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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/>



<span class="p1"><a href="euclid_4.html#K5" target="_self">|(</a><span class="default"><font color="Maroon">x</font>,<span class="p2">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">n</font></span>)</span></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"><font color="Maroon">x</font> <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"><font color="Maroon">x</font> <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"><font color="Maroon">x</font></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"><font color="Maroon">x</font> <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"><font color="Maroon">x</font></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"><font color="Maroon">x</font></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="E2:32"/>
<span class="p1"><a href="euclid_4.html#K5" target="_self">|(</a><span class="default"><font color="Maroon">x</font>,<span class="p2">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">n</font></span>)</span></span><a href="euclid_4.html#K5" target="_self">)|</a></span> <a href="hidden.html#R1">=</a> 0
 ;<br/>


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