<?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>, <font color="Maroon" title="c3">z</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="Red">::  thesis: </font></span></a><span class="hide"> <a href="metric_4.html#K13" title="METRIC_4:func.13">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="c3">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_4.html#K13" title="METRIC_4:func.13">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></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="metric_4.html#K13" title="METRIC_4:func.13">Eukl_dist3</a>  <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">z</font></span>)</span></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c4">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="c5">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="c6">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="c7">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="c8">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="c9">y3</font> = <font color="Maroon" title="c2">y</font> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a> , <font color="Maroon" title="c10">z1</font> = <font color="Maroon" title="c3">z</font> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> , <font color="Maroon" title="c11">z2</font> = <font color="Maroon" title="c3">z</font> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> , <font color="Maroon" title="c12">z3</font> = <font color="Maroon" title="c3">z</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="E2:43"/><span class="lab"><font color="Green" title="E30">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="c4">x1</font>,<font color="Maroon" title="c5">x2</font>,<font color="Maroon" title="c6">x3</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#T48" title="MCART_1:th.48">MCART_1:48</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c13">d9</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="c9">y3</font>,<font color="Maroon" title="c12">z3</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c14">d5</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">x2</font>,<font color="Maroon" title="c8">y2</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c15">d6</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="c8">y2</font>,<font color="Maroon" title="c11">z2</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c16">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="c4">x1</font>,<font color="Maroon" title="c10">z1</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c17">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">x1</font>,<font color="Maroon" title="c7">y1</font>;<br/>
<a NAME="E3:43"/><span class="lab"><font color="Green" title="E31">A2</font></span>: 
<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="c7">y1</font>,<font color="Maroon" title="c8">y2</font>,<font color="Maroon" title="c9">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#T48" title="MCART_1:th.48">MCART_1:48</a></span>;<br/>
<a NAME="E4:43"/>
<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="c9">y3</font>,<font color="Maroon" title="c12">z3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">y3</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c12">z3</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#D13" title="METRIC_1:def.13">METRIC_1:def 13</a></span>;<br/>
<a NAME="E5:43"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A3</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> <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="c9">y3</font>,<font color="Maroon" title="c12">z3</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T132" title="COMPLEX1:th.132">COMPLEX1:132</a></span>;<br/>
<a NAME="E6:43"/>
<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="c8">y2</font>,<font color="Maroon" title="c11">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">y2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c11">z2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#D13" title="METRIC_1:def.13">METRIC_1:def 13</a></span>;<br/>
<a NAME="E7:43"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">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> <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="c8">y2</font>,<font color="Maroon" title="c11">z2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T132" title="COMPLEX1:th.132">COMPLEX1:132</a></span>;<br/>
<a NAME="E8:43"/>
<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">x2</font>,<font color="Maroon" title="c8">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c8">y2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#D13" title="METRIC_1:def.13">METRIC_1:def 13</a></span>;<br/>
<a NAME="E9:43"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A5</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> <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">x2</font>,<font color="Maroon" title="c8">y2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T132" title="COMPLEX1:th.132">COMPLEX1:132</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c18">d7</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="c6">x3</font>,<font color="Maroon" title="c12">z3</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c19">d8</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="c6">x3</font>,<font color="Maroon" title="c9">y3</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c20">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="c7">y1</font>,<font color="Maroon" title="c10">z1</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c21">d4</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">x2</font>,<font color="Maroon" title="c11">z2</font>;<br/>
<a NAME="E10:43"/><span class="lab"><font color="Green" title="E35">A6</font></span>: 
<font color="Maroon" title="c3">z</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="c10">z1</font>,<font color="Maroon" title="c11">z2</font>,<font color="Maroon" title="c12">z3</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#T48" title="MCART_1:th.48">MCART_1:48</a></span>;<br/>
<a NAME="E11:43"/>
<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="c6">x3</font>,<font color="Maroon" title="c12">z3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x3</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c12">z3</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#D13" title="METRIC_1:def.13">METRIC_1:def 13</a></span>;<br/>
<a NAME="E12:43"/><span class="kw">then </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> <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="c6">x3</font>,<font color="Maroon" title="c12">z3</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T132" title="COMPLEX1:th.132">COMPLEX1:132</a></span>;<br/>
<a NAME="E13:43"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A7</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="c6">x3</font>,<font color="Maroon" title="c12">z3</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</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="c6">x3</font>,<font color="Maroon" title="c9">y3</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <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="c9">y3</font>,<font color="Maroon" title="c12">z3</font></span>)</span></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="metric_1.html#T11" title="METRIC_1:th.11">METRIC_1:11</a>, <a class="ref" href="square_1.html#T77" title="SQUARE_1:th.77">SQUARE_1:77</a></span>;<br/>
<a NAME="E14:43"/>
<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">x2</font>,<font color="Maroon" title="c11">z2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">x2</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c11">z2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#D13" title="METRIC_1:def.13">METRIC_1:def 13</a></span>;<br/>
<a NAME="E15:43"/><span class="kw">then </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> <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">x2</font>,<font color="Maroon" title="c11">z2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T132" title="COMPLEX1:th.132">COMPLEX1:132</a></span>;<br/>
<a NAME="E16:43"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A8</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="c5">x2</font>,<font color="Maroon" title="c11">z2</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</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="c5">x2</font>,<font color="Maroon" title="c8">y2</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <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="c8">y2</font>,<font color="Maroon" title="c11">z2</font></span>)</span></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="metric_1.html#T11" title="METRIC_1:th.11">METRIC_1:11</a>, <a class="ref" href="square_1.html#T77" title="SQUARE_1:th.77">SQUARE_1:77</a></span>;<br/>
<a NAME="E17:43"/>
<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">x1</font>,<font color="Maroon" title="c10">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c10">z1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#D13" title="METRIC_1:def.13">METRIC_1:def 13</a></span>;<br/>
<a NAME="E18:43"/><span class="kw">then </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> <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">x1</font>,<font color="Maroon" title="c10">z1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T132" title="COMPLEX1:th.132">COMPLEX1:132</a></span>;<br/>
<a NAME="E19:43"/><span class="kw">then </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="c4">x1</font>,<font color="Maroon" title="c10">z1</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</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">x1</font>,<font color="Maroon" title="c7">y1</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <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="c7">y1</font>,<font color="Maroon" title="c10">z1</font></span>)</span></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="metric_1.html#T11" title="METRIC_1:th.11">METRIC_1:11</a>, <a class="ref" href="square_1.html#T77" title="SQUARE_1:th.77">SQUARE_1:77</a></span>;<br/>
<a NAME="E20:43"/><span class="kw">then </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">x1</font>,<font color="Maroon" title="c10">z1</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">x2</font>,<font color="Maroon" title="c11">z2</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <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"><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">x1</font>,<font color="Maroon" title="c7">y1</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <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="c7">y1</font>,<font color="Maroon" title="c10">z1</font></span>)</span></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"><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">x2</font>,<font color="Maroon" title="c8">y2</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <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="c8">y2</font>,<font color="Maroon" title="c11">z2</font></span>)</span></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="txt" href="#E16:43"><span class="lab"><font color="Green" title="E37">A8</font></span></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/>
<a NAME="E21:43"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A9</font></span>: 
<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="c4">x1</font>,<font color="Maroon" title="c10">z1</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="c5">x2</font>,<font color="Maroon" title="c11">z2</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="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="c6">x3</font>,<font color="Maroon" title="c12">z3</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span> <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"><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">x1</font>,<font color="Maroon" title="c7">y1</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <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="c7">y1</font>,<font color="Maroon" title="c10">z1</font></span>)</span></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="c5">x2</font>,<font color="Maroon" title="c8">y2</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <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="c8">y2</font>,<font color="Maroon" title="c11">z2</font></span>)</span></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="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="c6">x3</font>,<font color="Maroon" title="c9">y3</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <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="c9">y3</font>,<font color="Maroon" title="c12">z3</font></span>)</span></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="txt" href="#E13:43"><span class="lab"><font color="Green" title="E36">A7</font></span></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/>
<a NAME="E22:43"/>
(  <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">x1</font>,<font color="Maroon" title="c10">z1</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">x2</font>,<font color="Maroon" title="c11">z2</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#T65" title="XREAL_1:th.65">XREAL_1:65</a></span>;<br/>
<a NAME="E23:43"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A10</font></span>: 
<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">x1</font>,<font color="Maroon" title="c10">z1</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">x2</font>,<font color="Maroon" title="c11">z2</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#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/>
<a NAME="E24:43"/>
 <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="c6">x3</font>,<font color="Maroon" title="c12">z3</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#T65" title="XREAL_1:th.65">XREAL_1:65</a></span>;<br/>
<a NAME="E25:43"/><span class="kw">then </span>
<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"><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">x1</font>,<font color="Maroon" title="c10">z1</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="c5">x2</font>,<font color="Maroon" title="c11">z2</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="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="c6">x3</font>,<font color="Maroon" title="c12">z3</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="txt" href="#E23:43"><span class="lab"><font color="Green" title="E39">A10</font></span></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/>
<a NAME="E26:43"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A11</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"><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">x1</font>,<font color="Maroon" title="c10">z1</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">x2</font>,<font color="Maroon" title="c11">z2</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="c6">x3</font>,<font color="Maroon" title="c12">z3</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <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"><span class="p5">(<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">x1</font>,<font color="Maroon" title="c7">y1</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<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="c7">y1</font>,<font color="Maroon" title="c10">z1</font></span>)</span></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"><span class="p5">(<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">x2</font>,<font color="Maroon" title="c8">y2</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<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="c8">y2</font>,<font color="Maroon" title="c11">z2</font></span>)</span></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"><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="c6">x3</font>,<font color="Maroon" title="c9">y3</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <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="c9">y3</font>,<font color="Maroon" title="c12">z3</font></span>)</span></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="#E21:43"><span class="lab"><font color="Green" title="E38">A9</font></span></a>, <a class="ref" href="square_1.html#T94" title="SQUARE_1:th.94">SQUARE_1:94</a></span>;<br/>
<a NAME="E27:43"/>
<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="c6">x3</font>,<font color="Maroon" title="c9">y3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">x3</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c9">y3</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#D13" title="METRIC_1:def.13">METRIC_1:def 13</a></span>;<br/>
<a NAME="E28:43"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A12</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> <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="c6">x3</font>,<font color="Maroon" title="c9">y3</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T132" title="COMPLEX1:th.132">COMPLEX1:132</a></span>;<br/>
<a NAME="E29:43"/>
<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="c7">y1</font>,<font color="Maroon" title="c10">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">y1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c10">z1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#D13" title="METRIC_1:def.13">METRIC_1:def 13</a></span>;<br/>
<a NAME="E30:43"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A13</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> <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="c7">y1</font>,<font color="Maroon" title="c10">z1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T132" title="COMPLEX1:th.132">COMPLEX1:132</a></span>;<br/>
<a NAME="E31:43"/>
<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">x1</font>,<font color="Maroon" title="c7">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">x1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c7">y1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#D13" title="METRIC_1:def.13">METRIC_1:def 13</a></span>;<br/>
<a NAME="E32:43"/><span class="kw">then </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> <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">x1</font>,<font color="Maroon" title="c7">y1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="complex1.html#T132" title="COMPLEX1:th.132">COMPLEX1:132</a></span>;<br/>
<a NAME="E33:43"/><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"><span class="p5">(<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">x1</font>,<font color="Maroon" title="c7">y1</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<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="c7">y1</font>,<font color="Maroon" title="c10">z1</font></span>)</span></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"><span class="p5">(<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">x2</font>,<font color="Maroon" title="c8">y2</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p5">(<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="c8">y2</font>,<font color="Maroon" title="c11">z2</font></span>)</span></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"><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="c6">x3</font>,<font color="Maroon" title="c9">y3</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <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="c9">y3</font>,<font color="Maroon" title="c12">z3</font></span>)</span></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<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">x1</font>,<font color="Maroon" title="c7">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="p4">(<span class="default"><span class="p5">(<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">x2</font>,<font color="Maroon" title="c8">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="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="c6">x3</font>,<font color="Maroon" title="c9">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="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<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="c7">y1</font>,<font color="Maroon" title="c10">z1</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="p4">(<span class="default"><span class="p5">(<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="c8">y2</font>,<font color="Maroon" title="c11">z2</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="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="c9">y3</font>,<font color="Maroon" title="c12">z3</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:43"><span class="lab"><font color="Green" title="E42">A13</font></span></a>, <a class="txt" href="#E9:43"><span class="lab"><font color="Green" title="E34">A5</font></span></a>, <a class="txt" href="#E7:43"><span class="lab"><font color="Green" title="E33">A4</font></span></a>, <a class="txt" href="#E28:43"><span class="lab"><font color="Green" title="E41">A12</font></span></a>, <a class="txt" href="#E5:43"><span class="lab"><font color="Green" title="E32">A3</font></span></a>, <a class="txt" href="metric_4.html#E20"><span class="lab"><font color="Green" title="E12">Lm1</font></span></a></span>;<br/>
<a NAME="E34:43"/><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="c4">x1</font>,<font color="Maroon" title="c10">z1</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">x2</font>,<font color="Maroon" title="c11">z2</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="c6">x3</font>,<font color="Maroon" title="c12">z3</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<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">x1</font>,<font color="Maroon" title="c7">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="p4">(<span class="default"><span class="p5">(<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">x2</font>,<font color="Maroon" title="c8">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="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="c6">x3</font>,<font color="Maroon" title="c9">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="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<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="c7">y1</font>,<font color="Maroon" title="c10">z1</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="p4">(<span class="default"><span class="p5">(<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="c8">y2</font>,<font color="Maroon" title="c11">z2</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="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="c9">y3</font>,<font color="Maroon" title="c12">z3</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:43"><span class="lab"><font color="Green" title="E40">A11</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E35:43"/><span class="kw">then </span>
<a href="metric_4.html#K13" title="METRIC_4:func.13">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="c3">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<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">x1</font>,<font color="Maroon" title="c7">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="p4">(<span class="default"><span class="p5">(<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">x2</font>,<font color="Maroon" title="c8">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="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="c6">x3</font>,<font color="Maroon" title="c9">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="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<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="c7">y1</font>,<font color="Maroon" title="c10">z1</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="p4">(<span class="default"><span class="p5">(<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="c8">y2</font>,<font color="Maroon" title="c11">z2</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="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="c9">y3</font>,<font color="Maroon" title="c12">z3</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43"><span class="lab"><font color="Green" title="E30">A1</font></span></a>, <a class="txt" href="#E10:43"><span class="lab"><font color="Green" title="E35">A6</font></span></a>, <a class="ref" href="metric_4.html#D13" target="_self" title="METRIC_4:def.13">Def13</a></span>;<br/>
<a NAME="E36:43"/><span class="kw">then </span>
<a href="metric_4.html#K13" title="METRIC_4:func.13">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="c3">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_4.html#K13" title="METRIC_4:func.13">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></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="square_1.html#K7" title="SQUARE_1:func.7">sqrt</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<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="c7">y1</font>,<font color="Maroon" title="c10">z1</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="p4">(<span class="default"><span class="p5">(<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="c8">y2</font>,<font color="Maroon" title="c11">z2</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="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="c9">y3</font>,<font color="Maroon" title="c12">z3</font></span>)</span> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> </span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43"><span class="lab"><font color="Green" title="E30">A1</font></span></a>, <a class="txt" href="#E3:43"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="ref" href="metric_4.html#D13" target="_self" title="METRIC_4:def.13">Def13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E37:43"/>
<a href="metric_4.html#K13" title="METRIC_4:func.13">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="c3">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_4.html#K13" title="METRIC_4:func.13">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></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="metric_4.html#K13" title="METRIC_4:func.13">Eukl_dist3</a>  <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">z</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="txt" href="#E10:43"><span class="lab"><font color="Green" title="E35">A6</font></span></a>, <a class="ref" href="metric_4.html#D13" target="_self" title="METRIC_4:def.13">Def13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
