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<b>let </b><font color="Maroon">X</font> be  non <a href="struct_0.html#V3">empty</a> <a href="metric_1.html#NM1">MetrSpace</a>;<br/><b>let </b><font color="Maroon">Y</font> be  non <a href="struct_0.html#V3">empty</a> <a href="metric_1.html#NM1">MetrSpace</a>;<br/><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font></span><a href="zfmisc_1.html#K2">:]</a></span>, <font color="Maroon">y</font> be    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font></span><a href="zfmisc_1.html#K2">:]</a></span>, <font color="Maroon">z</font> be    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/>





<b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x</font> <a href="mcart_1.html#K1">`1</a> , <font color="Maroon">y1</font> = <font color="Maroon">y</font> <a href="mcart_1.html#K1">`1</a> , <font color="Maroon">z1</font> = <font color="Maroon">z</font> <a href="mcart_1.html#K1">`1</a>  as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="mcart_1.html#T10">MCART_1:10</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x2</font> = <font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a> , <font color="Maroon">y2</font> = <font color="Maroon">y</font> <a href="mcart_1.html#K2">`2</a> , <font color="Maroon">z2</font> = <font color="Maroon">z</font> <a href="mcart_1.html#K2">`2</a>  as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font> <b>by </b><i><a class="ref" href="mcart_1.html#T10">MCART_1:10</a></i>;<br/>
<a NAME="E3:5"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">y1</font>,<font color="Maroon">y2</font></span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">z1</font>,<font color="Maroon">z2</font></span><a href="tarski.html#K4">]</a></span> )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T24">MCART_1:24</a></i>;<br/>
<b>set </b><font color="Maroon">d1</font> =  <a href="metric_1.html#K4">dist</a> <font color="Maroon">x1</font>,<font color="Maroon">z1</font>;<br/>
<b>set </b><font color="Maroon">d2</font> =  <a href="metric_1.html#K4">dist</a> <font color="Maroon">x1</font>,<font color="Maroon">y1</font>;<br/>
<b>set </b><font color="Maroon">d3</font> =  <a href="metric_1.html#K4">dist</a> <font color="Maroon">y1</font>,<font color="Maroon">z1</font>;<br/>
<b>set </b><font color="Maroon">d4</font> =  <a href="metric_1.html#K4">dist</a> <font color="Maroon">x2</font>,<font color="Maroon">z2</font>;<br/>
<b>set </b><font color="Maroon">d5</font> =  <a href="metric_1.html#K4">dist</a> <font color="Maroon">x2</font>,<font color="Maroon">y2</font>;<br/>
<b>set </b><font color="Maroon">d6</font> =  <a href="metric_1.html#K4">dist</a> <font color="Maroon">y2</font>,<font color="Maroon">z2</font>;<br/>
<i><font color="Green">E29</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x1</font>,<font color="Maroon">y1</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">y1</font>,<font color="Maroon">z1</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x2</font>,<font color="Maroon">y2</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">y2</font>,<font color="Maroon">z2</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x1</font>,<font color="Maroon">y1</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x2</font>,<font color="Maroon">y2</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">y1</font>,<font color="Maroon">z1</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">y2</font>,<font color="Maroon">z2</font></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_3.html#K1" target="_self">dist_cart2</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">y1</font>,<font color="Maroon">z1</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">y2</font>,<font color="Maroon">z2</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_3.html#K1" target="_self">dist_cart2</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_3.html#K1" target="_self">dist_cart2</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span>
<b>by </b><i>, </i>
;<br/>
<a NAME="E5:5"/><i><font color="Green">E33</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">x1</font>,<font color="Maroon">z1</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">x1</font>,<font color="Maroon">y1</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">y1</font>,<font color="Maroon">z1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="metric_1.html#T4">METRIC_1:4</a></i>;<br/>
<a NAME="E6:5"/><i><font color="Green">E34</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">x2</font>,<font color="Maroon">z2</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">x2</font>,<font color="Maroon">y2</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">y2</font>,<font color="Maroon">z2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="metric_1.html#T4">METRIC_1:4</a></i>;<br/>
<a NAME="E7:5"/>
<span class="p1">(<span class="default"><a href="metric_3.html#K1" target="_self">dist_cart2</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x1</font>,<font color="Maroon">z1</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">x2</font>,<font color="Maroon">z2</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E8:5"/>
<span class="p1">(<span class="default"><a href="metric_3.html#K1" target="_self">dist_cart2</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">z</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_3.html#K1" target="_self">dist_cart2</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_3.html#K1" target="_self">dist_cart2</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>


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