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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">c<sub>2</sub></font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:10"/><i><font color="Green">E10</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E2:10"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 ;<br/>

<a NAME="E3:10"/><i><font color="Green">E12</font></i>: 
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="euclid.html#K12">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="euclid.html#K12">.|</a></span>
 
<b>by </b><i><a class="ref" href="euclid.html#T14">EUCLID:14</a></i>;<br/>
<a NAME="E4:10"/><i><font color="Green">E13</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="euclid.html#K12">.|</a></span>
 
<b>by </b><i><a class="ref" href="euclid.html#T12">EUCLID:12</a></i>;<br/>
<a NAME="E5:10"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="topalg_1.html#E1:10"><i><font color="Green">E10</font></i></a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/>
<a NAME="E6:10"/><b>then </b>
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="euclid.html#K12">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K4">*</a> <span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="euclid.html#K12">.|</a></span>
 
<b>by </b><i><a class="txt" href="topalg_1.html#E1:10"><i><font color="Green">E10</font></i></a>, <a class="txt" href="topalg_1.html#E2:10"><i><font color="Green">E11</font></i></a>, <a class="txt" href="topalg_1.html#E3:10"><i><font color="Green">E12</font></i></a>, <a class="txt" href="topalg_1.html#E4:10"><i><font color="Green">E13</font></i></a>, <a class="ref" href="xreal_1.html#T68">XREAL_1:68</a></i>;<br/>
<b>hence </b><a NAME="E7:10"/>
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="euclid.html#K9">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="euclid.html#K12">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="euclid.html#K12">.|</a></span>
 ;<br/>


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