<?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">  <a href="metric_2.html#K9" title="METRIC_2:func.9">set_in_rel</a> <font color="Maroon" title="c1">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="mcart_1.html#K13" title="MCART_1:func.13">[:</a><span class="default"><span class="p2">(<span class="default"><a href="metric_2.html#K6" title="METRIC_2:func.6">elem_in_rel_1</a> <font color="Maroon" title="c1">M</font></span>)</span>,<span class="p2">(<span class="default"><a href="metric_2.html#K7" title="METRIC_2:func.7">elem_in_rel_2</a> <font color="Maroon" title="c1">M</font></span>)</span>,<span class="p2">(<span class="default"><a href="metric_2.html#K5" title="METRIC_2:func.5">real_in_rel</a> <font color="Maroon" title="c1">M</font></span>)</span></span><a href="mcart_1.html#K13" title="MCART_1:func.13">:]</a></span></span><br/>

<a NAME="E1:47"/>
for <font color="Olive" title="b1">VQv</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="metric_2.html#K2" title="METRIC_2:func.2">-neighbour</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="metric_2.html#K2" title="METRIC_2:func.2">-neighbour</a> </span>)</span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span>  st <font color="Olive" title="b1">VQv</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_2.html#K9" title="METRIC_2:func.9">set_in_rel</a> <font color="Maroon" title="c1">M</font> holds <br/><font color="Olive" title="b1">VQv</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="mcart_1.html#K13" title="MCART_1:func.13">[:</a><span class="default"><span class="p2">(<span class="default"><a href="metric_2.html#K6" title="METRIC_2:func.6">elem_in_rel_1</a> <font color="Maroon" title="c1">M</font></span>)</span>,<span class="p2">(<span class="default"><a href="metric_2.html#K7" title="METRIC_2:func.7">elem_in_rel_2</a> <font color="Maroon" title="c1">M</font></span>)</span>,<span class="p2">(<span class="default"><a href="metric_2.html#K5" title="METRIC_2:func.5">real_in_rel</a> <font color="Maroon" title="c1">M</font></span>)</span></span><a href="mcart_1.html#K13" title="MCART_1:func.13">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">VQv</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="metric_2.html#K2" title="METRIC_2:func.2">-neighbour</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">M</font> <a href="metric_2.html#K2" title="METRIC_2:func.2">-neighbour</a> </span>)</span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> </span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">VQv</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_2.html#K9" title="METRIC_2:func.9">set_in_rel</a> <font color="Maroon" title="c1">M</font> implies <font color="Maroon" title="c2">VQv</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="mcart_1.html#K13" title="MCART_1:func.13">[:</a><span class="default"><span class="p2">(<span class="default"><a href="metric_2.html#K6" title="METRIC_2:func.6">elem_in_rel_1</a> <font color="Maroon" title="c1">M</font></span>)</span>,<span class="p2">(<span class="default"><a href="metric_2.html#K7" title="METRIC_2:func.7">elem_in_rel_2</a> <font color="Maroon" title="c1">M</font></span>)</span>,<span class="p2">(<span class="default"><a href="metric_2.html#K5" title="METRIC_2:func.5">real_in_rel</a> <font color="Maroon" title="c1">M</font></span>)</span></span><a href="mcart_1.html#K13" title="MCART_1:func.13">:]</a></span> )</span><br/>

<b>assume </b><a NAME="E1:47_1"/>
<font color="Maroon" title="c2">VQv</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_2.html#K9" title="METRIC_2:func.9">set_in_rel</a> <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="c2">VQv</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="mcart_1.html#K13" title="MCART_1:func.13">[:</a><span class="default"><span class="p2">(<span class="default"><a href="metric_2.html#K6" title="METRIC_2:func.6">elem_in_rel_1</a> <font color="Maroon" title="c1">M</font></span>)</span>,<span class="p2">(<span class="default"><a href="metric_2.html#K7" title="METRIC_2:func.7">elem_in_rel_2</a> <font color="Maroon" title="c1">M</font></span>)</span>,<span class="p2">(<span class="default"><a href="metric_2.html#K5" title="METRIC_2:func.5">real_in_rel</a> <font color="Maroon" title="c1">M</font></span>)</span></span><a href="mcart_1.html#K13" title="MCART_1:func.13">:]</a></span></span><br/>

<b>then consider </b><font color="Maroon" title="c3">V</font>, <font color="Maroon" title="c4">Q</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font> <a href="metric_2.html#K2" title="METRIC_2:func.2">-neighbour</a> , <font color="Maroon" title="c5">v</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E3:47_1"/><i><font color="Green" title="E23">A1</font></i>: 
( <font color="Maroon" title="c2">VQv</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Maroon" title="c3">V</font>,<font color="Maroon" title="c4">Q</font>,<font color="Maroon" title="c5">v</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> &amp; <font color="Maroon" title="c3">V</font>,<font color="Maroon" title="c4">Q</font> <a href="metric_2.html#R4" title="METRIC_2:pred.4">is_dst</a> <font color="Maroon" title="c5">v</font> )
 <b>by </b><i><a class="ref" href="metric_2.html#T46" target="_self" title="METRIC_2:th.46">Th46</a></i>;<br/>
<a NAME="E4:47_1"/><i><font color="Green" title="E24">A2</font></i>: 
<font color="Maroon" title="c3">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_2.html#K6" title="METRIC_2:func.6">elem_in_rel_1</a> <font color="Maroon" title="c1">M</font>
 
<b>by </b><i><a class="txt" href="#E3:47_1"><i><font color="Green" title="E23">A1</font></i></a></i>;<br/>
<a NAME="E5:47_1"/><i><font color="Green" title="E25">A3</font></i>: 
<font color="Maroon" title="c4">Q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_2.html#K7" title="METRIC_2:func.7">elem_in_rel_2</a> <font color="Maroon" title="c1">M</font>
 
<b>by </b><i><a class="txt" href="#E3:47_1"><i><font color="Green" title="E23">A1</font></i></a></i>;<br/>
<a NAME="E6:47_1"/>
<font color="Maroon" title="c5">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_2.html#K5" title="METRIC_2:func.5">real_in_rel</a> <font color="Maroon" title="c1">M</font>
 
<b>by </b><i><a class="txt" href="#E3:47_1"><i><font color="Green" title="E23">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:47_1"/>
<font color="Maroon" title="c2">VQv</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="mcart_1.html#K13" title="MCART_1:func.13">[:</a><span class="default"><span class="p2">(<span class="default"><a href="metric_2.html#K6" title="METRIC_2:func.6">elem_in_rel_1</a> <font color="Maroon" title="c1">M</font></span>)</span>,<span class="p2">(<span class="default"><a href="metric_2.html#K7" title="METRIC_2:func.7">elem_in_rel_2</a> <font color="Maroon" title="c1">M</font></span>)</span>,<span class="p2">(<span class="default"><a href="metric_2.html#K5" title="METRIC_2:func.5">real_in_rel</a> <font color="Maroon" title="c1">M</font></span>)</span></span><a href="mcart_1.html#K13" title="MCART_1:func.13">:]</a></span>
 <b>by </b><i><a class="txt" href="#E3:47_1"><i><font color="Green" title="E23">A1</font></i></a>, <a class="txt" href="#E4:47_1"><i><font color="Green" title="E24">A2</font></i></a>, <a class="txt" href="#E5:47_1"><i><font color="Green" title="E25">A3</font></i></a>, <a class="ref" href="mcart_1.html#T73" title="MCART_1:th.73">MCART_1:73</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><b>end;</b></div>
<b>hence </b><a NAME="E2:47"/>
 <a href="metric_2.html#K9" title="METRIC_2:func.9">set_in_rel</a> <font color="Maroon" title="c1">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="mcart_1.html#K13" title="MCART_1:func.13">[:</a><span class="default"><span class="p2">(<span class="default"><a href="metric_2.html#K6" title="METRIC_2:func.6">elem_in_rel_1</a> <font color="Maroon" title="c1">M</font></span>)</span>,<span class="p2">(<span class="default"><a href="metric_2.html#K7" title="METRIC_2:func.7">elem_in_rel_2</a> <font color="Maroon" title="c1">M</font></span>)</span>,<span class="p2">(<span class="default"><a href="metric_2.html#K5" title="METRIC_2:func.5">real_in_rel</a> <font color="Maroon" title="c1">M</font></span>)</span></span><a href="mcart_1.html#K13" title="MCART_1:func.13">:]</a></span>
 <b>by </b><i><a class="ref" href="subset_1.html#T7" title="SUBSET_1:th.7">SUBSET_1:7</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>
