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<b>let </b><font color="Maroon" title="c1">X</font>, <font color="Maroon" title="c2">Y</font>, <font color="Maroon" title="c3">Z</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="metric_1.html#NM1" title="METRIC_1:NM.1">MetrSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</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></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span> holds  <span class="p1">(<span class="default"><a href="metric_3.html#K4" title="METRIC_3:func.4">dist_cart3</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font>,<font color="Maroon" title="c3">Z</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b3">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_3.html#K4" title="METRIC_3:func.4">dist_cart3</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font>,<font color="Maroon" title="c3">Z</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_3.html#K4" title="METRIC_3:func.4">dist_cart3</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font>,<font color="Maroon" title="c3">Z</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">z</font></span>)</span></span><br/><b>let </b><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</font>, <font color="Maroon" title="c6">z</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</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></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="metric_3.html#K4" title="METRIC_3:func.4">dist_cart3</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font>,<font color="Maroon" title="c3">Z</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c6">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_3.html#K4" title="METRIC_3:func.4">dist_cart3</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font>,<font color="Maroon" title="c3">Z</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_3.html#K4" title="METRIC_3:func.4">dist_cart3</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font>,<font color="Maroon" title="c3">Z</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span></span><br/>







<b>reconsider </b><font color="Maroon" title="c7">x1</font> = <font color="Maroon" title="c4">x</font> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> , <font color="Maroon" title="c8">y1</font> = <font color="Maroon" title="c5">y</font> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> , <font color="Maroon" title="c9">z1</font> = <font color="Maroon" title="c6">z</font> <a href="mcart_1.html#K5" title="MCART_1:func.5">`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="c10">x2</font> = <font color="Maroon" title="c4">x</font> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> , <font color="Maroon" title="c11">y2</font> = <font color="Maroon" title="c5">y</font> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> , <font color="Maroon" title="c12">z2</font> = <font color="Maroon" title="c6">z</font> <a href="mcart_1.html#K6" title="MCART_1:func.6">`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="c13">x3</font> = <font color="Maroon" title="c4">x</font> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a> , <font color="Maroon" title="c14">y3</font> = <font color="Maroon" title="c5">y</font> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a> , <font color="Maroon" title="c15">z3</font> = <font color="Maroon" title="c6">z</font> <a href="mcart_1.html#K7" title="MCART_1:func.7">`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/>
<a NAME="E4:13"/><i><font color="Green" title="E9">A1</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c10">x2</font>,<font color="Maroon" title="c13">x3</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> &amp; <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Maroon" title="c8">y1</font>,<font color="Maroon" title="c11">y2</font>,<font color="Maroon" title="c14">y3</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> &amp; <font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K3" title="MCART_1:func.3">[</a><span class="default"><font color="Maroon" title="c9">z1</font>,<font color="Maroon" title="c12">z2</font>,<font color="Maroon" title="c15">z3</font></span><a href="mcart_1.html#K3" title="MCART_1:func.3">]</a></span> )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T48" title="MCART_1:th.48">MCART_1:48</a></i>;<br/>
<b>set </b><font color="Maroon" title="c16">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="c9">z1</font>;<br/>
<b>set </b><font color="Maroon" title="c17">d2</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="c18">d3</font> =  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c8">y1</font>,<font color="Maroon" title="c9">z1</font>;<br/>
<b>set </b><font color="Maroon" title="c19">d4</font> =  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c10">x2</font>,<font color="Maroon" title="c12">z2</font>;<br/>
<b>set </b><font color="Maroon" title="c20">d5</font> =  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c10">x2</font>,<font color="Maroon" title="c11">y2</font>;<br/>
<b>set </b><font color="Maroon" title="c21">d6</font> =  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c11">y2</font>,<font color="Maroon" title="c12">z2</font>;<br/>
<b>set </b><font color="Maroon" title="c22">d7</font> =  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c13">x3</font>,<font color="Maroon" title="c15">z3</font>;<br/>
<b>set </b><font color="Maroon" title="c23">d8</font> =  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c13">x3</font>,<font color="Maroon" title="c14">y3</font>;<br/>
<b>set </b><font color="Maroon" title="c24">d9</font> =  <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c14">y3</font>,<font color="Maroon" title="c15">z3</font>;<br/>
<i><font color="Green" title="E10">A2</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c8">y1</font>,<font color="Maroon" title="c9">z1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c10">x2</font>,<font color="Maroon" title="c11">y2</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c11">y2</font>,<font color="Maroon" title="c12">z2</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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="c13">x3</font>,<font color="Maroon" title="c14">y3</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c14">y3</font>,<font color="Maroon" title="c15">z3</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c10">x2</font>,<font color="Maroon" title="c11">y2</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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">x3</font>,<font color="Maroon" title="c14">y3</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c8">y1</font>,<font color="Maroon" title="c9">z1</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c11">y2</font>,<font color="Maroon" title="c12">z2</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c14">y3</font>,<font color="Maroon" title="c15">z3</font></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_3.html#K4" title="METRIC_3:func.4">dist_cart3</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font>,<font color="Maroon" title="c3">Z</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c8">y1</font>,<font color="Maroon" title="c9">z1</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c11">y2</font>,<font color="Maroon" title="c12">z2</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c14">y3</font>,<font color="Maroon" title="c15">z3</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E4:13"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="metric_3.html#D4" target="_self" title="METRIC_3:def.4">Def4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_3.html#K4" title="METRIC_3:func.4">dist_cart3</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font>,<font color="Maroon" title="c3">Z</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_3.html#K4" title="METRIC_3:func.4">dist_cart3</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font>,<font color="Maroon" title="c3">Z</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span>
<b>by </b><i><a class="txt" href="#E4:13"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="metric_3.html#D4" target="_self" title="METRIC_3:def.4">Def4</a></i>
;<br/>
<a NAME="E6:13"/><i><font color="Green" title="E11">A3</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="c9">z1</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#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#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c8">y1</font>,<font color="Maroon" title="c9">z1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="metric_1.html#T4" title="METRIC_1:th.4">METRIC_1:4</a></i>;<br/>
<a NAME="E7:13"/><i><font color="Green" title="E12">A4</font></i>: 
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c10">x2</font>,<font color="Maroon" title="c12">z2</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#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c10">x2</font>,<font color="Maroon" title="c11">y2</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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">y2</font>,<font color="Maroon" title="c12">z2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="metric_1.html#T4" title="METRIC_1:th.4">METRIC_1:4</a></i>;<br/>
<b>set </b><font color="Maroon" title="c25">d10</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="c9">z1</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c10">x2</font>,<font color="Maroon" title="c12">z2</font></span>)</span>;<br/>
<a NAME="E8:13"/><i><font color="Green" title="E13">A5</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="c9">z1</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c10">x2</font>,<font color="Maroon" title="c12">z2</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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="c7">x1</font>,<font color="Maroon" title="c8">y1</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c8">y1</font>,<font color="Maroon" title="c9">z1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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="c10">x2</font>,<font color="Maroon" title="c11">y2</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c11">y2</font>,<font color="Maroon" title="c12">z2</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E6:13"><i><font color="Green" title="E11">A3</font></i></a>, <a class="txt" href="#E7:13"><i><font color="Green" title="E12">A4</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<a NAME="E9:13"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c13">x3</font>,<font color="Maroon" title="c15">z3</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#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c13">x3</font>,<font color="Maroon" title="c14">y3</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c14">y3</font>,<font color="Maroon" title="c15">z3</font></span>)</span>
 
<b>by </b><i><a class="ref" href="metric_1.html#T4" title="METRIC_1:th.4">METRIC_1:4</a></i>;<br/>
<a NAME="E10:13"/><b>then </b>
<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="c9">z1</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c10">x2</font>,<font color="Maroon" title="c12">z2</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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">x3</font>,<font color="Maroon" title="c15">z3</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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" 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#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c8">y1</font>,<font color="Maroon" title="c9">z1</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c10">x2</font>,<font color="Maroon" title="c11">y2</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c11">y2</font>,<font color="Maroon" title="c12">z2</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</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="c13">x3</font>,<font color="Maroon" title="c14">y3</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p2">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c14">y3</font>,<font color="Maroon" title="c15">z3</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E8:13"><i><font color="Green" title="E13">A5</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/>
<b>hence </b><a NAME="E11:13"/>
<span class="p1">(<span class="default"><a href="metric_3.html#K4" title="METRIC_3:func.4">dist_cart3</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font>,<font color="Maroon" title="c3">Z</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c6">z</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_3.html#K4" title="METRIC_3:func.4">dist_cart3</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font>,<font color="Maroon" title="c3">Z</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="metric_3.html#K4" title="METRIC_3:func.4">dist_cart3</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font>,<font color="Maroon" title="c3">Z</font></span>)</span> <a href="metric_1.html#K1" title="METRIC_1:func.1">.</a> <font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E4:13"><i><font color="Green" title="E9">A1</font></i></a>, <a class="txt" href="#E5:13"><i><font color="Green" title="E10">A2</font></i></a>, <a class="ref" href="metric_3.html#D4" target="_self" title="METRIC_3:def.4">Def4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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