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<b>let </b><font color="Maroon" title="c1">M</font> be    <a href="metric_1.html#L1" title="METRIC_1:struct.1">MetrStruct</a> ;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:15_1"/><i><font color="Green" title="E16">A1</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">M</font> holds   <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Olive" title="b2">b</font>,<font color="Olive" title="b1">a</font>
 ;<br/><a NAME="E2:15_1"/>
the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <font color="Maroon" title="c1">M</font> is <a href="metric_1.html#V4" title="METRIC_1:attr.4">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" title="c2">a</font>, <font color="Maroon" title="c3">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">M</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="metric_1.html#D5" target="_self" title="METRIC_1:def.5">METRIC_1:def 5</a><br/>

<b>thus </b>the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <font color="Maroon" title="c1">M</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> = 
 <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>

<br/>.= 
 <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font>
<b>by </b><i><a class="txt" href="metric_1.html#E1:15_1"><i><font color="Green" title="E16">A1</font></i></a></i>
<br/>.= 
the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <font color="Maroon" title="c1">M</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font>

;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:15_1"/>
<font color="Maroon" title="c1">M</font> is <a href="metric_1.html#V8" title="METRIC_1:attr.8">symmetric</a>
 <b>by </b><i><a class="ref" href="metric_1.html#D9" target="_self" title="METRIC_1:def.9">Def9</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:15"/>
<font color="Maroon" title="c1">M</font> is <a href="metric_1.html#V8" title="METRIC_1:attr.8">symmetric</a>
 ;<br/>

<a NAME="E3:15"/><b>then </b><i><font color="Green" title="E16">A2</font></i>: 
the <a href="metric_1.html#U1" title="METRIC_1:sel.1">distance</a> of <font color="Maroon" title="c1">M</font> is <a href="metric_1.html#V4" title="METRIC_1:attr.4">symmetric</a>
 
<b>by </b><i><a class="ref" href="metric_1.html#D9" target="_self" title="METRIC_1:def.9">Def9</a></i>;<br/>
<a NAME="E4:15"/>
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">M</font> holds   <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Olive" title="b2">b</font>,<font color="Olive" title="b1">a</font>
 
<b>by </b><i><a class="txt" href="metric_1.html#E3:15"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="metric_1.html#D5" target="_self" title="METRIC_1:def.5">Def5</a></i>;<br/>
<b>hence </b><a NAME="E5:15"/>
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">M</font> holds   <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Olive" title="b2">b</font>,<font color="Olive" title="b1">a</font>
 ;<br/>


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