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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="subset_1.html#M1">Element</a> of  <a href="euclid.html#K1">REAL</a> <font color="Maroon">n</font>;<br/>



<b>thus </b><a NAME="E1:31"/>
( <span class="p1"><a href="euclid_4.html#K5" target="_self">|(</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">x</font></span><a href="euclid_4.html#K5" target="_self">)|</a></span> <a href="hidden.html#R1">=</a> 0 implies <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K5">0*</a> <font color="Maroon">n</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:31_1"/>
<span class="p1"><a href="euclid_4.html#K5" target="_self">|(</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">x</font></span><a href="euclid_4.html#K5" target="_self">)|</a></span> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<a NAME="E2:31_1"/><b>then </b>
<span class="p1"><a href="euclid_4.html#K4" target="_self">|.</a><span class="default"><font color="Maroon">x</font></span><a href="euclid_4.html#K4" target="_self">.|</a></span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i/>;<br/>
<a NAME="E3:31_1"/><b>then </b><i><font color="Green">E41</font></i>: 
<span class="p1"><a href="euclid.html#K12">|.</a><span class="default"><span class="p2">(<span class="default"><a href="euclid_4.html#K3" target="_self">Rn2Fin</a> <font color="Maroon">x</font></span>)</span></span><a href="euclid.html#K12">.|</a></span> <a href="hidden.html#R1">=</a> 0
 
;<br/>
<b>thus </b><a NAME="E4:31_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K5">0*</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="euclid.html#T11">EUCLID:11</a></i>;<br/>


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

<b>thus </b><a NAME="E2:31"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K5">0*</a> <font color="Maroon">n</font> implies <span class="p1"><a href="euclid_4.html#K5" target="_self">|(</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">x</font></span><a href="euclid_4.html#K5" target="_self">)|</a></span> <a href="hidden.html#R1">=</a> 0 )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:31_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K5">0*</a> <font color="Maroon">n</font>
 ;<br/>

<b>then </b><span class="p1"><a href="euclid_4.html#K5" target="_self">|(</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">x</font></span><a href="euclid_4.html#K5" target="_self">)|</a></span> = 
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid_4.html#K4" target="_self">|.</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">n</font></span>)</span> <a href="euclid.html#K7">+</a> <span class="p5">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="euclid_4.html#K4" target="_self">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid_4.html#K4" target="_self">|.</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">n</font></span>)</span> <a href="euclid.html#K8">-</a> <span class="p5">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="euclid_4.html#K4" target="_self">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid_4.html#K4" target="_self">|.</a><span class="default"><span class="p4">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">n</font></span>)</span></span><a href="euclid_4.html#K4" target="_self">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid_4.html#K4" target="_self">|.</a><span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">n</font></span>)</span> <a href="euclid.html#K8">-</a> <span class="p5">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">n</font></span>)</span></span>)</span></span><a href="euclid_4.html#K4" target="_self">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid_4.html#K4" target="_self">|.</a><span class="default"><span class="p4">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">n</font></span>)</span></span><a href="euclid_4.html#K4" target="_self">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><span class="p3"><a href="euclid_4.html#K4" target="_self">|.</a><span class="default"><span class="p4">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">n</font></span>)</span></span><a href="euclid_4.html#K4" target="_self">.|</a></span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="real_1.html#K6">/</a> 4</span>)</span> <a href="real_1.html#K4">*</a> 0

;<br/>
<b>hence </b><a NAME="E3:31_2"/>
<span class="p1"><a href="euclid_4.html#K5" target="_self">|(</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">x</font></span><a href="euclid_4.html#K5" target="_self">)|</a></span> <a href="hidden.html#R1">=</a> 0
 ;<br/>


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


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