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<b>set </b><font color="Maroon" title="c5">M</font> =  <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# <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>,<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> #);<br/>
<a NAME="E1:21_1_1"/>
 <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# <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>,<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> #) is   <a href="metric_1.html#NM1" title="METRIC_1:NM.1">MetrSpace</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c7">y</font>, <font color="Maroon" title="c8">z</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# <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>,<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> #);<br/><b>reconsider </b><font color="Maroon" title="c9">x'</font> = <font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c10">y'</font> = <font color="Maroon" title="c7">y</font>, <font color="Maroon" title="c11">z'</font> = <font color="Maroon" title="c8">z</font> as    <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/><a NAME="E2:21_1_1_1_1"/><i><font color="Green" title="E13">A1</font></i>: 
 <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c9">x'</font>,<font color="Maroon" title="c10">y'</font>
 <b>by </b><i><a class="ref" href="metric_1.html#D1" title="METRIC_1:def.1">METRIC_1:def 1</a></i>;<br/><a NAME="E3:21_1_1_1_1"/><i><font color="Green" title="E14">A2</font></i>: 
 <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c8">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c9">x'</font>,<font color="Maroon" title="c11">z'</font>
 <b>by </b><i><a class="ref" href="metric_1.html#D1" title="METRIC_1:def.1">METRIC_1:def 1</a></i>;<br/><a NAME="E4:21_1_1_1_1"/><i><font color="Green" title="E15">A3</font></i>: 
 <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c8">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c10">y'</font>,<font color="Maroon" title="c11">z'</font>
 <b>by </b><i><a class="ref" href="metric_1.html#D1" title="METRIC_1:def.1">METRIC_1:def 1</a></i>;<br/><a NAME="E5:21_1_1_1_1"/><i><font color="Green" title="E16">A4</font></i>: 
 <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c10">y'</font>,<font color="Maroon" title="c9">x'</font>
 <b>by </b><i><a class="ref" href="metric_1.html#D1" title="METRIC_1:def.1">METRIC_1:def 1</a></i>;<br/><b>thus </b><a NAME="E6:21_1_1_1_1"/>
(  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 iff <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y</font> )
 <b>by </b><i><a class="txt" href="metric_3.html#E2:21_1_1_1_1"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="metric_3.html#T19" target="_self" title="METRIC_3:th.19">Th19</a></i>;<br/><b>thus </b><a NAME="E7:21_1_1_1_1"/>
 <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c6">x</font>
 <b>by </b><i><a class="txt" href="metric_3.html#E2:21_1_1_1_1"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="metric_3.html#E5:21_1_1_1_1"><i><font color="Green" title="E16">A4</font></i></a>, <a class="ref" href="metric_3.html#T20" target="_self" title="METRIC_3:th.20">Th20</a></i>;<br/><b>thus </b><a NAME="E8:21_1_1_1_1"/>
 <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c8">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">y</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#K2" title="METRIC_1:func.2">dist</a> <font color="Maroon" title="c7">y</font>,<font color="Maroon" title="c8">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="metric_3.html#E2:21_1_1_1_1"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="metric_3.html#E3:21_1_1_1_1"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="metric_3.html#E4:21_1_1_1_1"><i><font color="Green" title="E15">A3</font></i></a>, <a class="ref" href="metric_3.html#T21" target="_self" title="METRIC_3:th.21">Th21</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:21_1_1_1"/>
 <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# <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>,<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> #) is   <a href="metric_1.html#NM1" title="METRIC_1:NM.1">MetrSpace</a>
 <b>by </b><i><a class="ref" href="metric_1.html#T6" title="METRIC_1:th.6">METRIC_1:6</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:21_1_1"/>
 <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# <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>,<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> #) is  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="metric_1.html#V1" title="METRIC_1:attr.1">strict</a> <a href="metric_1.html#NM1" title="METRIC_1:NM.1">MetrSpace</a>
 ;<br/>


</div>
