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<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>, <font color="Maroon">y</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>;<br/><b>let </b><font color="Maroon">x'</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:48"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E2:48"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">y'</font> = <font color="Maroon">y</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> <font color="Maroon">n</font></span>)</span> <b>by </b><i><a class="ref" href="topreal3.html#T13">TOPREAL3:13</a></i>;<br/>
<b>reconsider </b><font color="Maroon">r</font> = <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x'</font>,<font color="Maroon">y'</font></span>)</span> <a href="real_1.html#K6">/</a> 2 as   <a href="real_1.html#NM1">Real</a> ;<br/>
<b>take </b>
<font color="Maroon">r</font>
;<br/>

<a NAME="E5:48"/><i><font color="Green">E54</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">x'</font>,<font color="Maroon">y'</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i>, , <a class="ref" href="metric_1.html#T2">METRIC_1:2</a></i>;<br/>
<a NAME="E6:48"/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">x'</font>,<font color="Maroon">y'</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="metric_1.html#T5">METRIC_1:5</a></i>;<br/>
<a NAME="E7:48"/><b>then </b>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">x'</font>,<font color="Maroon">y'</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">r</font>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T218">XREAL_1:218</a></i>;<br/>
<b>hence </b><a NAME="E8:48"/>
not <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x'</font>,<font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="metric_1.html#T12">METRIC_1:12</a></i>;<br/>


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