*</a> <font color="Maroon">n</font> &amp; <font color="Maroon">y1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">y0</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K5">0*</a> <font color="Maroon">n</font> )
 
<b>by </b><i><a class="ref" href="euclidlp.html#T18" target="_self">Th18</a>, </i>;<br/>
<b>consider </b><font color="Maroon">s</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E8:83"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">x1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="euclid.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">x3</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x2</font></span>)</span>
 <b>by </b><i><a class="ref" href="euclidlp.html#T18" target="_self">Th18</a>, <a class="ref" href="euclidlp.html#T19" target="_self">Th19</a>, </i>;<br/>
<b>consider </b><font color="Maroon">t</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E10:83"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">y1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">y0</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> <a href="euclid.html#K9">*</a> <span class="p1">(<span class="default"><font color="Maroon">y3</font> <a href="euclid.html#K8">-</a> <font color="Maroon">y2</font></span>)</span>
 <b>by </b><i><a class="ref" href="euclidlp.html#T18" target="_self">Th18</a>, <a class="ref" href="euclidlp.html#T19" target="_self">Th19</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">x1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x0</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">y1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">y0</font></span>)</span></span><a href="euclid_4.html#K5">)|</a></span> = 
<font color="Maroon">s</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">x3</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">y1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">y0</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="ref" href="euclidlp.html#T23" target="_self">Th23</a></i>
<br/>.= 
<font color="Maroon">s</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">t</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">x3</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x2</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">y3</font> <a href="euclid.html#K8">-</a> <font color="Maroon">y2</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="ref" href="euclidlp.html#T25" target="_self">Th25</a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="euclidlp.html#T21" target="_self">Th21</a></i>
;<br/>
<a NAME="E12:83"/><b>then </b>
<font color="Maroon">x1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x0</font>,<font color="Maroon">y1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">y0</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">x1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x0</font> <a href="euclidlp.html#R4" target="_self">_|_</a> <font color="Maroon">y1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">y0</font>
 <b>by </b><i><a class="ref" href="euclidlp.html#T22" target="_self">Th22</a>, </i>;<br/>


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