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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="sub_metr.html#NM1">PseudoMetricSpace</a>;<br/>

<a NAME="E1:46"/><i><font color="Green">E23</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <a href="metric_2.html#K2" target="_self">-neighbour</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="metric_2.html#K6" target="_self">elem_in_rel_1</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="metric_2.html#K7" target="_self">elem_in_rel_2</a> <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <a href="metric_2.html#K2" target="_self">-neighbour</a> ;<br/>

<b>assume </b><a NAME="E1:46_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="metric_2.html#K6" target="_self">elem_in_rel_1</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>3</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <a href="metric_2.html#K2" target="_self">-neighbour</a> , <font color="Maroon">c<sub>4</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E3:46_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="metric_2.html#R4" target="_self">is_dst</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="metric_2.html#T40" target="_self">Th40</a></i>;<br/>
<a NAME="E4:46_1"/>
<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="metric_2.html#R4" target="_self">is_dst</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="metric_2.html#E3:46_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="metric_2.html#T32" target="_self">Th32</a></i>;<br/>
<b>hence </b><a NAME="E5:46_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="metric_2.html#K7" target="_self">elem_in_rel_2</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:46"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <a href="metric_2.html#K2" target="_self">-neighbour</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="metric_2.html#K7" target="_self">elem_in_rel_2</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="metric_2.html#K6" target="_self">elem_in_rel_1</a> <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <a href="metric_2.html#K2" target="_self">-neighbour</a> ;<br/>

<b>assume </b><a NAME="E1:46_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="metric_2.html#K7" target="_self">elem_in_rel_2</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>3</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <a href="metric_2.html#K2" target="_self">-neighbour</a> , <font color="Maroon">c<sub>4</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E3:46_2"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="metric_2.html#R4" target="_self">is_dst</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="metric_2.html#T42" target="_self">Th42</a></i>;<br/>
<a NAME="E4:46_2"/>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="metric_2.html#R4" target="_self">is_dst</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="metric_2.html#E3:46_2"><i><font color="Green">E24</font></i></a>, <a class="ref" href="metric_2.html#T32" target="_self">Th32</a></i>;<br/>
<b>hence </b><a NAME="E5:46_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="metric_2.html#K6" target="_self">elem_in_rel_1</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:46"/><b>then </b>
(  <a href="metric_2.html#K6" target="_self">elem_in_rel_1</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="metric_2.html#K7" target="_self">elem_in_rel_2</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="metric_2.html#K7" target="_self">elem_in_rel_2</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="metric_2.html#K6" target="_self">elem_in_rel_1</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="metric_2.html#E1:46"><i><font color="Green">E23</font></i></a>, <a class="ref" href="subset_1.html#T7">SUBSET_1:7</a></i>;<br/>
<b>hence </b><a NAME="E4:46"/>
 <a href="metric_2.html#K6" target="_self">elem_in_rel_1</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="metric_2.html#K7" target="_self">elem_in_rel_2</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div>
