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<div class="add">

<b>let </b><font color="Maroon">T</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">t1</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">r</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="E1:26"/><i><font color="Green">E44</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 ;<br/>

<b>set </b><font color="Maroon">P</font> =  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">t1</font>,<font color="Maroon">r</font>;<br/>
<b>reconsider </b><font color="Maroon">r</font> = <font color="Maroon">r</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E3:26"/>
 ex <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a> ex <font color="Olive">t1</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> st <br/>( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r</font> &amp; <font color="Olive">t1</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">t1</font>,<font color="Maroon">r</font> &amp; ( for <font color="Olive">z</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>  st <font color="Olive">z</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">t1</font>,<font color="Maroon">r</font> holds <br/> <a href="metric_1.html#K4">dist</a> <font color="Olive">t1</font>,<font color="Olive">z</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">r</font>
;<br/>

<b>take </b>
<font color="Maroon">t1</font>
;<br/>

<b>thus </b><a NAME="E1:26_1"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E2:26_1"/>
<font color="Maroon">t1</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">t1</font>,<font color="Maroon">r</font>
 <b>by </b><i>, </i>;<br/>

<b>let </b><font color="Maroon">z</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font>;<br/>

<b>assume </b><a NAME="E3:26_1"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">t1</font>,<font color="Maroon">r</font>
 ;<br/>

<b>hence </b><a NAME="E4:26_1"/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">t1</font>,<font color="Maroon">z</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:26"/>
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">t1</font>,<font color="Maroon">r</font> is <a href="tbsp_1.html#V6" target="_self">bounded</a>
 <b>by </b><i/>;<br/>


</div>
