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<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">y</font>, <font color="Maroon" title="c6">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#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>,<span class="p1">(<span class="default"><a href="metric_3.html#K13" title="METRIC_3:func.13">dist_cart3S</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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y</font> ) &amp; ( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y</font> implies  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp;  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">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="c5">y</font>,<font color="Maroon" title="c4">x</font>) &amp;  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</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"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</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"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> (<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font>)</span>)</span> )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c7">x9</font> = <font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c8">y9</font> = <font color="Maroon" title="c5">y</font>, <font color="Maroon" title="c9">z9</font> = <font color="Maroon" title="c6">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#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> ;<br/><a NAME="E2:38_1_1_1"/><span class="lab"><font color="Green" title="E26">A1</font></span>: 
 <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="metric_3.html#K13" title="METRIC_3:func.13">dist_cart3S</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="c7">x9</font>,<font color="Maroon" title="c8">y9</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#D1" title="METRIC_1:def.1">METRIC_1:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E3:38_1_1_1"/>
(  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">y</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  iff <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="metric_3.html#T14" target="_self" title="METRIC_3:th.14">Th14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">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="c5">y</font>,<font color="Maroon" title="c4">x</font>) &amp;  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</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"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</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"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> (<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font>)</span>)</span> )</span><br/><a NAME="E4:38_1_1_1"/>
 <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> (<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c4">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="metric_3.html#K13" title="METRIC_3:func.13">dist_cart3S</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="c8">y9</font>,<font color="Maroon" title="c7">x9</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#D1" title="METRIC_1:def.1">METRIC_1:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E5:38_1_1_1"/>
 <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">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="c5">y</font>,<font color="Maroon" title="c4">x</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:38_1_1_1"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="ref" href="metric_3.html#T15" target="_self" title="METRIC_3:th.15">Th15</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</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"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</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"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> (<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font>)</span>)</span></span><br/><a NAME="E6:38_1_1_1"/>
(  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c6">z</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="metric_3.html#K13" title="METRIC_3:func.13">dist_cart3S</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="c7">x9</font>,<font color="Maroon" title="c9">z9</font>) &amp;  <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> (<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="metric_3.html#K13" title="METRIC_3:func.13">dist_cart3S</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="c8">y9</font>,<font color="Maroon" title="c9">z9</font>) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#D1" title="METRIC_1:def.1">METRIC_1:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E7:38_1_1_1"/>
 <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</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"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</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"><a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> (<font color="Maroon" title="c5">y</font>,<font color="Maroon" title="c6">z</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:38_1_1_1"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="ref" href="metric_3.html#T18" target="_self" title="METRIC_3:th.18">Th18</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:38_1_1"/>
 <a href="metric_1.html#G1" title="METRIC_1:aggr.1">MetrStruct</a>(# <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>,<span class="p1">(<span class="default"><a href="metric_3.html#K13" title="METRIC_3:func.13">dist_cart3S</a> (<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font>,<font color="Maroon" title="c3">Z</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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#T6" title="METRIC_1:th.6">METRIC_1:6</a></span>; <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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