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<b>let </b><font color="Maroon" title="c1">a</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>let </b><font color="Maroon" title="c2">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="c3">x</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="c2">n</font>;<br/><b>let </b><font color="Maroon" title="c4">L</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="euclidlp.html#K1" title="EUCLIDLP:func.1">line_of_REAL</a> <font color="Maroon" title="c2">n</font>;<br/>







<b>assume </b><a NAME="E1:77"/><i><font color="Green" title="E66">A1</font></i>: 
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">L</font> &amp; <font color="Maroon" title="c1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 &amp; <font color="Maroon" title="c1">a</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">L</font> )
 ;<br/>

<a NAME="E2:77"/><b>then </b><i><font color="Green" title="E67">A2</font></i>: 
 <a href="euclid_4.html#K2" title="EUCLID_4:func.2">Line</a> <font color="Maroon" title="c3">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">L</font>
 
<b>by </b><i><a class="ref" href="euclidlp.html#T53" target="_self" title="EUCLIDLP:th.53">Th53</a></i>;<br/>
<a NAME="E3:77"/><i><font color="Green" title="E68">A3</font></i>: 
1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="euclidlp.html#E1:77"><i><font color="Green" title="E66">A1</font></i></a></i>;<br/>
<i><font color="Green" title="E69">A4</font></i>: <span class="p1">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c1">a</font></span>)</span> = 
<span class="p1">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T100" title="XCMPLX_1:th.100">XCMPLX_1:100</a></i>
<br/>.= 
1 <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span></span>)</span>

<br/>.= 
1 <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T188" title="XCMPLX_1:th.188">XCMPLX_1:188</a></i>
<br/>.= 
1 <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T63" title="XCMPLX_1:th.63">XCMPLX_1:63</a></i>
<br/>.= 
1 <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 1</span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span>

<br/>.= 
1 <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_1.html#T188" title="XCMPLX_1:th.188">XCMPLX_1:188</a></i>
<br/>.= 
1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span>

<br/>.= 
1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
<b>by </b><i><a class="txt" href="euclidlp.html#E3:77"><i><font color="Green" title="E68">A3</font></i></a>, <a class="ref" href="xcmplx_1.html#T60" title="XCMPLX_1:th.60">XCMPLX_1:60</a></i>
<br/>.= 
0

;<br/>
 <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c2">n</font> = 
0 <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c3">x</font>
<b>by </b><i><a class="ref" href="euclid_4.html#T3" title="EUCLID_4:th.3">EUCLID_4:3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c3">x</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid_4.html#T7" title="EUCLID_4:th.7">EUCLID_4:7</a>, <a class="txt" href="euclidlp.html#E4:77"><i><font color="Green" title="E69">A4</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="euclid.html#K7" title="EUCLID:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c1">a</font></span>)</span></span>)</span> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c3">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="euclid_4.html#T4" title="EUCLID_4:th.4">EUCLID_4:4</a></i>
;<br/>
<a NAME="E6:77"/><b>then </b>
 <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="euclid_4.html#K2" title="EUCLID_4:func.2">Line</a> <font color="Maroon" title="c3">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="euclid.html#K9" title="EUCLID:func.9">*</a> <font color="Maroon" title="c3">x</font></span>)</span>
 
;<br/>
<b>hence </b><a NAME="E7:77"/>
 <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">L</font>
 <b>by </b><i><a class="txt" href="euclidlp.html#E2:77"><i><font color="Green" title="E67">A2</font></i></a></i>;<br/>


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