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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">  for <font color="Olive" title="b1">V</font>, <font color="Olive" title="b2">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> <br/>  for <font color="Olive" title="b3">p1</font>, <font color="Olive" title="b4">p2</font>, <font color="Olive" title="b5">q1</font>, <font color="Olive" title="b6">q2</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="b3">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">V</font> &amp; <font color="Olive" title="b5">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">Q</font> &amp; <font color="Olive" title="b4">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">V</font> &amp; <font color="Olive" title="b6">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">Q</font> holds <br/> <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Olive" title="b3">p1</font>,<font color="Olive" title="b5">q1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Olive" title="b4">p2</font>,<font color="Olive" title="b6">q2</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c2">V</font>, <font color="Maroon" title="c3">Q</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">  for <font color="Olive" title="b1">p1</font>, <font color="Olive" title="b2">p2</font>, <font color="Olive" title="b3">q1</font>, <font color="Olive" title="b4">q2</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">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V</font> &amp; <font color="Olive" title="b3">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font> &amp; <font color="Olive" title="b2">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V</font> &amp; <font color="Olive" title="b4">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font> holds <br/> <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Olive" title="b1">p1</font>,<font color="Olive" title="b3">q1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Olive" title="b2">p2</font>,<font color="Olive" title="b4">q2</font>)</span><br/><span class="kw">let </span><font color="Maroon" title="c4">p1</font>, <font color="Maroon" title="c5">p2</font>, <font color="Maroon" title="c6">q1</font>, <font color="Maroon" title="c7">q2</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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V</font> &amp; <font color="Maroon" title="c6">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font> &amp; <font color="Maroon" title="c5">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V</font> &amp; <font color="Maroon" title="c7">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font> implies  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c6">q1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c5">p2</font>,<font color="Maroon" title="c7">q2</font>) )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:28"/><span class="lab"><font color="Green" title="E17">A1</font></span>: 
<font color="Maroon" title="c4">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V</font>
 <span class="kw">and </span><br/><a NAME="E2:28"/><span class="lab"><font color="Green" title="E18">A2</font></span>: 
<font color="Maroon" title="c6">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font>
 <span class="kw">and </span><br/><a NAME="E3:28"/><span class="lab"><font color="Green" title="E19">A3</font></span>: 
<font color="Maroon" title="c5">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">V</font>
 <span class="kw">and </span><br/><a NAME="E4:28"/><span class="lab"><font color="Green" title="E20">A4</font></span>: 
<font color="Maroon" title="c7">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">Q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c6">q1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c5">p2</font>,<font color="Maroon" title="c7">q2</font>)</span><br/>

<a NAME="E5:28"/>
<font color="Maroon" title="c2">V</font> is    <a href="metric_2.html#M1" title="METRIC_2:mode.1">equivalence_class</a> of <font color="Maroon" title="c1">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="metric_2.html#T17" target="_self" title="METRIC_2:th.17">Th26</a></span>;<br/>
<a NAME="E6:28"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">x</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="Maroon" title="c2">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="metric_2.html#K1" title="METRIC_2:func.1">-neighbour</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="metric_2.html#D3" target="_self" title="METRIC_2:def.3">Def3</a></span>;<br/>
<a NAME="E7:28"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A5</font></span>: 
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="txt" href="#E3:28"><span class="lab"><font color="Green" title="E19">A3</font></span></a>, <a class="ref" href="metric_2.html#T10" target="_self" title="METRIC_2:th.10">Th17</a></span>;<br/>
<a NAME="E8:28"/>
<font color="Maroon" title="c3">Q</font> is    <a href="metric_2.html#M1" title="METRIC_2:mode.1">equivalence_class</a> of <font color="Maroon" title="c1">M</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="metric_2.html#T17" target="_self" title="METRIC_2:th.17">Th26</a></span>;<br/>
<a NAME="E9:28"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">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="Maroon" title="c3">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">y</font> <a href="metric_2.html#K1" title="METRIC_2:func.1">-neighbour</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="metric_2.html#D3" target="_self" title="METRIC_2:def.3">Def3</a></span>;<br/>
<a NAME="E10:28"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A6</font></span>: 
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c6">q1</font>,<font color="Maroon" title="c7">q2</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:28"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="txt" href="#E4:28"><span class="lab"><font color="Green" title="E20">A4</font></span></a>, <a class="ref" href="metric_2.html#T10" target="_self" title="METRIC_2:th.10">Th17</a></span>;<br/>
<a NAME="E11:28"/>
(  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c5">p2</font>,<font color="Maroon" title="c7">q2</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c5">p2</font>,<font color="Maroon" title="c4">p1</font>)</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c7">q2</font>)</span>)</span> &amp;  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c7">q2</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c6">q1</font>)</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c6">q1</font>,<font color="Maroon" title="c7">q2</font>)</span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#T4" title="METRIC_1:th.4">METRIC_1:4</a></span>;<br/>
<a NAME="E12:28"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A7</font></span>: 
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c5">p2</font>,<font color="Maroon" title="c7">q2</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c6">q1</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:28"><span class="lab"><font color="Green" title="E21">A5</font></span></a>, <a class="txt" href="#E10:28"><span class="lab"><font color="Green" title="E22">A6</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E13:28"/>
(  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c6">q1</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font>)</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c5">p2</font>,<font color="Maroon" title="c6">q1</font>)</span>)</span> &amp;  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c5">p2</font>,<font color="Maroon" title="c6">q1</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c5">p2</font>,<font color="Maroon" title="c7">q2</font>)</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c7">q2</font>,<font color="Maroon" title="c6">q1</font>)</span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#T4" title="METRIC_1:th.4">METRIC_1:4</a></span>;<br/>
<a NAME="E14:28"/><span class="kw">then </span>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c6">q1</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c5">p2</font>,<font color="Maroon" title="c7">q2</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:28"><span class="lab"><font color="Green" title="E21">A5</font></span></a>, <a class="txt" href="#E10:28"><span class="lab"><font color="Green" title="E22">A6</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:28"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c6">q1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c5">p2</font>,<font color="Maroon" title="c7">q2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:28"><span class="lab"><font color="Green" title="E23">A7</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</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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