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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="metric_1.html#L1" target="_self">MetrStruct</a> ;<br/>

<b>assume </b><a NAME="E1:20"/><i><font color="Green">E18</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>( (  <a href="metric_1.html#K2" target="_self">dist</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 implies <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> ) &amp; ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> implies  <a href="metric_1.html#K2" target="_self">dist</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0 ) &amp;  <a href="metric_1.html#K2" target="_self">dist</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K2" target="_self">dist</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> &amp;  <a href="metric_1.html#K2" target="_self">dist</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2" target="_self">dist</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2" target="_self">dist</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span>)</span> )
 ;<br/>

<a NAME="E2:20"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="metric_1.html#V6" target="_self">Reflexive</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="metric_1.html#V7" target="_self">discerning</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="metric_1.html#V8" target="_self">symmetric</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="metric_1.html#V9" target="_self">triangle</a> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:20_1"/>
the <a href="metric_1.html#U1" target="_self">distance</a> of <font color="Maroon">c<sub>1</sub></font> 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="20_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="metric_1.html#D3" target="_self">METRIC_1:def 3</a><br/>

the <a href="metric_1.html#U1" target="_self">distance</a> of <font color="Maroon">c<sub>1</sub></font> <a href="metric_1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font> = 
 <a href="metric_1.html#K2" target="_self">dist</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font>

<br/>.= 
0
<b>by </b><i><a class="txt" href="metric_1.html#E1:20"><i><font color="Green">E18</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E2:20_1_1"/>
the <a href="metric_1.html#U1" target="_self">distance</a> of <font color="Maroon">c<sub>1</sub></font> <a href="metric_1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/>


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

<b>thus </b><a NAME="E2:20_1"/>
the <a href="metric_1.html#U1" target="_self">distance</a> of <font color="Maroon">c<sub>1</sub></font> 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="20_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>; <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="E1:20_1_2"/>
the <a href="metric_1.html#U1" target="_self">distance</a> of <font color="Maroon">c<sub>1</sub></font> <a href="metric_1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/>

<a NAME="E2:20_1_2"/><b>then </b>
 <a href="metric_1.html#K2" target="_self">dist</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 0
 
;<br/>
<b>hence </b><a NAME="E3:20_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="metric_1.html#E1:20"><i><font color="Green">E18</font></i></a></i>;<br/>


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

<b>thus </b><a NAME="E3:20_1"/>
the <a href="metric_1.html#U1" target="_self">distance</a> of <font color="Maroon">c<sub>1</sub></font> 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="20_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="metric_1.html#D5" target="_self">METRIC_1:def 5</a><br/>

the <a href="metric_1.html#U1" target="_self">distance</a> of <font color="Maroon">c<sub>1</sub></font> <a href="metric_1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> = 
 <a href="metric_1.html#K2" target="_self">dist</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>

<br/>.= 
 <a href="metric_1.html#K2" target="_self">dist</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="txt" href="metric_1.html#E1:20"><i><font color="Green">E18</font></i></a></i>
<br/>.= 
the <a href="metric_1.html#U1" target="_self">distance</a> of <font color="Maroon">c<sub>1</sub></font> <a href="metric_1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>

;<br/>
<b>hence </b><a NAME="E2:20_1_3"/>
the <a href="metric_1.html#U1" target="_self">distance</a> of <font color="Maroon">c<sub>1</sub></font> <a href="metric_1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> the <a href="metric_1.html#U1" target="_self">distance</a> of <font color="Maroon">c<sub>1</sub></font> <a href="metric_1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 ;<br/>


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

<b>thus </b><a NAME="E4:20_1"/>
the <a href="metric_1.html#U1" target="_self">distance</a> of <font color="Maroon">c<sub>1</sub></font> 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="20_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font>, <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 <font color="Maroon">c<sub>1</sub></font>; <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="E1:20_1_4"/><i><font color="Green">E19</font></i>: 
the <a href="metric_1.html#U1" target="_self">distance</a> of <font color="Maroon">c<sub>1</sub></font> <a href="metric_1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K2" target="_self">dist</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 
;<br/>
<a NAME="E2:20_1_4"/><i><font color="Green">E20</font></i>: 
the <a href="metric_1.html#U1" target="_self">distance</a> of <font color="Maroon">c<sub>1</sub></font> <a href="metric_1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K2" target="_self">dist</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 
;<br/>
<a NAME="E3:20_1_4"/>
the <a href="metric_1.html#U1" target="_self">distance</a> of <font color="Maroon">c<sub>1</sub></font> <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>  <a href="metric_1.html#K2" target="_self">dist</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 
;<br/>
<b>hence </b><a NAME="E4:20_1_4"/>
the <a href="metric_1.html#U1" target="_self">distance</a> of <font color="Maroon">c<sub>1</sub></font> <a href="metric_1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</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 <font color="Maroon">c<sub>1</sub></font> <a href="metric_1.html#K1" target="_self">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</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 <font color="Maroon">c<sub>1</sub></font> <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>
 <b>by </b><i><a class="txt" href="metric_1.html#E1:20"><i><font color="Green">E18</font></i></a>, <a class="txt" href="metric_1.html#E1:20_1_4"><i><font color="Green">E19</font></i></a>, <a class="txt" href="metric_1.html#E2:20_1_4"><i><font color="Green">E20</font></i></a></i>;<br/>


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


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:20"/>
<font color="Maroon">c<sub>1</sub></font> is   <a href="metric_1.html#NM1">MetrSpace</a>
 ;<br/>


</div>
