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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1" target="_self">REAL</a> <font color="Maroon">c<sub>1</sub></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">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be    <a href="finseq_2.html#M2">Element</a> of  <a href="euclid.html#K1" target="_self">REAL</a> <font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0 iff <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></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">c<sub>1</sub></font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></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">c<sub>2</sub></font> <a href="euclid.html#K8" target="_self">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="euclid.html#K12" target="_self">.|</a></span>
 
<b>by </b><i><a class="ref" href="euclid.html#D6" target="_self">Def6</a></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">c<sub>1</sub></font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0 iff <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="ref" href="euclid.html#T19" target="_self">Th19</a></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">c<sub>1</sub></font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> = 
<span class="p1"><a href="euclid.html#K12" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="euclid.html#K8" target="_self">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="euclid.html#K12" target="_self">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#D6" target="_self">Def6</a></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">c<sub>3</sub></font> <a href="euclid.html#K8" target="_self">-</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="euclid.html#K12" target="_self">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T21" target="_self">Th21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="euclid.html#K13" target="_self">Pitag_dist</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="euclid.html#D6" target="_self">Def6</a></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">c<sub>1</sub></font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></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">c<sub>1</sub></font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></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">c<sub>1</sub></font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></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">c<sub>1</sub></font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></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">c<sub>2</sub></font> <a href="euclid.html#K8" target="_self">-</a> <font color="Maroon">c<sub>3</sub></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">c<sub>1</sub></font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></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">c<sub>2</sub></font> <a href="euclid.html#K8" target="_self">-</a> <font color="Maroon">c<sub>4</sub></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">c<sub>1</sub></font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></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">c<sub>3</sub></font> <a href="euclid.html#K8" target="_self">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="euclid.html#K12" target="_self">.|</a></span> )
 
<b>by </b><i><a class="ref" href="euclid.html#D6" target="_self">Def6</a></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">c<sub>1</sub></font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></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">c<sub>1</sub></font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></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">c<sub>1</sub></font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="euclid.html#T22" target="_self">Th22</a></i>;<br/>


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


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