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<span class="kw">take </span><font color="Maroon" title="c2">M0</font> =  <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# 1,<a href="metric_1.html#K3" title="METRIC_1:func.3">Empty^2-to-zero</a> #); <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="c2">M0</font> is  <a href="metric_1.html#V1" title="METRIC_1:attr.1">strict</a>  &amp; <font color="Maroon" title="c2">M0</font> is  <a href="metric_1.html#V12" title="METRIC_1:attr.12">ultra</a>  &amp; <font color="Maroon" title="c2">M0</font> is  <a href="metric_1.html#V6" title="METRIC_1:attr.6">Reflexive</a>  &amp; <font color="Maroon" title="c2">M0</font> is  <a href="metric_1.html#V8" title="METRIC_1:attr.8">symmetric</a>  &amp; <font color="Maroon" title="c2">M0</font> is  <a href="metric_1.html#V11" title="METRIC_1:attr.11">Discerning</a>  &amp; <font color="Maroon" title="c2">M0</font> is  <a href="metric_1.html#V9" title="METRIC_1:attr.9">triangle</a>  )</span><br/>

<a NAME="E1:57_1_1"/>
<font color="Maroon" title="c2">M0</font> is  <a href="metric_1.html#V12" title="METRIC_1:attr.12">ultra</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font>, <font color="Maroon" title="c5">z</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">M0</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="metric_1.html#D19" target="_self" title="METRIC_1:def.19">METRIC_1:def 19</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">z</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> (<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">z</font>)</span>)</span>)</span><br/>

<span class="kw">thus </span><a NAME="E1:57_1_1_1"/>
 <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">z</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xxreal_0.html#K4" title="XXREAL_0:func.4">max</a> (<span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> (<font color="Maroon" title="c4">y</font>,<font color="Maroon" title="c5">z</font>)</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#T24" target="_self" title="METRIC_1:th.24">Th25</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/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:57_1_1"/>
( <font color="Maroon" title="c2">M0</font> is  <a href="metric_1.html#V1" title="METRIC_1:attr.1">strict</a>  &amp; <font color="Maroon" title="c2">M0</font> is  <a href="metric_1.html#V12" title="METRIC_1:attr.12">ultra</a>  &amp; <font color="Maroon" title="c2">M0</font> is  <a href="metric_1.html#V6" title="METRIC_1:attr.6">Reflexive</a>  &amp; <font color="Maroon" title="c2">M0</font> is  <a href="metric_1.html#V8" title="METRIC_1:attr.8">symmetric</a>  &amp; <font color="Maroon" title="c2">M0</font> is  <a href="metric_1.html#V11" title="METRIC_1:attr.11">Discerning</a>  &amp; <font color="Maroon" title="c2">M0</font> is  <a href="metric_1.html#V9" title="METRIC_1:attr.9">triangle</a>  )
 ; <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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