<?xml version="1.0"?>
<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">x</font> be    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K3">[:</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></span><a href="zfmisc_1.html#K3">:]</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#K3">[:</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></span><a href="zfmisc_1.html#K3">:]</a></span>;<br/>







<b>reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x</font> <a href="mcart_1.html#K5">`1</a> , <font color="Maroon">y1</font> = <font color="Maroon">y</font> <a href="mcart_1.html#K5">`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#K6">`2</a> , <font color="Maroon">y2</font> = <font color="Maroon">y</font> <a href="mcart_1.html#K6">`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#K7">`3</a> , <font color="Maroon">y3</font> = <font color="Maroon">y</font> <a href="mcart_1.html#K7">`3</a>  as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Z</font> ;<br/>
<a NAME="E4:10"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">x1</font>,<font color="Maroon">x2</font>,<font color="Maroon">x3</font></span><a href="mcart_1.html#K3">]</a></span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">y1</font>,<font color="Maroon">y2</font>,<font color="Maroon">y3</font></span><a href="mcart_1.html#K3">]</a></span> )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T48">MCART_1:48</a></i>;<br/>
<b>thus </b><a NAME="E5:10"/>
( <span class="p1">(<span class="default"><a href="metric_4.html#K4" target="_self">dist_cart3S</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font>,<font color="Maroon">Z</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="10_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:10_1"/><i><font color="Green">E28</font></i>: 
<span class="p1">(<span class="default"><a href="metric_4.html#K4" target="_self">dist_cart3S</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font>,<font color="Maroon">Z</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/>
<a NAME="E2:10_1"/>
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<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="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<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="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<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="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E3:10_1"/><b>then </b><i><font color="Green">E31</font></i>: 
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<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="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<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="square_1.html#K5">^2</a> </span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 0
 
;<br/>
<a NAME="E4:10_1"/><i><font color="Green">E32</font></i>: 
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="square_1.html#K5">^2</a> 
 
<b>by </b><i><a class="ref" href="xreal_1.html#T65">XREAL_1:65</a></i>;<br/>
<a NAME="E5:10_1"/><i><font color="Green">E33</font></i>: 
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">x2</font>,<font color="Maroon">y2</font></span>)</span> <a href="square_1.html#K5">^2</a> 
 
<b>by </b><i><a class="ref" href="xreal_1.html#T65">XREAL_1:65</a></i>;<br/>
<a NAME="E6:10_1"/>
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="square_1.html#K5">^2</a> 
 
<b>by </b><i><a class="ref" href="xreal_1.html#T65">XREAL_1:65</a></i>;<br/>
<a NAME="E7:10_1"/><b>then </b><i><font color="Green">E34</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"><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="square_1.html#K5">^2</a> </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="square_1.html#K5">^2</a> </span>)</span>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E8:10_1"/><b>then </b>
( <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="square_1.html#K5">^2</a>  <a href="hidden.html#R1">=</a> 0 &amp; <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="square_1.html#K5">^2</a> </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="square_1.html#K5">^2</a> </span>)</span> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E9:10_1"/><b>then </b>
 <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
 
<b>by </b><i><a class="ref" href="xcmplx_1.html#T6">XCMPLX_1:6</a></i>;<br/>
<a NAME="E10:10_1"/><b>then </b><i><font color="Green">E41</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y1</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#T2">METRIC_1:2</a></i>;<br/>
<a NAME="E11:10_1"/><i><font color="Green">E42</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">x2</font>,<font color="Maroon">y2</font></span>)</span> <a href="square_1.html#K5">^2</a> </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="square_1.html#K5">^2</a> </span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E12:10_1"/><i><font color="Green">E43</font></i>: 
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">x2</font>,<font color="Maroon">y2</font></span>)</span> <a href="square_1.html#K5">^2</a> 
 
<b>by </b><i><a class="ref" href="xreal_1.html#T65">XREAL_1:65</a></i>;<br/>
<a NAME="E13:10_1"/><i><font color="Green">E44</font></i>: 
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="square_1.html#K5">^2</a> 
 
<b>by </b><i><a class="ref" href="xreal_1.html#T65">XREAL_1:65</a></i>;<br/>
<a NAME="E14:10_1"/><b>then </b><i><font color="Green">E49</font></i>: 
( <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="square_1.html#K5">^2</a>  <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="square_1.html#K5">^2</a>  <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="E15:10_1"/>
<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="square_1.html#K5">^2</a>  <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i>, , <a class="ref" href="metric_4.html#D1" target="_self">Def1</a>, <a class="ref" href="xreal_1.html#T29">XREAL_1:29</a></i>;<br/>
<a NAME="E16:10_1"/><b>then </b>
 <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="xcmplx_1.html#T6">XCMPLX_1:6</a></i>;<br/>
<a NAME="E17:10_1"/><b>then </b><i><font color="Green">E50</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="E18:10_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
 
<b>by </b><i>, <a class="ref" href="xcmplx_1.html#T6">XCMPLX_1:6</a></i>;<br/>
<b>hence </b><a NAME="E19:10_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="E6:10"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 ;<br/>

<a NAME="E7:10"/><b>then </b><i><font color="Green">E68</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="square_1.html#K5">^2</a>  <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="metric_1.html#T1">METRIC_1:1</a>, <a class="ref" href="square_1.html#T60">SQUARE_1:60</a></i>;<br/>
<a NAME="E8:10"/><i><font color="Green">E69</font></i>: 
<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="square_1.html#K5">^2</a>  <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i>, <a class="ref" href="metric_1.html#T1">METRIC_1:1</a>, <a class="ref" href="square_1.html#T60">SQUARE_1:60</a></i>;<br/>
<span class="p1">(<span class="default"><a href="metric_4.html#K4" target="_self">dist_cart3S</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font>,<font color="Maroon">Z</font></span>)</span> <a href="metric_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> = 
 <a href="square_1.html#K7">sqrt</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<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="square_1.html#K5">^2</a> </span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<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="square_1.html#K5">^2</a> </span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3">(<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="square_1.html#K5">^2</a> </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>, <a class="ref" href="square_1.html#T60">SQUARE_1:60</a>, <a class="ref" href="square_1.html#T82">SQUARE_1:82</a></i>
;<br/>
<b>hence </b><a NAME="E10:10"/>
<span class="p1">(<span class="default"><a href="metric_4.html#K4" target="_self">dist_cart3S</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font>,<font color="Maroon">Z</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>
