<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,<a href="numbers.html#K1">REAL</a> </span><a href="zfmisc_1.html#K2">:]</a></span>, <font color="Maroon">y</font> be    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><a href="numbers.html#K1">REAL</a> ,<a href="numbers.html#K1">REAL</a> </span><a href="zfmisc_1.html#K2">:]</a></span>;<br/>

<b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a> , <font color="Maroon">x2</font> = <font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a> , <font color="Maroon">y1</font> = <font color="Maroon">y</font> <a href="mcart_1.html#K1">`1</a> , <font color="Maroon">y2</font> = <font color="Maroon">y</font> <a href="mcart_1.html#K2">`2</a>  as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="mcart_1.html#T10">MCART_1:10</a></i>;<br/>
<a NAME="E2:24"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y1</font>,<font color="Maroon">y2</font></span><a href="domain_1.html#K1">]</a></span> )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T24">MCART_1:24</a></i>;<br/>
<b>thus </b><a NAME="E3:24"/>
( <a href="metric_4.html#K9" target="_self">Eukl_dist2</a>  <a href="metric_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 0 implies <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="24_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:24_1"/><i><font color="Green">E28</font></i>: 
<a href="metric_4.html#K9" target="_self">Eukl_dist2</a>  <a href="metric_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<b>set </b><font color="Maroon">d1</font> = <a href="metric_1.html#K7">real_dist</a>  <a href="metric_1.html#K1">.</a> <font color="Maroon">x1</font>,<font color="Maroon">y1</font>;<br/>
<b>set </b><font color="Maroon">d2</font> = <a href="metric_1.html#K7">real_dist</a>  <a href="metric_1.html#K1">.</a> <font color="Maroon">x2</font>,<font color="Maroon">y2</font>;<br/>
<a NAME="E2:24_1"/><i><font color="Green">E31</font></i>: 
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="metric_1.html#K7">real_dist</a>  <a href="metric_1.html#K1">.</a> <font color="Maroon">x1</font>,<font color="Maroon">y1</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="metric_1.html#K7">real_dist</a>  <a href="metric_1.html#K1">.</a> <font color="Maroon">x2</font>,<font color="Maroon">y2</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i>, , <a class="ref" href="metric_4.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E3:24_1"/><i><font color="Green">E32</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K7">real_dist</a>  <a href="metric_1.html#K1">.</a> <font color="Maroon">x1</font>,<font color="Maroon">y1</font></span>)</span> <a href="square_1.html#K5">^2</a> 
 
<b>by </b><i><a class="ref" href="xreal_1.html#T65">XREAL_1:65</a></i>;<br/>
<a NAME="E4:24_1"/><i><font color="Green">E33</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K7">real_dist</a>  <a href="metric_1.html#K1">.</a> <font color="Maroon">x2</font>,<font color="Maroon">y2</font></span>)</span> <a href="square_1.html#K5">^2</a> 
 
<b>by </b><i><a class="ref" href="xreal_1.html#T65">XREAL_1:65</a></i>;<br/>
<a NAME="E5:24_1"/><b>then </b><i><font color="Green">E34</font></i>: 
( <span class="p1">(<span class="default"><a href="metric_1.html#K7">real_dist</a>  <a href="metric_1.html#K1">.</a> <font color="Maroon">x1</font>,<font color="Maroon">y1</font></span>)</span> <a href="square_1.html#K5">^2</a>  <a href="hidden.html#R1">=</a> 0 &amp; <span class="p1">(<span class="default"><a href="metric_1.html#K7">real_dist</a>  <a href="metric_1.html#K1">.</a> <font color="Maroon">x2</font>,<font color="Maroon">y2</font></span>)</span> <a href="square_1.html#K5">^2</a>  <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E6:24_1"/>
<span class="p1">(<span class="default"><a href="metric_1.html#K7">real_dist</a>  <a href="metric_1.html#K1">.</a> <font color="Maroon">x1</font>,<font color="Maroon">y1</font></span>)</span> <a href="square_1.html#K5">^2</a>  <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E7:24_1"/><b>then </b>
<a href="metric_1.html#K7">real_dist</a>  <a href="metric_1.html#K1">.</a> <font color="Maroon">x1</font>,<font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="xcmplx_1.html#T6">XCMPLX_1:6</a></i>;<br/>
<a NAME="E8:24_1"/><b>then </b><i><font color="Green">E41</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#T9">METRIC_1:9</a></i>;<br/>
<a NAME="E9:24_1"/>
<a href="metric_1.html#K7">real_dist</a>  <a href="metric_1.html#K1">.</a> <font color="Maroon">x2</font>,<font color="Maroon">y2</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T6">XCMPLX_1:6</a></i>;<br/>
<b>hence </b><a NAME="E10:24_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i>, , <a class="ref" href="metric_1.html#T9">METRIC_1:9</a></i>;<br/>


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

<b>assume </b><a NAME="E4:24"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 ;<br/>

<a NAME="E5:24"/><b>then </b><i><font color="Green">E43</font></i>: 
<span class="p1">(<span class="default"><a href="metric_1.html#K7">real_dist</a>  <a href="metric_1.html#K1">.</a> <font color="Maroon">x1</font>,<font color="Maroon">y1</font></span>)</span> <a href="square_1.html#K5">^2</a>  <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="metric_1.html#T9">METRIC_1:9</a>, <a class="ref" href="square_1.html#T60">SQUARE_1:60</a></i>;<br/>
<a NAME="E6:24"/><i><font color="Green">E44</font></i>: 
<span class="p1">(<span class="default"><a href="metric_1.html#K7">real_dist</a>  <a href="metric_1.html#K1">.</a> <font color="Maroon">x2</font>,<font color="Maroon">y2</font></span>)</span> <a href="square_1.html#K5">^2</a>  <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i>, <a class="ref" href="metric_1.html#T9">METRIC_1:9</a>, <a class="ref" href="square_1.html#T60">SQUARE_1:60</a></i>;<br/>
<a href="metric_4.html#K9" target="_self">Eukl_dist2</a>  <a href="metric_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> = 
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="metric_1.html#K7">real_dist</a>  <a href="metric_1.html#K1">.</a> <font color="Maroon">x1</font>,<font color="Maroon">y1</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="metric_1.html#K7">real_dist</a>  <a href="metric_1.html#K1">.</a> <font color="Maroon">x2</font>,<font color="Maroon">y2</font></span>)</span> <a href="square_1.html#K5">^2</a> </span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="metric_4.html#D1" target="_self">Def1</a></i>
<br/>.= 
0
<b>by </b><i>, , </i>
;<br/>
<b>hence </b><a NAME="E8:24"/>
<a href="metric_4.html#K9" target="_self">Eukl_dist2</a>  <a href="metric_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 0
 ;<br/>


</div>
