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

<b>take </b>
 <a href="metric_1.html#G1">MetrStruct</a>(# <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>,<a href="metric_1.html#K3">Empty^2-to-zero</a>  #)
;<br/>

<a NAME="E1:19_1_1"/>
 <a href="metric_1.html#G1">MetrStruct</a>(# <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>,<a href="metric_1.html#K3">Empty^2-to-zero</a>  #) is <a href="sub_metr.html#V3" target="_self">ultra</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <a href="metric_1.html#G1">MetrStruct</a>(# <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>,<a href="metric_1.html#K3">Empty^2-to-zero</a>  #); <i><font color="Red">:: according to </font></i><a class="ref" href="sub_metr.html#D4" target="_self">SUB_METR:def 4</a><br/>

<a NAME="E1:19_1_1_1"/>
(  <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <a href="metric_1.html#K3">Empty^2-to-zero</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp;  <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <a href="metric_1.html#K3">Empty^2-to-zero</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> &amp;  <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <a href="metric_1.html#K3">Empty^2-to-zero</a>  <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="ref" href="metric_1.html#D1">METRIC_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E2:19_1_1_1"/>
 <a href="metric_1.html#K2">dist</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="xxreal_0.html#K4">max</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2">dist</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="metric_1.html#K2">dist</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="sub_metr.html#T8" target="_self">Th8</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:19_1_1"/>
(  <a href="metric_1.html#G1">MetrStruct</a>(# <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>,<a href="metric_1.html#K3">Empty^2-to-zero</a>  #) is <a href="metric_1.html#V1">strict</a> &amp;  <a href="metric_1.html#G1">MetrStruct</a>(# <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>,<a href="metric_1.html#K3">Empty^2-to-zero</a>  #) is <a href="sub_metr.html#V3" target="_self">ultra</a> &amp;  <a href="metric_1.html#G1">MetrStruct</a>(# <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>,<a href="metric_1.html#K3">Empty^2-to-zero</a>  #) is <a href="metric_1.html#V6">Reflexive</a> &amp;  <a href="metric_1.html#G1">MetrStruct</a>(# <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>,<a href="metric_1.html#K3">Empty^2-to-zero</a>  #) is <a href="metric_1.html#V8">symmetric</a> &amp;  <a href="metric_1.html#G1">MetrStruct</a>(# <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>,<a href="metric_1.html#K3">Empty^2-to-zero</a>  #) is <a href="sub_metr.html#V2" target="_self">Discerning</a> )
 ;<br/>


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