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<span class="kw">let </span><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#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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being    <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> holds  <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="Olive" title="b1">x</font>,<font color="Olive" title="b2">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="Olive" title="b2">y</font>,<font color="Olive" title="b1">x</font>)</span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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> <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="c6">y</font>,<font color="Maroon" title="c5">x</font>)</span><br/>









<span class="kw">reconsider </span><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/>
<span class="kw">reconsider </span><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/>
<span class="kw">reconsider </span><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/>
<span class="kw">reconsider </span><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="E5:19"/><span class="lab"><font color="Green" title="E12">A1</font></span>: 
( <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> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="mcart_1.html#T56" title="MCART_1:th.56">MCART_1:56</a></span>;<br/>
<span class="kw">then </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> <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="c8">y1</font>,<font color="Maroon" title="c7">x1</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">y2</font>,<font color="Maroon" title="c9">x2</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="c12">y3</font>,<font color="Maroon" title="c11">x3</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="c13">x4</font>,<font color="Maroon" title="c14">y4</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="metric_3.html#D7" target="_self" title="METRIC_3:def.7">Def7</a></span>
<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="c6">y</font>,<font color="Maroon" title="c5">x</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:19"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="ref" href="metric_3.html#D7" target="_self" title="METRIC_3:def.7">Def7</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E7:19"/>
<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> <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="c6">y</font>,<font color="Maroon" title="c5">x</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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