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

<b>set </b><font color="Maroon">c<sub>2</sub></font> =  <a href="metric_1.html#G1" target="_self">MetrStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" target="_self">discrete_dist</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #);<br/>
<a NAME="E1:24_1_1"/>
(  <a href="metric_1.html#G1" target="_self">MetrStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" target="_self">discrete_dist</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) is <a href="metric_1.html#V6" target="_self">Reflexive</a> &amp;  <a href="metric_1.html#G1" target="_self">MetrStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" target="_self">discrete_dist</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) is <a href="metric_1.html#V7" target="_self">discerning</a> &amp;  <a href="metric_1.html#G1" target="_self">MetrStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" target="_self">discrete_dist</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) is <a href="metric_1.html#V8" target="_self">symmetric</a> &amp;  <a href="metric_1.html#G1" target="_self">MetrStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" target="_self">discrete_dist</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) is <a href="metric_1.html#V9" target="_self">triangle</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:24_1_1_1"/>
the <a href="metric_1.html#U1" target="_self">distance</a> of <a href="metric_1.html#G1" target="_self">MetrStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" target="_self">discrete_dist</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) is <a href="metric_1.html#V2" target="_self">Reflexive</a>
  <i><font color="Red">:: according to </font></i><a class="ref" href="metric_1.html#D7" target="_self">METRIC_1:def 7</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <a href="metric_1.html#G1" target="_self">MetrStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" target="_self">discrete_dist</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #); <i><font color="Red">:: according to </font></i><a class="ref" href="metric_1.html#D3" target="_self">METRIC_1:def 3</a><br/>

<b>thus </b><a NAME="E1:24_1_1_1_1"/>
the <a href="metric_1.html#U1" target="_self">distance</a> of <a href="metric_1.html#G1" target="_self">MetrStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" target="_self">discrete_dist</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) <a href="metric_1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="metric_1.html#D11" target="_self">Def11</a></i>;<br/>


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

<b>thus </b><a NAME="E2:24_1_1_1"/>
the <a href="metric_1.html#U1" target="_self">distance</a> of <a href="metric_1.html#G1" target="_self">MetrStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" target="_self">discrete_dist</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) is <a href="metric_1.html#V3" target="_self">discerning</a>
  <i><font color="Red">:: according to </font></i><a class="ref" href="metric_1.html#D8" target="_self">METRIC_1:def 8</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <a href="metric_1.html#G1" target="_self">MetrStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" target="_self">discrete_dist</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #); <i><font color="Red">:: according to </font></i><a class="ref" href="metric_1.html#D4" target="_self">METRIC_1:def 4</a><br/>

<b>assume </b><a NAME="E1:24_1_1_1_2"/>
the <a href="metric_1.html#U1" target="_self">distance</a> of <a href="metric_1.html#G1" target="_self">MetrStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" target="_self">discrete_dist</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) <a href="metric_1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<b>hence </b><a NAME="E2:24_1_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="metric_1.html#D11" target="_self">Def11</a></i>;<br/>


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

<b>thus </b><a NAME="E3:24_1_1_1"/>
the <a href="metric_1.html#U1" target="_self">distance</a> of <a href="metric_1.html#G1" target="_self">MetrStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" target="_self">discrete_dist</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) is <a href="metric_1.html#V4" target="_self">symmetric</a>
  <i><font color="Red">:: according to </font></i><a class="ref" href="metric_1.html#D9" target="_self">METRIC_1:def 9</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <a href="metric_1.html#G1" target="_self">MetrStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" target="_self">discrete_dist</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #); <i><font color="Red">:: according to </font></i><a class="ref" href="metric_1.html#D5" target="_self">METRIC_1:def 5</a><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:24_1_1_1_3_1_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font> or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:24_1_1_1_3_1_1_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><b>hence </b>the <a href="metric_1.html#U1" target="_self">distance</a> of <a href="metric_1.html#G1" target="_self">MetrStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" target="_self">discrete_dist</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) <a href="metric_1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> = 
1
<b>by </b><i><a class="ref" href="metric_1.html#D11" target="_self">Def11</a></i>
<br/>.= 
the <a href="metric_1.html#U1" target="_self">distance</a> of <a href="metric_1.html#G1" target="_self">MetrStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" target="_self">discrete_dist</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) <a href="metric_1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="metric_1.html#E1:24_1_1_1_3_1_1_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="metric_1.html#D11" target="_self">Def11</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:24_1_1_1_3_1_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:24_1_1_1_3_1_1_2"/>
the <a href="metric_1.html#U1" target="_self">distance</a> of <a href="metric_1.html#G1" target="_self">MetrStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" target="_self">discrete_dist</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) <a href="metric_1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> the <a href="metric_1.html#U1" target="_self">distance</a> of <a href="metric_1.html#G1" target="_self">MetrStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" target="_self">discrete_dist</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) <a href="metric_1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E2:24_1_1_1_3"/>
the <a href="metric_1.html#U1" target="_self">distance</a> of <a href="metric_1.html#G1" target="_self">MetrStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" target="_self">discrete_dist</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) <a href="metric_1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> the <a href="metric_1.html#U1" target="_self">distance</a> of <a href="metric_1.html#G1" target="_self">MetrStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" target="_self">discrete_dist</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) <a href="metric_1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 ;<br/>


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

<b>thus </b><a NAME="E4:24_1_1_1"/>
the <a href="metric_1.html#U1" target="_self">distance</a> of <a href="metric_1.html#G1" target="_self">MetrStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" target="_self">discrete_dist</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) is <a href="metric_1.html#V5" target="_self">triangle</a>
  <i><font color="Red">:: according to </font></i><a class="ref" href="metric_1.html#D10" target="_self">METRIC_1:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <a href="metric_1.html#G1" target="_self">MetrStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" target="_self">discrete_dist</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #); <i><font color="Red">:: according to </font></i><a class="ref" href="metric_1.html#D6" target="_self">METRIC_1:def 6</a><br/>

<a NAME="E1:24_1_1_1_4"/><i><font color="Green">E19</font></i>: 
( the <a href="metric_1.html#U1" target="_self">distance</a> of <a href="metric_1.html#G1" target="_self">MetrStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" target="_self">discrete_dist</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) <a href="metric_1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0 iff <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="ref" href="metric_1.html#D11" target="_self">Def11</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_1_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:24_1_1_1_4_1"/>
( ( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> ) or ( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font> ) or ( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font> ) or ( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font> ) or ( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_1_4_1_1"><b>suppose </b></a><a NAME="E1:24_1_1_1_4_1_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:24_1_1_1_4_1_1"/>
the <a href="metric_1.html#U1" target="_self">distance</a> of <a href="metric_1.html#G1" target="_self">MetrStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" target="_self">discrete_dist</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) <a href="metric_1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">the <a href="metric_1.html#U1" target="_self">distance</a> of <a href="metric_1.html#G1" target="_self">MetrStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" target="_self">discrete_dist</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) <a href="metric_1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">the <a href="metric_1.html#U1" target="_self">distance</a> of <a href="metric_1.html#G1" target="_self">MetrStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" target="_self">discrete_dist</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) <a href="metric_1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="metric_1.html#E1:24_1_1_1_4"><i><font color="Green">E19</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_1_4_1_2"><b>suppose </b></a><a NAME="E1:24_1_1_1_4_1_2"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:24_1_1_1_4_1_2"/>
the <a href="metric_1.html#U1" target="_self">distance</a> of <a href="metric_1.html#G1" target="_self">MetrStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" target="_self">discrete_dist</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) <a href="metric_1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">the <a href="metric_1.html#U1" target="_self">distance</a> of <a href="metric_1.html#G1" target="_self">MetrStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" target="_self">discrete_dist</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) <a href="metric_1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">the <a href="metric_1.html#U1" target="_self">distance</a> of <a href="metric_1.html#G1" target="_self">MetrStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" target="_self">discrete_dist</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) <a href="metric_1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="metric_1.html#E1:24_1_1_1_4"><i><font color="Green">E19</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_1_4_1_3"><b>suppose </b></a><a NAME="E1:24_1_1_1_4_1_3"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/><div class="add"><a NAME="E2:24_1_1_1_4_1_3"/><b>then </b><i><font color="Green">E21</font></i>: 
the <a href="metric_1.html#U1" target="_self">distance</a> of <a href="metric_1.html#G1" target="_self">MetrStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" target="_self">discrete_dist</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) <a href="metric_1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="metric_1.html#D11" target="_self">Def11</a></i>;<br/><a NAME="E3:24_1_1_1_4_1_3"/><i><font color="Green">E22</font></i>: 
the <a href="metric_1.html#U1" target="_self">distance</a> of <a href="metric_1.html#G1" target="_self">MetrStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" target="_self">discrete_dist</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) <a href="metric_1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="metric_1.html#E1:24_1_1_1_4_1_3"><i><font color="Green">E20</font></i></a>, <a class="ref" href="metric_1.html#D11" target="_self">Def11</a></i>;<br/><a NAME="E4:24_1_1_1_4_1_3"/>
the <a href="metric_1.html#U1" target="_self">distance</a> of <a href="metric_1.html#G1" target="_self">MetrStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" target="_self">discrete_dist</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) <a href="metric_1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="metric_1.html#E1:24_1_1_1_4_1_3"><i><font color="Green">E20</font></i></a>, <a class="ref" href="metric_1.html#D11" target="_self">Def11</a></i>;<br/><b>hence </b><a NAME="E5:24_1_1_1_4_1_3"/>
the <a href="metric_1.html#U1" target="_self">distance</a> of <a href="metric_1.html#G1" target="_self">MetrStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" target="_self">discrete_dist</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) <a href="metric_1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">the <a href="metric_1.html#U1" target="_self">distance</a> of <a href="metric_1.html#G1" target="_self">MetrStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" target="_self">discrete_dist</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) <a href="metric_1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">the <a href="metric_1.html#U1" target="_self">distance</a> of <a href="metric_1.html#G1" target="_self">MetrStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" target="_self">discrete_dist</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) <a href="metric_1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="metric_1.html#E2:24_1_1_1_4_1_3"><i><font color="Green">E21</font></i></a>, <a class="txt" href="metric_1.html#E3:24_1_1_1_4_1_3"><i><font color="Green">E22</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_1_4_1_4"><b>suppose </b></a><a NAME="E1:24_1_1_1_4_1_4"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/><div class="add"><a NAME="E2:24_1_1_1_4_1_4"/><b>then </b>
the <a href="metric_1.html#U1" target="_self">distance</a> of <a href="metric_1.html#G1" target="_self">MetrStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" target="_self">discrete_dist</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) <a href="metric_1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="metric_1.html#D11" target="_self">Def11</a></i>;<br/><b>hence </b><a NAME="E3:24_1_1_1_4_1_4"/>
the <a href="metric_1.html#U1" target="_self">distance</a> of <a href="metric_1.html#G1" target="_self">MetrStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" target="_self">discrete_dist</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) <a href="metric_1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">the <a href="metric_1.html#U1" target="_self">distance</a> of <a href="metric_1.html#G1" target="_self">MetrStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" target="_self">discrete_dist</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) <a href="metric_1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">the <a href="metric_1.html#U1" target="_self">distance</a> of <a href="metric_1.html#G1" target="_self">MetrStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" target="_self">discrete_dist</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) <a href="metric_1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="metric_1.html#E1:24_1_1_1_4_1_4"><i><font color="Green">E20</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1_1_4_1_5"><b>suppose </b></a><a NAME="E1:24_1_1_1_4_1_5"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/><div class="add"><a NAME="E2:24_1_1_1_4_1_5"/><b>then </b>
( the <a href="metric_1.html#U1" target="_self">distance</a> of <a href="metric_1.html#G1" target="_self">MetrStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" target="_self">discrete_dist</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) <a href="metric_1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; the <a href="metric_1.html#U1" target="_self">distance</a> of <a href="metric_1.html#G1" target="_self">MetrStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" target="_self">discrete_dist</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) <a href="metric_1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; the <a href="metric_1.html#U1" target="_self">distance</a> of <a href="metric_1.html#G1" target="_self">MetrStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" target="_self">discrete_dist</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) <a href="metric_1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="ref" href="metric_1.html#D11" target="_self">Def11</a></i>;<br/><b>hence </b><a NAME="E3:24_1_1_1_4_1_5"/>
the <a href="metric_1.html#U1" target="_self">distance</a> of <a href="metric_1.html#G1" target="_self">MetrStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" target="_self">discrete_dist</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) <a href="metric_1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default">the <a href="metric_1.html#U1" target="_self">distance</a> of <a href="metric_1.html#G1" target="_self">MetrStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" target="_self">discrete_dist</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) <a href="metric_1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default">the <a href="metric_1.html#U1" target="_self">distance</a> of <a href="metric_1.html#G1" target="_self">MetrStruct</a>(# <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="metric_1.html#K5" target="_self">discrete_dist</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #) <a href="metric_1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>
 ;<br/><b>thus </b><a NAME="E4:24_1_1_1_4_1_5"/>
verum
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

<b>thus </b><a NAME="E5:24_1_1_1"/>
verum
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:24_1_1"/>
(  <a href="metric_1.html#K6" target="_self">DiscreteSpace</a> <font color="Maroon">c<sub>1</sub></font> is <a href="metric_1.html#V6" target="_self">Reflexive</a> &amp;  <a href="metric_1.html#K6" target="_self">DiscreteSpace</a> <font color="Maroon">c<sub>1</sub></font> is <a href="metric_1.html#V7" target="_self">discerning</a> &amp;  <a href="metric_1.html#K6" target="_self">DiscreteSpace</a> <font color="Maroon">c<sub>1</sub></font> is <a href="metric_1.html#V8" target="_self">symmetric</a> &amp;  <a href="metric_1.html#K6" target="_self">DiscreteSpace</a> <font color="Maroon">c<sub>1</sub></font> is <a href="metric_1.html#V9" target="_self">triangle</a> )
 ;<br/>


</div>
