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<span class="kw">let </span><font color="Maroon" title="c1">M</font> be    <a href="metric_1.html#L1" title="METRIC_1:struct.1">MetrStruct</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  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>) ) implies <font color="Maroon" title="c1">M</font> is  <a href="metric_1.html#V8" title="METRIC_1:attr.8">symmetric</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:19"/><span class="lab"><font color="Green" title="E16">A1</font></span>: 
 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>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">M</font> is  <a href="metric_1.html#V8" title="METRIC_1:attr.8">symmetric</a> </span><br/>

<a NAME="E2:19"/>
 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="19_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="metric_1.html#D4" target="_self" title="METRIC_1:def.4">METRIC_1:def 4</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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="hidden.html#R1" title="HIDDEN:pred.1">=</a>  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>)</span><br/>

<span class="kw">thus </span> 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>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19"><span class="lab"><font color="Green" title="E16">A1</font></span></a></span>
<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>)

; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:19"/>
<font color="Maroon" title="c1">M</font> is  <a href="metric_1.html#V8" title="METRIC_1:attr.8">symmetric</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#D8" target="_self" title="METRIC_1:def.8">Def9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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