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<div class="add">

<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">Z</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">W</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#K4">[:</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>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Z</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W</font></span><a href="zfmisc_1.html#K4">:]</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#K4">[:</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>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Z</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">W</font></span><a href="zfmisc_1.html#K4">:]</a></span>;<br/>









<b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x</font> <a href="mcart_1.html#K8">`1</a> , <font color="Maroon">y1</font> = <font color="Maroon">y</font> <a href="mcart_1.html#K8">`1</a>  as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font> ;<br/>
<b>reconsider </b><font color="Maroon">x2</font> = <font color="Maroon">x</font> <a href="mcart_1.html#K9">`2</a> , <font color="Maroon">y2</font> = <font color="Maroon">y</font> <a href="mcart_1.html#K9">`2</a>  as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font> ;<br/>
<b>reconsider </b><font color="Maroon">x3</font> = <font color="Maroon">x</font> <a href="mcart_1.html#K10">`3</a> , <font color="Maroon">y3</font> = <font color="Maroon">y</font> <a href="mcart_1.html#K10">`3</a>  as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Z</font> ;<br/>
<b>reconsider </b><font color="Maroon">x4</font> = <font color="Maroon">x</font> <a href="mcart_1.html#K11">`4</a> , <font color="Maroon">y4</font> = <font color="Maroon">y</font> <a href="mcart_1.html#K11">`4</a>  as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">W</font> ;<br/>
<a NAME="E5:18"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="mcart_1.html#K4">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font>,<font color="Maroon">x3</font>,<font color="Maroon">x4</font></span><a href="mcart_1.html#K4">]</a></span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="mcart_1.html#K4">[</a><span class="default"><font color="Maroon">y1</font>,<font color="Maroon">y2</font>,<font color="Maroon">y3</font>,<font color="Maroon">y4</font></span><a href="mcart_1.html#K4">]</a></span> )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T60">MCART_1:60</a></i>;<br/>
<b>thus </b><a NAME="E6:18"/>
( <span class="p1">(<span class="default"><a href="metric_3.html#K7" target="_self">dist_cart4</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font>,<font color="Maroon">Z</font>,<font color="Maroon">W</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 0 implies <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_1"/><i><font color="Green">E29</font></i>: 
<span class="p1">(<span class="default"><a href="metric_3.html#K7" target="_self">dist_cart4</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font>,<font color="Maroon">Z</font>,<font color="Maroon">W</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 0
 ;<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">y1</font>;<br/>
<b>set </b><font color="Maroon">d2</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">d3</font> =  <a href="metric_1.html#K4">dist</a> <font color="Maroon">x3</font>,<font color="Maroon">y3</font>;<br/>
<b>set </b><font color="Maroon">d5</font> =  <a href="metric_1.html#K4">dist</a> <font color="Maroon">x4</font>,<font color="Maroon">y4</font>;<br/>
<b>set </b><font color="Maroon">d4</font> = <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">x2</font>,<font color="Maroon">y2</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">d6</font> = <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x3</font>,<font color="Maroon">y3</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">x4</font>,<font color="Maroon">y4</font></span>)</span>;<br/>
<a NAME="E2:18_1"/><i><font color="Green">E33</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">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">x3</font>,<font color="Maroon">y3</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">x4</font>,<font color="Maroon">y4</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E3:18_1"/><i><font color="Green">E34</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="metric_1.html#K4">dist</a> <font color="Maroon">x1</font>,<font color="Maroon">y1</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="metric_1.html#K4">dist</a> <font color="Maroon">x2</font>,<font color="Maroon">y2</font> )
 
<b>by </b><i><a class="ref" href="metric_1.html#T5">METRIC_1:5</a></i>;<br/>
<a NAME="E4:18_1"/><b>then </b><i><font color="Green">E35</font></i>: 
0 <a href="real_1.html#K3">+</a> 0 <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">x2</font>,<font color="Maroon">y2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E5:18_1"/><i><font color="Green">E36</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="metric_1.html#K4">dist</a> <font color="Maroon">x3</font>,<font color="Maroon">y3</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="metric_1.html#K4">dist</a> <font color="Maroon">x4</font>,<font color="Maroon">y4</font> )
 
<b>by </b><i><a class="ref" href="metric_1.html#T5">METRIC_1:5</a></i>;<br/>
<a NAME="E6:18_1"/><b>then </b>
0 <a href="real_1.html#K3">+</a> 0 <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">x3</font>,<font color="Maroon">y3</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">x4</font>,<font color="Maroon">y4</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E7:18_1"/><b>then </b><i><font color="Green">E40</font></i>: 
( <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">x2</font>,<font color="Maroon">y2</font></span>)</span> <a href="hidden.html#R1">=</a> 0 &amp; <span class="p1">(<span class="default"><a href="metric_1.html#K4">dist</a> <font color="Maroon">x3</font>,<font color="Maroon">y3</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">x4</font>,<font color="Maroon">y4</font></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i>, , <a class="ref" href="xreal_1.html#T29">XREAL_1:29</a></i>;<br/>
<a NAME="E8:18_1"/><b>then </b><i><font color="Green">E59</font></i>: 
(  <a href="metric_1.html#K4">dist</a> <font color="Maroon">x1</font>,<font color="Maroon">y1</font> <a href="hidden.html#R1">=</a> 0 &amp;  <a href="metric_1.html#K4">dist</a> <font color="Maroon">x2</font>,<font color="Maroon">y2</font> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T29">XREAL_1:29</a></i>;<br/>
<a NAME="E9:18_1"/><b>then </b><i><font color="Green">E60</font></i>: 
<font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y2</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#T2">METRIC_1:2</a></i>;<br/>
<a NAME="E10:18_1"/>
(  <a href="metric_1.html#K4">dist</a> <font color="Maroon">x3</font>,<font color="Maroon">y3</font> <a href="hidden.html#R1">=</a> 0 &amp;  <a href="metric_1.html#K4">dist</a> <font color="Maroon">x4</font>,<font color="Maroon">y4</font> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i>, , <a class="ref" href="xreal_1.html#T29">XREAL_1:29</a></i>;<br/>
<a NAME="E11:18_1"/><b>then </b>
( <font color="Maroon">x3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y3</font> &amp; <font color="Maroon">x4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y4</font> )
 
<b>by </b><i><a class="ref" href="metric_1.html#T2">METRIC_1:2</a></i>;<br/>
<b>hence </b><a NAME="E12:18_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i>, , , <a class="ref" href="metric_1.html#T2">METRIC_1:2</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E7:18"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 ;<br/>

<a NAME="E8:18"/><b>then </b><i><font color="Green">E62</font></i>: 
 <a href="metric_1.html#K4">dist</a> <font color="Maroon">x2</font>,<font color="Maroon">y2</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="metric_1.html#T1">METRIC_1:1</a></i>;<br/>
<a NAME="E9:18"/><i><font color="Green">E63</font></i>: 
(  <a href="metric_1.html#K4">dist</a> <font color="Maroon">x3</font>,<font color="Maroon">y3</font> <a href="hidden.html#R1">=</a> 0 &amp;  <a href="metric_1.html#K4">dist</a> <font color="Maroon">x4</font>,<font color="Maroon">y4</font> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i>, <a class="ref" href="metric_1.html#T1">METRIC_1:1</a></i>;<br/>
<span class="p1">(<span class="default"><a href="metric_3.html#K7" target="_self">dist_cart4</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font>,<font color="Maroon">Z</font>,<font color="Maroon">W</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> = 
<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">x3</font>,<font color="Maroon">y3</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">x4</font>,<font color="Maroon">y4</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
0
<b>by </b><i>, , , <a class="ref" href="metric_1.html#T1">METRIC_1:1</a></i>
;<br/>
<b>hence </b><a NAME="E11:18"/>
<span class="p1">(<span class="default"><a href="metric_3.html#K7" target="_self">dist_cart4</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font>,<font color="Maroon">Z</font>,<font color="Maroon">W</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 0
 ;<br/>


</div>
