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<div class="add">

<b>reconsider </b><font color="Maroon" title="c1">M</font> =  <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ,<a href="metric_1.html#K7" title="METRIC_1:func.7">real_dist</a>  #) as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="metric_1.html#L1" title="METRIC_1:struct.1">MetrStruct</a>  <b>by </b><i><a class="ref" href="struct_0.html#D1" title="STRUCT_0:def.1">STRUCT_0:def 1</a></i>;<br/>
<a NAME="E2:30_1_1"/>
<font color="Maroon" title="c1">M</font> is   <a href="metric_1.html#NM1" title="METRIC_1:NM.1">MetrSpace</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:30_1_1_1"/>
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">M</font> holds <br/> ( (  <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="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> ) &amp; ( <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> implies  <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="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp;  <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> &amp;  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">c</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><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></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="metric_1.html#T9" target="_self" title="METRIC_1:th.9">Th9</a>, <a class="ref" href="metric_1.html#T10" target="_self" title="METRIC_1:th.10">Th10</a>, <a class="ref" href="metric_1.html#T11" target="_self" title="METRIC_1:th.11">Th11</a></i>;<br/>
<b>hence </b><a NAME="E2:30_1_1_1"/>
<font color="Maroon" title="c1">M</font> is   <a href="metric_1.html#NM1" title="METRIC_1:NM.1">MetrSpace</a>
 <b>by </b><i><a class="ref" href="metric_1.html#T6" target="_self" title="METRIC_1:th.6">Th6</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:30_1_1"/>
(  <a href="metric_1.html#K8" title="METRIC_1:func.8">RealSpace</a>  is <a href="metric_1.html#V6" title="METRIC_1:attr.6">Reflexive</a> &amp;  <a href="metric_1.html#K8" title="METRIC_1:func.8">RealSpace</a>  is <a href="metric_1.html#V7" title="METRIC_1:attr.7">discerning</a> &amp;  <a href="metric_1.html#K8" title="METRIC_1:func.8">RealSpace</a>  is <a href="metric_1.html#V8" title="METRIC_1:attr.8">symmetric</a> &amp;  <a href="metric_1.html#K8" title="METRIC_1:func.8">RealSpace</a>  is <a href="metric_1.html#V9" title="METRIC_1:attr.9">triangle</a> )
 ;<br/>


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