<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c2">Y</font>, <font color="Maroon" title="c3">Z</font>, <font color="Maroon" title="c4">W</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="metric_1.html#NM1" title="METRIC_1:NM.1">MetrSpace</a>;<br/><b>let </b><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">y</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K4" title="ZFMISC_1:func.4">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">Z</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">W</font></span><a href="zfmisc_1.html#K4" title="ZFMISC_1:func.4">:]</a></span>;<br/>









<b>reconsider </b><font color="Maroon" title="c7">x1</font> = <font color="Maroon" title="c5">x</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a> , <font color="Maroon" title="c8">y1</font> = <font color="Maroon" title="c6">y</font> <a href="mcart_1.html#K8" title="MCART_1:func.8">`1</a>  as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">X</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c9">x2</font> = <font color="Maroon" title="c5">x</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a> , <font color="Maroon" title="c10">y2</font> = <font color="Maroon" title="c6">y</font> <a href="mcart_1.html#K9" title="MCART_1:func.9">`2</a>  as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">Y</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c11">x3</font> = <font color="Maroon" title="c5">x</font> <a href="mcart_1.html#K10" title="MCART_1:func.10">`3</a> , <font color="Maroon" title="c12">y3</font> = <font color="Maroon" title="c6">y</font> <a href="mcart_1.html#K10" title="MCART_1:func.10">`3</a>  as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">Z</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c13">x4</font> = <font color="Maroon" title="c5">x</font> <a href="mcart_1.html#K11" title="MCART_1:func.11">`4</a> , <font color="Maroon" title="c14">y4</font> = <font color="Maroon" title="c6">y</font> <a href="mcart_1.html#K11" title="MCART_1:func.11">`4</a>  as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c4">W</font> ;<br/>
<a NAME="E5:18"/><i><font color="Green" title="E11">A1</font></i>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c9">x2</font>,<font color="Maroon" title="c11">x3</font>,<font color="Maroon" title="c13">x4</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> &amp; <font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Maroon" title="c8">y1</font>,<font color="Maroon" title="c10">y2</font>,<font color="Maroon" title="c12">y3</font>,<font color="Maroon" title="c14">y4</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T60" title="MCART_1:th.60">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" title="METRIC_3:func.7">dist_cart4</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font>,<font color="Maroon" title="c3">Z</font>,<font color="Maroon" title="c4">W</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 implies <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">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" title="E12">A2</font></i>: 
<span class="p1">(<span class="default"><a href="metric_3.html#K7" title="METRIC_3:func.7">dist_cart4</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font>,<font color="Maroon" title="c3">Z</font>,<font color="Maroon" title="c4">W</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/>

<b>set </b><font color="Maroon" title="c15">d1</font> =  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c8">y1</font>;<br/>
<b>set </b><font color="Maroon" title="c16">d2</font> =  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c9">x2</font>,<font color="Maroon" title="c10">y2</font>;<br/>
<b>set </b><font color="Maroon" title="c17">d3</font> =  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c11">x3</font>,<font color="Maroon" title="c12">y3</font>;<br/>
<b>set </b><font color="Maroon" title="c18">d5</font> =  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c13">x4</font>,<font color="Maroon" title="c14">y4</font>;<br/>
<b>set </b><font color="Maroon" title="c19">d4</font> = <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c8">y1</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c9">x2</font>,<font color="Maroon" title="c10">y2</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c20">d6</font> = <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c11">x3</font>,<font color="Maroon" title="c12">y3</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c13">x4</font>,<font color="Maroon" title="c14">y4</font></span>)</span>;<br/>
<a NAME="E2:18_1"/><i><font color="Green" title="E13">A3</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c8">y1</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c9">x2</font>,<font color="Maroon" title="c10">y2</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c11">x3</font>,<font color="Maroon" title="c12">y3</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c13">x4</font>,<font color="Maroon" title="c14">y4</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 
<b>by </b><i><a class="txt" href="metric_3.html#E5:18"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="metric_3.html#E1:18_1"><i><font color="Green" title="E12">A2</font></i></a>, <a class="ref" href="metric_3.html#D7" target="_self" title="METRIC_3:def.7">Def7</a></i>;<br/>
<a NAME="E3:18_1"/><i><font color="Green" title="E14">A4</font></i>: 
( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c8">y1</font> &amp; 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c9">x2</font>,<font color="Maroon" title="c10">y2</font> )
 
<b>by </b><i><a class="ref" href="metric_1.html#T5" title="METRIC_1:th.5">METRIC_1:5</a></i>;<br/>
<a NAME="E4:18_1"/><b>then </b><i><font color="Green" title="E15">A5</font></i>: 
0 <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c8">y1</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c9">x2</font>,<font color="Maroon" title="c10">y2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<a NAME="E5:18_1"/><i><font color="Green" title="E16">A6</font></i>: 
( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c11">x3</font>,<font color="Maroon" title="c12">y3</font> &amp; 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c13">x4</font>,<font color="Maroon" title="c14">y4</font> )
 
<b>by </b><i><a class="ref" href="metric_1.html#T5" title="METRIC_1:th.5">METRIC_1:5</a></i>;<br/>
<a NAME="E6:18_1"/><b>then </b>
0 <a href="real_1.html#K3" title="REAL_1:func.3">+</a> 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c11">x3</font>,<font color="Maroon" title="c12">y3</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c13">x4</font>,<font color="Maroon" title="c14">y4</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<a NAME="E7:18_1"/><b>then </b><i><font color="Green" title="E17">A7</font></i>: 
( <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c8">y1</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c9">x2</font>,<font color="Maroon" title="c10">y2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c11">x3</font>,<font color="Maroon" title="c12">y3</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c13">x4</font>,<font color="Maroon" title="c14">y4</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 
<b>by </b><i><a class="txt" href="metric_3.html#E2:18_1"><i><font color="Green" title="E13">A3</font></i></a>, <a class="txt" href="metric_3.html#E4:18_1"><i><font color="Green" title="E15">A5</font></i></a>, <a class="ref" href="xreal_1.html#T29" title="XREAL_1:th.29">XREAL_1:29</a></i>;<br/>
<a NAME="E8:18_1"/><b>then </b><i><font color="Green" title="E18">A8</font></i>: 
(  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c8">y1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp;  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c9">x2</font>,<font color="Maroon" title="c10">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 
<b>by </b><i><a class="txt" href="metric_3.html#E3:18_1"><i><font color="Green" title="E14">A4</font></i></a>, <a class="ref" href="xreal_1.html#T29" title="XREAL_1:th.29">XREAL_1:29</a></i>;<br/>
<a NAME="E9:18_1"/><b>then </b><i><font color="Green" title="E19">A9</font></i>: 
<font color="Maroon" title="c9">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">y2</font>
 
<b>by </b><i><a class="ref" href="metric_1.html#T2" title="METRIC_1:th.2">METRIC_1:2</a></i>;<br/>
<a NAME="E10:18_1"/>
(  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c11">x3</font>,<font color="Maroon" title="c12">y3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp;  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c13">x4</font>,<font color="Maroon" title="c14">y4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 
<b>by </b><i><a class="txt" href="metric_3.html#E5:18_1"><i><font color="Green" title="E16">A6</font></i></a>, <a class="txt" href="metric_3.html#E7:18_1"><i><font color="Green" title="E17">A7</font></i></a>, <a class="ref" href="xreal_1.html#T29" title="XREAL_1:th.29">XREAL_1:29</a></i>;<br/>
<a NAME="E11:18_1"/><b>then </b>
( <font color="Maroon" title="c11">x3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">y3</font> &amp; <font color="Maroon" title="c13">x4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">y4</font> )
 
<b>by </b><i><a class="ref" href="metric_1.html#T2" title="METRIC_1:th.2">METRIC_1:2</a></i>;<br/>
<b>hence </b><a NAME="E12:18_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y</font>
 <b>by </b><i><a class="txt" href="metric_3.html#E5:18"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="metric_3.html#E8:18_1"><i><font color="Green" title="E18">A8</font></i></a>, <a class="txt" href="metric_3.html#E9:18_1"><i><font color="Green" title="E19">A9</font></i></a>, <a class="ref" href="metric_1.html#T2" title="METRIC_1:th.2">METRIC_1:2</a></i>;<br/>


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

<b>assume </b><a NAME="E7:18"/><i><font color="Green" title="E12">A10</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y</font>
 ;<br/>

<a NAME="E8:18"/><b>then </b><i><font color="Green" title="E13">A11</font></i>: 
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c9">x2</font>,<font color="Maroon" title="c10">y2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 
<b>by </b><i><a class="ref" href="metric_1.html#T1" title="METRIC_1:th.1">METRIC_1:1</a></i>;<br/>
<a NAME="E9:18"/><i><font color="Green" title="E14">A12</font></i>: 
(  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c11">x3</font>,<font color="Maroon" title="c12">y3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp;  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c13">x4</font>,<font color="Maroon" title="c14">y4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 )
 
<b>by </b><i><a class="txt" href="metric_3.html#E7:18"><i><font color="Green" title="E12">A10</font></i></a>, <a class="ref" href="metric_1.html#T1" title="METRIC_1:th.1">METRIC_1:1</a></i>;<br/>
<span class="p1">(<span class="default"><a href="metric_3.html#K7" title="METRIC_3:func.7">dist_cart4</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font>,<font color="Maroon" title="c3">Z</font>,<font color="Maroon" title="c4">W</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c8">y1</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c9">x2</font>,<font color="Maroon" title="c10">y2</font></span>)</span></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c11">x3</font>,<font color="Maroon" title="c12">y3</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p2">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c13">x4</font>,<font color="Maroon" title="c14">y4</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="metric_3.html#E5:18"><i><font color="Green" title="E11">A1</font></i></a>, <a class="ref" href="metric_3.html#D7" target="_self" title="METRIC_3:def.7">Def7</a></i>
<br/>.= 
0
<b>by </b><i><a class="txt" href="metric_3.html#E7:18"><i><font color="Green" title="E12">A10</font></i></a>, <a class="txt" href="metric_3.html#E8:18"><i><font color="Green" title="E13">A11</font></i></a>, <a class="txt" href="metric_3.html#E9:18"><i><font color="Green" title="E14">A12</font></i></a>, <a class="ref" href="metric_1.html#T1" title="METRIC_1:th.1">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" title="METRIC_3:func.7">dist_cart4</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font>,<font color="Maroon" title="c3">Z</font>,<font color="Maroon" title="c4">W</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/>


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