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

<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="E1:63"/>
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="63_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<span class="kw">assume </span><a NAME="E1:63_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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<span class="kw">then </span><span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E3:63_1"/><span class="lab"><font color="Green" title="E23">A1</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E4:63_1"/><span class="lab"><font color="Green" title="E24">A2</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="metric_2.html#T46" target="_self" title="METRIC_2:th.46">Th46</a></span>;<br/>
<a NAME="E5:63_1"/><span class="lab"><font color="Green" title="E25">A3</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:63_1"><span class="lab"><font color="Green" title="E24">A2</font></span></a></span>;<br/>
<a NAME="E6:63_1"/>
( <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> &amp; <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> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:63_1"><span class="lab"><font color="Green" title="E24">A2</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:63_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:63_1"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="txt" href="#E5:63_1"><span class="lab"><font color="Green" title="E25">A3</font></span></a>, <a class="ref" href="mcart_1.html#T73" title="MCART_1:th.73">MCART_1:73</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:63"/>
 <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 class="kw">by</span> <span class="lab"><a class="ref" href="subset_1.html#T7" title="SUBSET_1:th.7">SUBSET_1:7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
