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<b>let </b><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y0</font>, <font color="Maroon" title="c4">y1</font> be    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of  <a href="euclid.html#K1" title="EUCLID:func.1">REAL</a> <font color="Maroon" title="c1">n</font>;<br/>







<b>assume </b><a NAME="E1:50"/><i><font color="Green" title="E50">A1</font></i>: 
( <font color="Maroon" title="c2">x</font> <a href="euclidlp.html#R4" title="EUCLIDLP:pred.4">_|_</a> <font color="Maroon" title="c3">y0</font> &amp; <font color="Maroon" title="c3">y0</font> <a href="euclidlp.html#R2" title="EUCLIDLP:pred.2">//</a> <font color="Maroon" title="c4">y1</font> )
 ;<br/>

<a NAME="E2:50"/><b>then </b><i><font color="Green" title="E51">A2</font></i>: 
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c3">y0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y0</font> <a href="euclid_4.html#R1" title="EUCLID_4:pred.1">are_orthogonal</a>  )
 
<b>by </b><i><a class="ref" href="euclidlp.html#D3" target="_self" title="EUCLIDLP:def.3">Def3</a></i>;<br/>
<a NAME="E3:50"/><i><font color="Green" title="E52">A3</font></i>: 
( <font color="Maroon" title="c3">y0</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c4">y1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font> &amp;  ex <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <font color="Maroon" title="c4">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c3">y0</font> )
 
<b>by </b><i><a class="txt" href="euclidlp.html#E1:50"><i><font color="Green" title="E50">A1</font></i></a>, <a class="ref" href="euclidlp.html#D1" target="_self" title="EUCLIDLP:def.1">Def1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c5">r</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E5:50"/><i><font color="Green" title="E53">A4</font></i>: 
<font color="Maroon" title="c4">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">r</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c3">y0</font>
 ;<br/>
<span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">y1</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span> = 
<font color="Maroon" title="c5">r</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1"><a href="euclid_2.html#K2" title="EUCLID_2:func.2">|(</a><span class="default"><font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y0</font></span><a href="euclid_2.html#K2" title="EUCLID_2:func.2">)|</a></span>
<b>by </b><i><a class="ref" href="euclid_4.html#T27" title="EUCLID_4:th.27">EUCLID_4:27</a>, <a class="txt" href="euclidlp.html#E5:50"><i><font color="Green" title="E53">A4</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c5">r</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> 0
<b>by </b><i><a class="txt" href="euclidlp.html#E2:50"><i><font color="Green" title="E51">A2</font></i></a>, <a class="ref" href="euclid_4.html#D6" title="EUCLID_4:def.6">EUCLID_4:def 6</a></i>
<br/>.= 
0

;<br/>
<a NAME="E7:50"/><b>then </b>
<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c4">y1</font> <a href="euclid_4.html#R1" title="EUCLID_4:pred.1">are_orthogonal</a> 
 
<b>by </b><i><a class="ref" href="euclid_4.html#D6" title="EUCLID_4:def.6">EUCLID_4:def 6</a></i>;<br/>
<b>hence </b><a NAME="E8:50"/>
<font color="Maroon" title="c2">x</font> <a href="euclidlp.html#R4" title="EUCLIDLP:pred.4">_|_</a> <font color="Maroon" title="c4">y1</font>
 <b>by </b><i><a class="txt" href="euclidlp.html#E2:50"><i><font color="Green" title="E51">A2</font></i></a>, <a class="txt" href="euclidlp.html#E3:50"><i><font color="Green" title="E52">A3</font></i></a>, <a class="ref" href="euclidlp.html#D3" target="_self" title="EUCLIDLP:def.3">Def3</a></i>;<br/>


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