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<b>let </b><font color="Maroon" title="c2">F1</font>, <font color="Maroon" title="c3">F2</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</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></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:50_1_2"/><i><font color="Green" title="E24">A4</font></i>: 
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">p</font>, <font color="Olive" title="b4">q</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">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">V</font> &amp; <font color="Olive" title="b4">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">Q</font> holds <br/><font color="Maroon" title="c2">F1</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Olive" title="b1">V</font>,<font color="Olive" title="b2">Q</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="b3">p</font>,<font color="Olive" title="b4">q</font>
 <b>and </b><br/><a NAME="E2:50_1_2"/><i><font color="Green" title="E25">A5</font></i>: 
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">p</font>, <font color="Olive" title="b4">q</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">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">V</font> &amp; <font color="Olive" title="b4">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">Q</font> holds <br/><font color="Maroon" title="c3">F2</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Olive" title="b1">V</font>,<font color="Olive" title="b2">Q</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="b3">p</font>,<font color="Olive" title="b4">q</font>
 ;<br/>

<a NAME="E3:50_1_2"/>
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>  holds  <font color="Maroon" title="c2">F1</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Olive" title="b1">V</font>,<font color="Olive" title="b2">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F2</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Olive" title="b1">V</font>,<font color="Olive" title="b2">Q</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">V</font>, <font color="Maroon" title="c5">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> ;<br/>

<b>consider </b><font color="Maroon" title="c6">p</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">M</font><b> such that </b><br/><a NAME="E2:50_1_2_1"/><i><font color="Green" title="E26">A6</font></i>: 
<font color="Maroon" title="c4">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p</font> <a href="metric_2.html#K1" title="METRIC_2:func.1">-neighbour</a> 
 <b>by </b><i><a class="ref" href="metric_2.html#T23" target="_self" title="METRIC_2:th.23">Th23</a></i>;<br/>
<a NAME="E3:50_1_2_1"/><i><font color="Green" title="E27">A7</font></i>: 
<font color="Maroon" title="c6">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">V</font>
 
<b>by </b><i><a class="txt" href="metric_2.html#E2:50_1_2_1"><i><font color="Green" title="E26">A6</font></i></a>, <a class="ref" href="metric_2.html#T9" target="_self" title="METRIC_2:th.9">Th9</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">q</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">M</font><b> such that </b><br/><a NAME="E5:50_1_2_1"/><i><font color="Green" title="E28">A8</font></i>: 
<font color="Maroon" title="c5">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">q</font> <a href="metric_2.html#K1" title="METRIC_2:func.1">-neighbour</a> 
 <b>by </b><i><a class="ref" href="metric_2.html#T23" target="_self" title="METRIC_2:th.23">Th23</a></i>;<br/>
<a NAME="E6:50_1_2_1"/><i><font color="Green" title="E29">A9</font></i>: 
<font color="Maroon" title="c7">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Q</font>
 
<b>by </b><i><a class="txt" href="metric_2.html#E5:50_1_2_1"><i><font color="Green" title="E28">A8</font></i></a>, <a class="ref" href="metric_2.html#T9" target="_self" title="METRIC_2:th.9">Th9</a></i>;<br/>
<b>then </b><font color="Maroon" title="c2">F1</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c4">V</font>,<font color="Maroon" title="c5">Q</font> = 
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c6">p</font>,<font color="Maroon" title="c7">q</font>
<b>by </b><i><a class="txt" href="metric_2.html#E1:50_1_2"><i><font color="Green" title="E24">A4</font></i></a>, <a class="txt" href="metric_2.html#E3:50_1_2_1"><i><font color="Green" title="E27">A7</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c3">F2</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c4">V</font>,<font color="Maroon" title="c5">Q</font>
<b>by </b><i><a class="txt" href="metric_2.html#E2:50_1_2"><i><font color="Green" title="E25">A5</font></i></a>, <a class="txt" href="metric_2.html#E3:50_1_2_1"><i><font color="Green" title="E27">A7</font></i></a>, <a class="txt" href="metric_2.html#E6:50_1_2_1"><i><font color="Green" title="E29">A9</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E8:50_1_2_1"/>
<font color="Maroon" title="c2">F1</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c4">V</font>,<font color="Maroon" title="c5">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F2</font> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c4">V</font>,<font color="Maroon" title="c5">Q</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:50_1_2"/>
<font color="Maroon" title="c2">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F2</font>
 <b>by </b><i><a class="ref" href="binop_1.html#T2" title="BINOP_1:th.2">BINOP_1:2</a></i>;<br/>


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