<?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/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:12"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="metric_2.html#K1" target="_self">-neighbour</a> 
 ;<br/>

<a NAME="E2:12"/><i><font color="Green">E9</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="metric_2.html#K1" target="_self">-neighbour</a>  holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="metric_2.html#K1" target="_self">-neighbour</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E1:12_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="metric_2.html#K1" target="_self">-neighbour</a> 
 ;<br/>

<a NAME="E2:12_1"/><b>then </b>
( <font color="Maroon">c<sub>4</sub></font> <a href="metric_2.html#R3" target="_self">tolerates</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="metric_2.html#R3" target="_self">tolerates</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="metric_2.html#E1:12"><i><font color="Green">E8</font></i></a>, <a class="ref" href="metric_2.html#T7" target="_self">Th7</a></i>;<br/>
<a NAME="E3:12_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="metric_2.html#R3" target="_self">tolerates</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="metric_2.html#T6" target="_self">Th6</a></i>;<br/>
<b>hence </b><a NAME="E4:12_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="metric_2.html#K1" target="_self">-neighbour</a> 
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:12"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="metric_2.html#K1" target="_self">-neighbour</a>  holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="metric_2.html#K1" target="_self">-neighbour</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E1:12_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> <a href="metric_2.html#K1" target="_self">-neighbour</a> 
 ;<br/>

<a NAME="E2:12_2"/><b>then </b>
( <font color="Maroon">c<sub>4</sub></font> <a href="metric_2.html#R3" target="_self">tolerates</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="metric_2.html#R3" target="_self">tolerates</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="metric_2.html#E1:12"><i><font color="Green">E8</font></i></a>, <a class="ref" href="metric_2.html#T7" target="_self">Th7</a></i>;<br/>
<a NAME="E3:12_2"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="metric_2.html#R3" target="_self">tolerates</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="metric_2.html#T6" target="_self">Th6</a></i>;<br/>
<b>hence </b><a NAME="E4:12_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> <a href="metric_2.html#K1" target="_self">-neighbour</a> 
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:12"/><b>then </b>
( <font color="Maroon">c<sub>2</sub></font> <a href="metric_2.html#K1" target="_self">-neighbour</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> <a href="metric_2.html#K1" target="_self">-neighbour</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="metric_2.html#K1" target="_self">-neighbour</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> <a href="metric_2.html#K1" target="_self">-neighbour</a>  )
 
<b>by </b><i><a class="txt" href="metric_2.html#E2:12"><i><font color="Green">E9</font></i></a>, <a class="ref" href="subset_1.html#T7">SUBSET_1:7</a></i>;<br/>
<b>hence </b><a NAME="E5:12"/>
<font color="Maroon">c<sub>2</sub></font> <a href="metric_2.html#K1" target="_self">-neighbour</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="metric_2.html#K1" target="_self">-neighbour</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div>
