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<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> ;<br/>

<b>thus </b><a NAME="E1:16"/>
( <span class="p1"><a href="euclid_2.html#K1" target="_self">|(</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="euclid_2.html#K1" target="_self">)|</a></span> <a href="hidden.html#R1">=</a> 0 implies <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K5">0*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:16_1"/>
<span class="p1"><a href="euclid_2.html#K1" target="_self">|(</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="euclid_2.html#K1" target="_self">)|</a></span> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<a NAME="E2:16_1"/><b>then </b>
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="euclid.html#K12">.|</a></span> <a href="square_1.html#K5">^2</a>  <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="euclid_2.html#T12" target="_self">Th12</a></i>;<br/>
<a NAME="E3:16_1"/><b>then </b><i><font color="Green">E16</font></i>: 
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="euclid.html#K12">.|</a></span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="xcmplx_1.html#T6">XCMPLX_1:6</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>2</sub></font> = <font color="Maroon">c<sub>1</sub></font> as    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="ref" href="jordan2c.html#T52">JORDAN2C:52</a></i>;<br/>
<a NAME="E5:16_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K5">0*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="euclid_2.html#E3:16_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="euclid.html#T11">EUCLID:11</a></i>;<br/>
<b>hence </b><a NAME="E6:16_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K5">0*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:16"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K5">0*</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/>

<a NAME="E3:16"/><b>then </b>
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="euclid.html#K12">.|</a></span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="euclid.html#T10">EUCLID:10</a></i>;<br/>
<b>hence </b><a NAME="E4:16"/>
<span class="p1"><a href="euclid_2.html#K1" target="_self">|(</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span><a href="euclid_2.html#K1" target="_self">)|</a></span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="euclid_2.html#T12" target="_self">Th12</a>, <a class="ref" href="square_1.html#T60">SQUARE_1:60</a></i>;<br/>


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