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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="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1" target="_self">REAL</a> <font color="Maroon">n</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="pcomps_1.html#D7">PCOMPS_1:def 7</a><br/><b>let </b><font color="Maroon">y</font> be    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1" target="_self">REAL</a> <font color="Maroon">n</font>;<br/><b>let </b><font color="Maroon">z</font> be    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1" target="_self">REAL</a> <font color="Maroon">n</font>;<br/>







<b>thus </b><a NAME="E1:35"/>
( <span class="p1">(<span class="default"><a href="euclid.html#K13" target="_self">Pitag_dist</a> <font color="Maroon">n</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 0 iff <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><b>proof </b></a><div class="add">

<a NAME="E1:35_1"/>
<span class="p1">(<span class="default"><a href="euclid.html#K13" target="_self">Pitag_dist</a> <font color="Maroon">n</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K12" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="euclid.html#K8" target="_self">-</a> <font color="Maroon">y</font></span>)</span></span><a href="euclid.html#K12" target="_self">.|</a></span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:35_1"/>
( <span class="p1">(<span class="default"><a href="euclid.html#K13" target="_self">Pitag_dist</a> <font color="Maroon">n</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 0 iff <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 <b>by </b><i/>;<br/>


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

<b>thus </b><span class="p1">(<span class="default"><a href="euclid.html#K13" target="_self">Pitag_dist</a> <font color="Maroon">n</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> = 
<span class="p1"><a href="euclid.html#K12" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="euclid.html#K8" target="_self">-</a> <font color="Maroon">y</font></span>)</span></span><a href="euclid.html#K12" target="_self">.|</a></span>
<b>by </b><i/>
<br/>.= 
<span class="p1"><a href="euclid.html#K12" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K8" target="_self">-</a> <font color="Maroon">x</font></span>)</span></span><a href="euclid.html#K12" target="_self">.|</a></span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="euclid.html#K13" target="_self">Pitag_dist</a> <font color="Maroon">n</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">y</font>,<font color="Maroon">x</font>
<b>by </b><i/>
;<br/>

<b>thus </b><a NAME="E3:35"/>
<span class="p1">(<span class="default"><a href="euclid.html#K13" target="_self">Pitag_dist</a> <font color="Maroon">n</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">z</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K13" target="_self">Pitag_dist</a> <font color="Maroon">n</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K13" target="_self">Pitag_dist</a> <font color="Maroon">n</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_3"><b>proof </b></a><div class="add">

<a NAME="E1:35_3"/>
( <span class="p1">(<span class="default"><a href="euclid.html#K13" target="_self">Pitag_dist</a> <font color="Maroon">n</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K12" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="euclid.html#K8" target="_self">-</a> <font color="Maroon">y</font></span>)</span></span><a href="euclid.html#K12" target="_self">.|</a></span> &amp; <span class="p1">(<span class="default"><a href="euclid.html#K13" target="_self">Pitag_dist</a> <font color="Maroon">n</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K12" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="euclid.html#K8" target="_self">-</a> <font color="Maroon">z</font></span>)</span></span><a href="euclid.html#K12" target="_self">.|</a></span> &amp; <span class="p1">(<span class="default"><a href="euclid.html#K13" target="_self">Pitag_dist</a> <font color="Maroon">n</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">y</font>,<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K12" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">y</font> <a href="euclid.html#K8" target="_self">-</a> <font color="Maroon">z</font></span>)</span></span><a href="euclid.html#K12" target="_self">.|</a></span> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:35_3"/>
<span class="p1">(<span class="default"><a href="euclid.html#K13" target="_self">Pitag_dist</a> <font color="Maroon">n</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">z</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K13" target="_self">Pitag_dist</a> <font color="Maroon">n</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K13" target="_self">Pitag_dist</a> <font color="Maroon">n</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span>
 <b>by </b><i/>;<br/>


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


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