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<span class="kw">let </span><font color="Maroon" title="c2">S</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="metric_1.html#V6" title="METRIC_1:attr.6">Reflexive</a>   <a href="metric_1.html#V8" title="METRIC_1:attr.8">symmetric</a>   <a href="metric_1.html#V9" title="METRIC_1:attr.9">triangle</a>   <a href="metric_1.html#L1" title="METRIC_1:struct.1">MetrStruct</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">p</font>, <font color="Olive" title="b2">q</font>, <font color="Olive" title="b3">r</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">S</font>  st <font color="Olive" title="b2">q</font> <a href="metric_1.html#R1" title="METRIC_1:pred.1">is_between</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b3">r</font> holds <br/><font color="Olive" title="b2">q</font> <a href="metric_1.html#R1" title="METRIC_1:pred.1">is_between</a> <font color="Olive" title="b3">r</font>,<font color="Olive" title="b1">p</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">p</font>, <font color="Maroon" title="c4">q</font>, <font color="Maroon" title="c5">r</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">q</font> <a href="metric_1.html#R1" title="METRIC_1:pred.1">is_between</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c5">r</font> implies <font color="Maroon" title="c4">q</font> <a href="metric_1.html#R1" title="METRIC_1:pred.1">is_between</a> <font color="Maroon" title="c5">r</font>,<font color="Maroon" title="c3">p</font> )</span><br/>



<span class="kw">assume </span><a NAME="E1:70"/><span class="lab"><font color="Green" title="E50">A1</font></span>: 
<font color="Maroon" title="c4">q</font> <a href="metric_1.html#R1" title="METRIC_1:pred.1">is_between</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c5">r</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">q</font> <a href="metric_1.html#R1" title="METRIC_1:pred.1">is_between</a> <font color="Maroon" title="c5">r</font>,<font color="Maroon" title="c3">p</font></span><br/>

<span class="kw">hence </span><a NAME="E2:70"/>
( <font color="Maroon" title="c5">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">q</font> &amp; <font color="Maroon" title="c5">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">p</font> &amp; <font color="Maroon" title="c4">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">p</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#D22" target="_self" title="METRIC_1:def.22">Def23</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="metric_1.html#D22" target="_self" title="METRIC_1:def.22">METRIC_1:def 22</a> <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="c5">r</font>,<font color="Maroon" title="c3">p</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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">r</font>,<font color="Maroon" title="c4">q</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="c4">q</font>,<font color="Maroon" title="c3">p</font>)</span>)</span></span><br/>

<a NAME="E3:70"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c5">r</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> (<font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</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="c4">q</font>,<font color="Maroon" title="c5">r</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70"><span class="lab"><font color="Green" title="E50">A1</font></span></a>, <a class="ref" href="metric_1.html#D22" target="_self" title="METRIC_1:def.22">Def23</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:70"/>
 <a href="metric_1.html#K2" title="METRIC_1:func.2">dist</a> (<font color="Maroon" title="c5">r</font>,<font color="Maroon" title="c3">p</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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">r</font>,<font color="Maroon" title="c4">q</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="c4">q</font>,<font color="Maroon" title="c3">p</font>)</span>)</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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