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<div class="add">

<b>set </b><font color="Maroon">M</font> =  <a href="metric_1.html#G1">MetrStruct</a>(# <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>,<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> #);<br/>
<a NAME="E1:8_1_1"/>
 <a href="metric_1.html#G1">MetrStruct</a>(# <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>,<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> #) is   <a href="metric_1.html#NM1">MetrSpace</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="metric_1.html#G1">MetrStruct</a>(# <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>,<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> #), <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="metric_1.html#G1">MetrStruct</a>(# <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>,<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> #), <font color="Maroon">z</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="metric_1.html#G1">MetrStruct</a>(# <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>,<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> #);<br/><a NAME="E1:8_1_1_1_1"/><i><font color="Green">E28</font></i>: 
 <a href="metric_1.html#K2">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <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="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="metric_1.html#D1">METRIC_1:def 1</a></i>;<br/><a NAME="E2:8_1_1_1_1"/><i><font color="Green">E29</font></i>: 
 <a href="metric_1.html#K2">dist</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_3.html#K1" target="_self">dist_cart2</a> <font color="Maroon">X</font>,<font color="Maroon">Y</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="metric_1.html#D1">METRIC_1:def 1</a></i>;<br/><a NAME="E3:8_1_1_1_1"/><i><font color="Green">E33</font></i>: 
 <a href="metric_1.html#K2">dist</a> <font color="Maroon">y</font>,<font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <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="binop_1.html#K1">.</a> <font color="Maroon">y</font>,<font color="Maroon">z</font>
 <b>by </b><i><a class="ref" href="metric_1.html#D1">METRIC_1:def 1</a></i>;<br/><a NAME="E4:8_1_1_1_1"/><i><font color="Green">E34</font></i>: 
 <a href="metric_1.html#K2">dist</a> <font color="Maroon">y</font>,<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <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="binop_1.html#K1">.</a> <font color="Maroon">y</font>,<font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="metric_1.html#D1">METRIC_1:def 1</a></i>;<br/><b>thus </b><a NAME="E5:8_1_1_1_1"/>
(  <a href="metric_1.html#K2">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> 0 iff <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 <b>by </b><i>, </i>;<br/><b>thus </b><a NAME="E6:8_1_1_1_1"/>
 <a href="metric_1.html#K2">dist</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K2">dist</a> <font color="Maroon">y</font>,<font color="Maroon">x</font>
 <b>by </b><i>, , </i>;<br/><b>thus </b><a NAME="E7:8_1_1_1_1"/>
 <a href="metric_1.html#K2">dist</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"><a href="metric_1.html#K2">dist</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"><a href="metric_1.html#K2">dist</a> <font color="Maroon">y</font>,<font color="Maroon">z</font></span>)</span>
 <b>by </b><i>, , , </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:8_1_1_1"/>
 <a href="metric_1.html#G1">MetrStruct</a>(# <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>,<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> #) is   <a href="metric_1.html#NM1">MetrSpace</a>
 <b>by </b><i><a class="ref" href="metric_1.html#T6">METRIC_1:6</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:8_1_1"/>
 <a href="metric_1.html#G1">MetrStruct</a>(# <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>,<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> #) is  non <a href="struct_0.html#V3">empty</a> <a href="metric_1.html#V1">strict</a> <a href="metric_1.html#NM1">MetrSpace</a>
 ;<br/>


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