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

<b>let </b><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">y</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <a href="metric_4.html#K9" title="METRIC_4:func.9">Eukl_dist2</a>  <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  iff <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> )</span><br/>

<b>reconsider </b><font color="Maroon" title="c3">x1</font> = <font color="Maroon" title="c1">x</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> , <font color="Maroon" title="c4">x2</font> = <font color="Maroon" title="c1">x</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> , <font color="Maroon" title="c5">y1</font> = <font color="Maroon" title="c2">y</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> , <font color="Maroon" title="c6">y2</font> = <font color="Maroon" title="c2">y</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <b>by </b><i><a class="ref" href="mcart_1.html#T10" title="MCART_1:th.10">MCART_1:10</a></i>;<br/>
<a NAME="E2:24"/><i><font color="Green" title="E20">A1</font></i>: 
( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c4">x2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> &amp; <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c5">y1</font>,<font color="Maroon" title="c6">y2</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T24" title="MCART_1:th.24">MCART_1:24</a></i>;<br/>
<b>thus </b><a NAME="E3:24"/>
( <a href="metric_4.html#K9" title="METRIC_4:func.9">Eukl_dist2</a>  <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> implies <a href="metric_4.html#K9" title="METRIC_4:func.9">Eukl_dist2</a>  <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><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" title="E21">A2</font></i>: 
<a href="metric_4.html#K9" title="METRIC_4:func.9">Eukl_dist2</a>  <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font></span><br/>

<b>set </b><font color="Maroon" title="c7">d1</font> = <a href="metric_1.html#K7" title="METRIC_1:func.7">real_dist</a>  <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c5">y1</font>;<br/>
<b>set </b><font color="Maroon" title="c8">d2</font> = <a href="metric_1.html#K7" title="METRIC_1:func.7">real_dist</a>  <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c6">y2</font>;<br/>
<a NAME="E2:24_1"/><i><font color="Green" title="E22">A3</font></i>: 
 <a href="square_1.html#K7" title="SQUARE_1:func.7">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" title="METRIC_1:func.7">real_dist</a>  <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c5">y1</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="metric_1.html#K7" title="METRIC_1:func.7">real_dist</a>  <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c6">y2</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="#E2:24"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="#E1:24_1"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="metric_4.html#D9" target="_self" title="METRIC_4:def.9">Def9</a></i>;<br/>
<a NAME="E3:24_1"/><i><font color="Green" title="E23">A4</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K7" title="METRIC_1:func.7">real_dist</a>  <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c5">y1</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
 
<b>by </b><i><a class="ref" href="xreal_1.html#T65" title="XREAL_1:th.65">XREAL_1:65</a></i>;<br/>
<a NAME="E4:24_1"/><i><font color="Green" title="E24">A5</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K7" title="METRIC_1:func.7">real_dist</a>  <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c6">y2</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
 
<b>by </b><i><a class="ref" href="xreal_1.html#T65" title="XREAL_1:th.65">XREAL_1:65</a></i>;<br/>
<a NAME="E5:24_1"/><b>then </b><i><font color="Green" title="E25">A6</font></i>: 
( <span class="p1">(<span class="default"><a href="metric_1.html#K7" title="METRIC_1:func.7">real_dist</a>  <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c5">y1</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <span class="p1">(<span class="default"><a href="metric_1.html#K7" title="METRIC_1:func.7">real_dist</a>  <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c6">y2</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="txt" href="#E2:24_1"><i><font color="Green" title="E22">A3</font></i></a>, <a class="txt" href="#E3:24_1"><i><font color="Green" title="E23">A4</font></i></a>, <a class="ref" href="metric_4.html#T2" target="_self" title="METRIC_4:th.2">Th2</a></i>;<br/>
<a NAME="E6:24_1"/>
<span class="p1">(<span class="default"><a href="metric_1.html#K7" title="METRIC_1:func.7">real_dist</a>  <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c5">y1</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="#E2:24_1"><i><font color="Green" title="E22">A3</font></i></a>, <a class="txt" href="#E3:24_1"><i><font color="Green" title="E23">A4</font></i></a>, <a class="txt" href="#E4:24_1"><i><font color="Green" title="E24">A5</font></i></a>, <a class="ref" href="metric_4.html#T2" target="_self" title="METRIC_4:th.2">Th2</a></i>;<br/>
<a NAME="E7:24_1"/><b>then </b>
<a href="metric_1.html#K7" title="METRIC_1:func.7">real_dist</a>  <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c5">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
;<br/>
<a NAME="E8:24_1"/><b>then </b><i><font color="Green" title="E26">A7</font></i>: 
<font color="Maroon" title="c3">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y1</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#T9" title="METRIC_1:th.9">METRIC_1:9</a></i>;<br/>
<a NAME="E9:24_1"/>
<a href="metric_1.html#K7" title="METRIC_1:func.7">real_dist</a>  <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c6">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="#E5:24_1"><i><font color="Green" title="E25">A6</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:24_1"/>
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font>
 <b>by </b><i><a class="txt" href="#E2:24"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="#E8:24_1"><i><font color="Green" title="E26">A7</font></i></a>, <a class="ref" href="metric_1.html#T9" title="METRIC_1:th.9">METRIC_1:9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>assume </b><a NAME="E4:24"/><i><font color="Green" title="E21">A8</font></i>: 
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <a href="metric_4.html#K9" title="METRIC_4:func.9">Eukl_dist2</a>  <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/>

<a NAME="E5:24"/><b>then </b><i><font color="Green" title="E22">A9</font></i>: 
<span class="p1">(<span class="default"><a href="metric_1.html#K7" title="METRIC_1:func.7">real_dist</a>  <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c5">y1</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
 
<b>by </b><i><a class="ref" href="metric_1.html#T9" title="METRIC_1:th.9">METRIC_1:9</a></i>;<br/>
<a NAME="E6:24"/><i><font color="Green" title="E23">A10</font></i>: 
<span class="p1">(<span class="default"><a href="metric_1.html#K7" title="METRIC_1:func.7">real_dist</a>  <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c6">y2</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
 
<b>by </b><i><a class="txt" href="#E4:24"><i><font color="Green" title="E21">A8</font></i></a>, <a class="ref" href="metric_1.html#T9" title="METRIC_1:th.9">METRIC_1:9</a></i>;<br/>
<a href="metric_4.html#K9" title="METRIC_4:func.9">Eukl_dist2</a>  <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font> = 
 <a href="square_1.html#K7" title="SQUARE_1:func.7">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" title="METRIC_1:func.7">real_dist</a>  <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c5">y1</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="metric_1.html#K7" title="METRIC_1:func.7">real_dist</a>  <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c6">y2</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E2:24"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="metric_4.html#D9" target="_self" title="METRIC_4:def.9">Def9</a></i>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="txt" href="#E5:24"><i><font color="Green" title="E22">A9</font></i></a>, <a class="txt" href="#E6:24"><i><font color="Green" title="E23">A10</font></i></a>, <a class="ref" href="metric_4.html#T2" target="_self" title="METRIC_4:th.2">Th2</a></i>
;<br/>
<b>hence </b><a NAME="E8:24"/>
<a href="metric_4.html#K9" title="METRIC_4:func.9">Eukl_dist2</a>  <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
