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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">a</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">b</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> , <font color="Maroon">c</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</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#K14">Euclid</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#K14">Euclid</a> <font color="Maroon">n</font></span>)</span>, <font color="Maroon">z</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="E1:51"/><i><font color="Green">E43</font></i>: 
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">x</font>,<font color="Maroon">a</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">z</font>,<font color="Maroon">c</font>
 <b>and </b><br/><a NAME="E2:51"/><i><font color="Green">E49</font></i>: 
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">z</font>,<font color="Maroon">c</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">y</font>,<font color="Maroon">b</font>
 ;<br/>

<a NAME="E3:51"/><i><font color="Green">E54</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">z</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:51"/><i><font color="Green">E56</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">z</font>,<font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">b</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:51"/><i><font color="Green">E57</font></i>: 
<span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">z</font>,<font color="Maroon">y</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">z</font>,<font color="Maroon">y</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E6:51"/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">z</font>,<font color="Maroon">y</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">b</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<a NAME="E7:51"/><b>then </b><i><font color="Green">E72</font></i>: 
<span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">z</font>,<font color="Maroon">y</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">b</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E8:51"/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">z</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">z</font>,<font color="Maroon">y</font></span>)</span>
 
<b>by </b><i><a class="ref" href="metric_1.html#T4">METRIC_1:4</a></i>;<br/>
<b>hence </b><a NAME="E9:51"/>
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">b</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default">2 <a href="xcmplx_0.html#K3">*</a> <font color="Maroon">c</font></span>)</span>
 <b>by </b><i><a class="ref" href="kurato_2.html#T1" target="_self">Th1</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


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