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

<span class="kw">let </span><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#K3" title="ZFMISC_1:func.3">[:</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>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <a href="metric_3.html#K22" title="METRIC_3:func.22">Eukl_dist3</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/>

<span class="kw">reconsider </span><font color="Maroon" title="c3">x1</font> = <font color="Maroon" title="c1">x</font> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> , <font color="Maroon" title="c4">x2</font> = <font color="Maroon" title="c1">x</font> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> , <font color="Maroon" title="c5">x3</font> = <font color="Maroon" title="c1">x</font> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a> , <font color="Maroon" title="c6">y1</font> = <font color="Maroon" title="c2">y</font> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> , <font color="Maroon" title="c7">y2</font> = <font color="Maroon" title="c2">y</font> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> , <font color="Maroon" title="c8">y3</font> = <font color="Maroon" title="c2">y</font> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</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>  ;<br/>
<a NAME="E2:55"/><span class="lab"><font color="Green" title="E40">A1</font></span>: 
( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Maroon" title="c3">x1</font>,<font color="Maroon" title="c4">x2</font>,<font color="Maroon" title="c5">x3</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</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="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Maroon" title="c6">y1</font>,<font color="Maroon" title="c7">y2</font>,<font color="Maroon" title="c8">y3</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T44" title="MCART_1:th.44">MCART_1:44</a></span>;<br/>
<span class="kw">thus </span><a NAME="E3:55"/>
( <a href="metric_3.html#K22" title="METRIC_3:func.22">Eukl_dist3</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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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_3.html#K22" title="METRIC_3:func.22">Eukl_dist3</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="55_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c9">d3</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="c5">x3</font>,<font color="Maroon" title="c8">y3</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c10">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="c7">y2</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c11">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="c6">y1</font>);<br/>
<a NAME="E1:55_1"/><span class="lab"><font color="Green" title="E41">A2</font></span>: 
(  <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="c7">y2</font>)</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  &amp;  <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="c5">x3</font>,<font color="Maroon" title="c8">y3</font>)</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T63" title="XREAL_1:th.63">XREAL_1:63</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:55_1"/>
<a href="metric_3.html#K22" title="METRIC_3:func.22">Eukl_dist3</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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<a NAME="E3:55_1"/><span class="kw">then </span>
 <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"><span class="p4">(<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="c6">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="p3">(<span class="default"><span class="p4">(<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="c7">y2</font>)</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a></span>)</span></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="c5">x3</font>,<font color="Maroon" title="c8">y3</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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:55"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="ref" href="metric_3.html#D22" target="_self" title="METRIC_3:def.22">Def22</a></span>;<br/>
<a NAME="E4:55_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A3</font></span>: 
 <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="c6">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"><span class="p4">(<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="c7">y2</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="p3">(<span class="default"><span class="p4">(<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="c5">x3</font>,<font color="Maroon" title="c8">y3</font>)</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a></span>)</span></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> 
 
;<br/>
<a NAME="E5:55_1"/>
(  <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="c7">y2</font>)</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  &amp;  <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="c5">x3</font>,<font color="Maroon" title="c8">y3</font>)</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T63" title="XREAL_1:th.63">XREAL_1:63</a></span>;<br/>
<a NAME="E6:55_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A4</font></span>: 
(  <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="c6">y1</font>)</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a>  &amp; <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <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"><span class="p2">(<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="c7">y2</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="p1">(<span class="default"><span class="p2">(<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="c5">x3</font>,<font color="Maroon" title="c8">y3</font>)</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T7" title="XREAL_1:th.7">XREAL_1:7</a>, <a class="ref" href="xreal_1.html#T63" title="XREAL_1:th.63">XREAL_1:63</a></span>;<br/>
<a NAME="E7:55_1"/><span class="kw">then </span>
<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="c6">y1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:55_1"><span class="lab"><font color="Green" title="E42">A3</font></span></a>, <a class="txt" href="metric_3.html#E20"><span class="lab"><font color="Green" title="E14">Lm1</font></span></a></span>;<br/>
<a NAME="E8:55_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A5</font></span>: 
<font color="Maroon" title="c3">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#T8" title="METRIC_1:th.8">METRIC_1:8</a></span>;<br/>
<a NAME="E9:55_1"/><span class="lab"><font color="Green" title="E45">A6</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<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="c7">y2</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="p1">(<span class="default"><span class="p2">(<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="c5">x3</font>,<font color="Maroon" title="c8">y3</font>)</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:55_1"><span class="lab"><font color="Green" title="E42">A3</font></span></a>, <a class="txt" href="#E6:55_1"><span class="lab"><font color="Green" title="E43">A4</font></span></a>, <a class="txt" href="metric_3.html#E20"><span class="lab"><font color="Green" title="E14">Lm1</font></span></a></span>;<br/>
<a NAME="E10:55_1"/><span class="kw">then </span>
<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="c7">y2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:55_1"><span class="lab"><font color="Green" title="E41">A2</font></span></a>, <a class="ref" href="xreal_1.html#T27" title="XREAL_1:th.27">XREAL_1:27</a></span>;<br/>
<a NAME="E11:55_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A7</font></span>: 
<font color="Maroon" title="c4">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#T8" title="METRIC_1:th.8">METRIC_1:8</a></span>;<br/>
<a NAME="E12:55_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="c5">x3</font>,<font color="Maroon" title="c8">y3</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:55_1"><span class="lab"><font color="Green" title="E45">A6</font></span></a>, <a class="txt" href="#E1:55_1"><span class="lab"><font color="Green" title="E41">A2</font></span></a>, <a class="ref" href="xreal_1.html#T27" title="XREAL_1:th.27">XREAL_1:27</a></span>;<br/>
<span class="kw">hence </span><a NAME="E13:55_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:55"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="txt" href="#E8:55_1"><span class="lab"><font color="Green" title="E44">A5</font></span></a>, <a class="txt" href="#E11:55_1"><span class="lab"><font color="Green" title="E46">A7</font></span></a>, <a class="ref" href="metric_1.html#T8" title="METRIC_1:th.8">METRIC_1:8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E4:55"/><span class="lab"><font color="Green" title="E41">A8</font></span>: 
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <a href="metric_3.html#K22" title="METRIC_3:func.22">Eukl_dist3</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="E5:55"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A9</font></span>: 
( <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="c6">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>  &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="c7">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>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#T8" title="METRIC_1:th.8">METRIC_1:8</a></span>;<br/>
<a href="metric_3.html#K22" title="METRIC_3:func.22">Eukl_dist3</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"><span class="p4">(<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="c6">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="p3">(<span class="default"><span class="p4">(<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="c7">y2</font>)</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a></span>)</span></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="c5">x3</font>,<font color="Maroon" title="c8">y3</font>)</span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:55"><span class="lab"><font color="Green" title="E40">A1</font></span></a>, <a class="ref" href="metric_3.html#D22" target="_self" title="METRIC_3:def.22">Def22</a></span>
<br/>.= 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:55"><span class="lab"><font color="Green" title="E41">A8</font></span></a>, <a class="txt" href="#E5:55"><span class="lab"><font color="Green" title="E42">A9</font></span></a>, <a class="ref" href="metric_1.html#T8" title="METRIC_1:th.8">METRIC_1:8</a>, <a class="ref" href="square_1.html#T17" title="SQUARE_1:th.17">SQUARE_1:17</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E7:55"/>
<a href="metric_3.html#K22" title="METRIC_3:func.22">Eukl_dist3</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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
