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<b>let </b><font color="Maroon">F1</font> be   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="metric_2.html#K2" target="_self">-neighbour</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="metric_2.html#K2" target="_self">-neighbour</a> </span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>, <a href="numbers.html#K1">REAL</a> , <font color="Maroon">F2</font> be   <a href="funct_2.html#NM1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="metric_2.html#K2" target="_self">-neighbour</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">M</font> <a href="metric_2.html#K2" target="_self">-neighbour</a> </span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>, <a href="numbers.html#K1">REAL</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:50_1_2"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">V</font>, <font color="Olive">Q</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">M</font> <a href="metric_2.html#K2" target="_self">-neighbour</a> <br/> for <font color="Olive">p</font>, <font color="Olive">q</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font>  st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Q</font> holds <br/><font color="Maroon">F1</font> <a href="metric_1.html#K1">.</a> <font color="Olive">V</font>,<font color="Olive">Q</font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K4">dist</a> <font color="Olive">p</font>,<font color="Olive">q</font>
 <b>and </b><br/><a NAME="E2:50_1_2"/><i><font color="Green">E54</font></i>: 
for <font color="Olive">V</font>, <font color="Olive">Q</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">M</font> <a href="metric_2.html#K2" target="_self">-neighbour</a> <br/> for <font color="Olive">p</font>, <font color="Olive">q</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font>  st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Olive">Q</font> holds <br/><font color="Maroon">F2</font> <a href="metric_1.html#K1">.</a> <font color="Olive">V</font>,<font color="Olive">Q</font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K4">dist</a> <font color="Olive">p</font>,<font color="Olive">q</font>
 ;<br/>

<a NAME="E3:50_1_2"/>
for <font color="Olive">V</font>, <font color="Olive">Q</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">M</font> <a href="metric_2.html#K2" target="_self">-neighbour</a>  holds  <font color="Maroon">F1</font> <a href="metric_1.html#K1">.</a> <font color="Olive">V</font>,<font color="Olive">Q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F2</font> <a href="metric_1.html#K1">.</a> <font color="Olive">V</font>,<font color="Olive">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">V</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">M</font> <a href="metric_2.html#K2" target="_self">-neighbour</a> , <font color="Maroon">Q</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">M</font> <a href="metric_2.html#K2" target="_self">-neighbour</a> ;<br/>

<b>consider </b><font color="Maroon">p</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font><b> such that </b><br/><a NAME="E2:50_1_2_1"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">V</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="metric_2.html#K1" target="_self">-neighbour</a> 
 <b>by </b><i/>;<br/>
<a NAME="E3:50_1_2_1"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 
<b>by </b><i><a class="txt" href="metric_2.html#E4"><i><font color="Green">Lemma26</font></i></a>, </i>;<br/>
<b>consider </b><font color="Maroon">q</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font><b> such that </b><br/><a NAME="E5:50_1_2_1"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">Q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="metric_2.html#K1" target="_self">-neighbour</a> 
 <b>by </b><i/>;<br/>
<a NAME="E6:50_1_2_1"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 
<b>by </b><i>, </i>;<br/>
<b>then </b><font color="Maroon">F1</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">V</font>,<font color="Maroon">Q</font> = 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>
<b>by </b><i><a class="ref" href="metric_2.html#D3" target="_self">Def3</a>, <a class="ref" href="metric_2.html#T6" target="_self">Th6</a></i>
<br/>.= 
<font color="Maroon">F2</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">V</font>,<font color="Maroon">Q</font>
<b>by </b><i>, <a class="ref" href="metric_2.html#T6" target="_self">Th6</a>, </i>
;<br/>
<b>hence </b><a NAME="E8:50_1_2_1"/>
<font color="Maroon">F1</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">V</font>,<font color="Maroon">Q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F2</font> <a href="metric_1.html#K1">.</a> <font color="Maroon">V</font>,<font color="Maroon">Q</font>
 ;<br/>


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


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