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

<b>let </b><font color="Maroon" title="c1">M</font> be   <a href="sub_metr.html#NM1" title="SUB_METR:NM.1">PseudoMetricSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">M</font>  st <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">x</font> <a href="metric_2.html#K1" title="METRIC_2:func.1">-neighbour</a>  holds <br/><font color="Olive" title="b1">x</font> <a href="metric_2.html#K1" title="METRIC_2:func.1">-neighbour</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font> <a href="metric_2.html#K1" title="METRIC_2:func.1">-neighbour</a> </span><br/><b>let </b><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">M</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">x</font> <a href="metric_2.html#K1" title="METRIC_2:func.1">-neighbour</a>  implies <font color="Maroon" title="c2">x</font> <a href="metric_2.html#K1" title="METRIC_2:func.1">-neighbour</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y</font> <a href="metric_2.html#K1" title="METRIC_2:func.1">-neighbour</a>  )</span><br/>



<b>assume </b><a NAME="E1:12"/><i><font color="Green" title="E8">A1</font></i>: 
<font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">x</font> <a href="metric_2.html#K1" title="METRIC_2:func.1">-neighbour</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">x</font> <a href="metric_2.html#K1" title="METRIC_2:func.1">-neighbour</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y</font> <a href="metric_2.html#K1" title="METRIC_2:func.1">-neighbour</a> </span><br/>

<a NAME="E2:12"/><i><font color="Green" title="E9">A2</font></i>: 
for <font color="Olive" title="b1">p</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">M</font>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">x</font> <a href="metric_2.html#K1" title="METRIC_2:func.1">-neighbour</a>  holds <br/><font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">y</font> <a href="metric_2.html#K1" title="METRIC_2:func.1">-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" title="c4">p</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">M</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">x</font> <a href="metric_2.html#K1" title="METRIC_2:func.1">-neighbour</a>  implies <font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">y</font> <a href="metric_2.html#K1" title="METRIC_2:func.1">-neighbour</a>  )</span><br/>

<b>assume </b><a NAME="E1:12_1"/>
<font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">x</font> <a href="metric_2.html#K1" title="METRIC_2:func.1">-neighbour</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">y</font> <a href="metric_2.html#K1" title="METRIC_2:func.1">-neighbour</a> </span><br/>

<a NAME="E2:12_1"/><b>then </b>
( <font color="Maroon" title="c4">p</font> <a href="metric_2.html#R3" title="METRIC_2:pred.3">tolerates</a> <font color="Maroon" title="c2">x</font> &amp; <font color="Maroon" title="c2">x</font> <a href="metric_2.html#R3" title="METRIC_2:pred.3">tolerates</a> <font color="Maroon" title="c3">y</font> )
 
<b>by </b><i><a class="txt" href="metric_2.html#E1:12"><i><font color="Green" title="E8">A1</font></i></a>, <a class="ref" href="metric_2.html#T7" target="_self" title="METRIC_2:th.7">Th7</a></i>;<br/>
<a NAME="E3:12_1"/><b>then </b>
<font color="Maroon" title="c4">p</font> <a href="metric_2.html#R3" title="METRIC_2:pred.3">tolerates</a> <font color="Maroon" title="c3">y</font>
 
<b>by </b><i><a class="ref" href="metric_2.html#T6" target="_self" title="METRIC_2:th.6">Th6</a></i>;<br/>
<b>hence </b><a NAME="E4:12_1"/>
<font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">y</font> <a href="metric_2.html#K1" title="METRIC_2:func.1">-neighbour</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E3:12"/>
for <font color="Olive" title="b1">p</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">M</font>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">y</font> <a href="metric_2.html#K1" title="METRIC_2:func.1">-neighbour</a>  holds <br/><font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">x</font> <a href="metric_2.html#K1" title="METRIC_2:func.1">-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" title="c4">p</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">M</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">y</font> <a href="metric_2.html#K1" title="METRIC_2:func.1">-neighbour</a>  implies <font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">x</font> <a href="metric_2.html#K1" title="METRIC_2:func.1">-neighbour</a>  )</span><br/>

<b>assume </b><a NAME="E1:12_2"/>
<font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">y</font> <a href="metric_2.html#K1" title="METRIC_2:func.1">-neighbour</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">x</font> <a href="metric_2.html#K1" title="METRIC_2:func.1">-neighbour</a> </span><br/>

<a NAME="E2:12_2"/><b>then </b>
( <font color="Maroon" title="c4">p</font> <a href="metric_2.html#R3" title="METRIC_2:pred.3">tolerates</a> <font color="Maroon" title="c3">y</font> &amp; <font color="Maroon" title="c3">y</font> <a href="metric_2.html#R3" title="METRIC_2:pred.3">tolerates</a> <font color="Maroon" title="c2">x</font> )
 
<b>by </b><i><a class="txt" href="metric_2.html#E1:12"><i><font color="Green" title="E8">A1</font></i></a>, <a class="ref" href="metric_2.html#T7" target="_self" title="METRIC_2:th.7">Th7</a></i>;<br/>
<a NAME="E3:12_2"/><b>then </b>
<font color="Maroon" title="c4">p</font> <a href="metric_2.html#R3" title="METRIC_2:pred.3">tolerates</a> <font color="Maroon" title="c2">x</font>
 
<b>by </b><i><a class="ref" href="metric_2.html#T6" target="_self" title="METRIC_2:th.6">Th6</a></i>;<br/>
<b>hence </b><a NAME="E4:12_2"/>
<font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">x</font> <a href="metric_2.html#K1" title="METRIC_2:func.1">-neighbour</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E4:12"/><b>then </b>
( <font color="Maroon" title="c2">x</font> <a href="metric_2.html#K1" title="METRIC_2:func.1">-neighbour</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">y</font> <a href="metric_2.html#K1" title="METRIC_2:func.1">-neighbour</a>  &amp; <font color="Maroon" title="c3">y</font> <a href="metric_2.html#K1" title="METRIC_2:func.1">-neighbour</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">x</font> <a href="metric_2.html#K1" title="METRIC_2:func.1">-neighbour</a>  )
 
<b>by </b><i><a class="txt" href="metric_2.html#E2:12"><i><font color="Green" title="E9">A2</font></i></a>, <a class="ref" href="subset_1.html#T7" title="SUBSET_1:th.7">SUBSET_1:7</a></i>;<br/>
<b>hence </b><a NAME="E5:12"/>
<font color="Maroon" title="c2">x</font> <a href="metric_2.html#K1" title="METRIC_2:func.1">-neighbour</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y</font> <a href="metric_2.html#K1" title="METRIC_2:func.1">-neighbour</a> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
