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

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

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:15_1"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font> holds   <a href="metric_1.html#K2" target="_self">dist</a> <font color="Olive">a</font>,<font color="Olive">b</font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K2" target="_self">dist</a> <font color="Olive">b</font>,<font color="Olive">a</font>
 ;<br/><a NAME="E2:15_1"/>
the <a href="metric_1.html#U1" target="_self">distance</a> of <font color="Maroon">M</font> is <a href="metric_1.html#V4" target="_self">symmetric</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font>, <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</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/>

<b>thus </b>the <a href="metric_1.html#U1" target="_self">distance</a> of <font color="Maroon">M</font> <a href="metric_1.html#K1" target="_self">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> = 
 <a href="metric_1.html#K2" target="_self">dist</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>

<br/>.= 
 <a href="metric_1.html#K2" target="_self">dist</a> <font color="Maroon">b</font>,<font color="Maroon">a</font>
<b>by </b><i/>
<br/>.= 
the <a href="metric_1.html#U1" target="_self">distance</a> of <font color="Maroon">M</font> <a href="metric_1.html#K1" target="_self">.</a> <font color="Maroon">b</font>,<font color="Maroon">a</font>

;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:15_1"/>
<font color="Maroon">M</font> is <a href="metric_1.html#V8" target="_self">symmetric</a>
 <b>by </b><i/>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:15"/>
<font color="Maroon">M</font> is <a href="metric_1.html#V8" target="_self">symmetric</a>
 ;<br/>

<a NAME="E3:15"/><b>then </b><i><font color="Green">E40</font></i>: 
the <a href="metric_1.html#U1" target="_self">distance</a> of <font color="Maroon">M</font> is <a href="metric_1.html#V4" target="_self">symmetric</a>
 
<b>by </b><i/>;<br/>
<a NAME="E4:15"/>
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font> holds   <a href="metric_1.html#K2" target="_self">dist</a> <font color="Olive">a</font>,<font color="Olive">b</font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K2" target="_self">dist</a> <font color="Olive">b</font>,<font color="Olive">a</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E5:15"/>
for <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font> holds   <a href="metric_1.html#K2" target="_self">dist</a> <font color="Olive">a</font>,<font color="Olive">b</font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K2" target="_self">dist</a> <font color="Olive">b</font>,<font color="Olive">a</font>
 ;<br/>


</div>
