<?xml version="1.0"?>
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<span class="kw">let </span><font color="Maroon" title="c1">M</font> be   <a href="metric_1.html#NM2" title="METRIC_1:NM.2">PseudoMetricSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="metric_2.html#K6" title="METRIC_2:func.6">elem_in_rel_1</a> <font color="Maroon" title="c1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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><br/>

<a NAME="E1:46"/>
 for <font color="Olive" title="b1">V</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>   st <font color="Olive" title="b1">V</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> holds <br/><font color="Olive" title="b1">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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">V</font> be    <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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">V</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> implies <font color="Maroon" title="c2">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> )</span><br/>

<span class="kw">assume </span><a NAME="E1:46_1"/>
<font color="Maroon" title="c2">V</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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">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></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">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="c4">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="E3:46_1"/><span class="lab"><font color="Green" title="E23">A1</font></span>: 
<font color="Maroon" title="c3">Q</font>,<font color="Maroon" title="c2">V</font> <a href="metric_2.html#R4" title="METRIC_2:pred.4">is_dst</a> <font color="Maroon" title="c4">v</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="metric_2.html#T27" target="_self" title="METRIC_2:th.27">Th42</a></span>;<br/>
<a NAME="E4:46_1"/>
<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">Q</font> <a href="metric_2.html#R4" title="METRIC_2:pred.4">is_dst</a> <font color="Maroon" title="c4">v</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:46_1"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="ref" href="metric_2.html#T22" target="_self" title="METRIC_2:th.22">Th32</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:46_1"/>
<font color="Maroon" title="c2">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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E2:46"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A2</font></span>: 
 <a href="metric_2.html#K7" title="METRIC_2:func.7">elem_in_rel_2</a> <font color="Maroon" title="c1">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="metric_2.html#K6" title="METRIC_2:func.6">elem_in_rel_1</a> <font color="Maroon" title="c1">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="subset_1.html#T2" title="SUBSET_1:th.2">SUBSET_1:2</a></span>;<br/>
<a NAME="E3:46"/>
 for <font color="Olive" title="b1">V</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>   st <font color="Olive" title="b1">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> holds <br/><font color="Olive" title="b1">V</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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">V</font> be    <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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">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> implies <font color="Maroon" title="c2">V</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><br/>

<span class="kw">assume </span><a NAME="E1:46_2"/>
<font color="Maroon" title="c2">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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">V</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><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">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="c4">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="E3:46_2"/><span class="lab"><font color="Green" title="E24">A3</font></span>: 
<font color="Maroon" title="c2">V</font>,<font color="Maroon" title="c3">Q</font> <a href="metric_2.html#R4" title="METRIC_2:pred.4">is_dst</a> <font color="Maroon" title="c4">v</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="metric_2.html#T26" target="_self" title="METRIC_2:th.26">Th40</a></span>;<br/>
<a NAME="E4:46_2"/>
<font color="Maroon" title="c3">Q</font>,<font color="Maroon" title="c2">V</font> <a href="metric_2.html#R4" title="METRIC_2:pred.4">is_dst</a> <font color="Maroon" title="c4">v</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:46_2"><span class="lab"><font color="Green" title="E24">A3</font></span></a>, <a class="ref" href="metric_2.html#T22" target="_self" title="METRIC_2:th.22">Th32</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:46_2"/>
<font color="Maroon" title="c2">V</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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E4:46"/><span class="kw">then </span>
 <a href="metric_2.html#K6" title="METRIC_2:func.6">elem_in_rel_1</a> <font color="Maroon" title="c1">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="ref" href="subset_1.html#T2" title="SUBSET_1:th.2">SUBSET_1:2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:46"/>
 <a href="metric_2.html#K6" title="METRIC_2:func.6">elem_in_rel_1</a> <font color="Maroon" title="c1">M</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="#E2:46"><span class="lab"><font color="Green" title="E23">A2</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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