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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">p</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">q</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">r</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/>



<a NAME="E1:109"/>
(  ex <font color="Olive">a</font>, <font color="Olive">b</font> being  <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> st <br/>( <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> &amp;  <a href="metric_1.html#K4">dist</a> <font color="Olive">a</font>,<font color="Olive">b</font> <a href="hidden.html#R1">=</a>  <a href="topreal6.html#K1" target="_self">dist</a> <font color="Maroon">p</font>,<font color="Maroon">q</font> ) &amp;  ex <font color="Olive">a</font>, <font color="Olive">b</font> being  <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> st <br/>( <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> &amp;  <a href="metric_1.html#K4">dist</a> <font color="Olive">a</font>,<font color="Olive">b</font> <a href="hidden.html#R1">=</a>  <a href="topreal6.html#K1" target="_self">dist</a> <font color="Maroon">p</font>,<font color="Maroon">r</font> ) &amp;  ex <font color="Olive">a</font>, <font color="Olive">b</font> being  <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> st <br/>( <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> &amp;  <a href="metric_1.html#K4">dist</a> <font color="Olive">a</font>,<font color="Olive">b</font> <a href="hidden.html#R1">=</a>  <a href="topreal6.html#K1" target="_self">dist</a> <font color="Maroon">q</font>,<font color="Maroon">r</font> ) )
 
<b>by </b><i><a class="txt" href="topreal6.html#E1"><i><font color="Green">Lemma54</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:109"/>
 <a href="topreal6.html#K1" target="_self">dist</a> <font color="Maroon">p</font>,<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="topreal6.html#K1" target="_self">dist</a> <font color="Maroon">p</font>,<font color="Maroon">q</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="topreal6.html#K1" target="_self">dist</a> <font color="Maroon">q</font>,<font color="Maroon">r</font></span>)</span>
 <b>by </b><i><a class="ref" href="metric_1.html#T4">METRIC_1:4</a></i>;<br/>


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