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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">x</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">L</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>consider </b><font color="Maroon">x1</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">x2</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="E2:64"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">L</font> <a href="hidden.html#R1">=</a>  <a href="euclid_4.html#K2">Line</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font>
 <b>by </b><i><a class="ref" href="euclidlp.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E3:64"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="euclid_4.html#K4">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="euclid.html#K8">-</a> <font color="Olive">x'</font></span>)</span></span><a href="euclid_4.html#K4">.|</a></span> where <font color="Olive">x'</font> is    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font> : <font color="Olive">x'</font> <a href="hidden.html#R2">in</a>  <a href="euclid_4.html#K2">Line</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font> </span>}</span>  <a href="hidden.html#R1">=</a>  <a href="euclidlp.html#K2" target="_self">dist_v</a> <font color="Maroon">x</font>,<font color="Maroon">L</font>
 
<b>by </b><i><a class="txt" href="euclidlp.html#E8"><i><font color="Green">Lemma60</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">R</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="euclid_4.html#K4">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">x</font> <a href="euclid.html#K8">-</a> <font color="Olive">x'</font></span>)</span></span><a href="euclid_4.html#K4">.|</a></span> where <font color="Olive">x'</font> is    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font> : <font color="Olive">x'</font> <a href="hidden.html#R2">in</a>  <a href="euclid_4.html#K2">Line</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font> </span>}</span>  as   <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  ;<br/>
<b>consider </b><font color="Maroon">x0</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="E6:64"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a>  <a href="euclid_4.html#K2">Line</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font> &amp; <span class="p1"><a href="euclid_4.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x0</font></span>)</span></span><a href="euclid_4.html#K4">.|</a></span> <a href="hidden.html#R1">=</a>  <a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">R</font> &amp; <font color="Maroon">x1</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x2</font>,<font color="Maroon">x</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x0</font> <a href="euclid_4.html#R1">are_orthogonal</a>  )
 <b>by </b><i><a class="ref" href="euclid_4.html#T45">EUCLID_4:45</a></i>;<br/>
<b>take </b>
<font color="Maroon">x0</font>
;<br/>

<b>thus </b><a NAME="E7:64"/>
( <font color="Maroon">x0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">L</font> &amp; <span class="p1"><a href="euclid_4.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="euclid.html#K8">-</a> <font color="Maroon">x0</font></span>)</span></span><a href="euclid_4.html#K4">.|</a></span> <a href="hidden.html#R1">=</a>  <a href="euclidlp.html#K3" target="_self">dist</a> <font color="Maroon">x</font>,<font color="Maroon">L</font> )
 <b>by </b><i><a class="txt" href="euclidlp.html#E8"><i><font color="Green">Lemma60</font></i></a>, <a class="ref" href="euclidlp.html#T8" target="_self">Th8</a></i>;<br/>


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