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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">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>assume </b><a NAME="E1:50"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">x</font> <a href="euclidlp.html#R4" target="_self">_|_</a> <font color="Maroon">y0</font> &amp; <font color="Maroon">y0</font> <a href="euclidlp.html#R2" target="_self">//</a> <font color="Maroon">y1</font> )
 ;<br/>

<a NAME="E2:50"/><b>then </b><i><font color="Green">E66</font></i>: 
( <font color="Maroon">x</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">y0</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">x</font>,<font color="Maroon">y0</font> <a href="euclid_4.html#R1">are_orthogonal</a>  )
 
<b>by </b><i/>;<br/>
<a NAME="E3:50"/><i><font color="Green">E88</font></i>: 
( <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> &amp; <font color="Maroon">y1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="euclid.html#K5">0*</a> <font color="Maroon">n</font> &amp;  ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> st <font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Olive">r</font> <a href="euclid.html#K9">*</a> <font color="Maroon">y0</font> )
 
<b>by </b><i>, </i>;<br/>
<b>then consider </b><font color="Maroon">r</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E5:50"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> <a href="euclid.html#K9">*</a> <font color="Maroon">y0</font>
 ;<br/>
<span class="p1"><a href="euclid_4.html#K5">|(</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y1</font></span><a href="euclid_4.html#K5">)|</a></span> = 
<font color="Maroon">r</font> <a href="real_1.html#K4">*</a> <span class="p1"><a href="euclid_4.html#K5">|(</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y0</font></span><a href="euclid_4.html#K5">)|</a></span>
<b>by </b><i><a class="ref" href="euclid_4.html#T27">EUCLID_4:27</a>, <a class="ref" href="euclidlp.html#T3" target="_self">Th3</a></i>
<br/>.= 
<font color="Maroon">r</font> <a href="real_1.html#K4">*</a> 0
<b>by </b><i><a class="ref" href="euclidlp.html#T1" target="_self">Th1</a>, <a class="ref" href="euclid_4.html#D6">EUCLID_4:def 6</a></i>
<br/>.= 
0

;<br/>
<a NAME="E7:50"/><b>then </b>
<font color="Maroon">x</font>,<font color="Maroon">y1</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="E8:50"/>
<font color="Maroon">x</font> <a href="euclidlp.html#R4" target="_self">_|_</a> <font color="Maroon">y1</font>
 <b>by </b><i><a class="ref" href="euclidlp.html#T1" target="_self">Th1</a>, <a class="ref" href="euclidlp.html#T2" target="_self">Th2</a>, </i>;<br/>


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