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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="metric_1.html#V6">Reflexive</a> <a href="metric_1.html#V8">symmetric</a> <a href="metric_1.html#V9">triangle</a>  <a href="metric_1.html#L1">MetrStruct</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>





<b>assume </b><a NAME="E1:25"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/>

<b>assume </b><a NAME="E2:25"/><i><font color="Green">E18</font></i>: 
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="tbsp_1.html#E2:25"><i><font color="Green">E18</font></i></a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E5:25"/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="tbsp_1.html#E2:25"><i><font color="Green">E18</font></i></a>, <a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/>
<b>hence </b><a NAME="E6:25"/>
contradiction
 <b>by </b><i><a class="txt" href="tbsp_1.html#E1:25"><i><font color="Green">E17</font></i></a>, <a class="ref" href="metric_1.html#T5">METRIC_1:5</a></i>;<br/>


</div>
