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<span class="kw">let </span><font color="Maroon" title="c2">S</font> be   <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">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/>(  not <font color="Olive" title="b1">p</font> <a href="metric_1.html#R1" title="METRIC_1:pred.1">is_between</a> <font color="Olive" title="b2">q</font>,<font color="Olive" title="b3">r</font> &amp;  not <font color="Olive" title="b3">r</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="b2">q</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 (  not <font color="Maroon" title="c3">p</font> <a href="metric_1.html#R1" title="METRIC_1:pred.1">is_between</a> <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c5">r</font> &amp;  not <font color="Maroon" title="c5">r</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="c4">q</font> ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:71"/><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"> (  not <font color="Maroon" title="c3">p</font> <a href="metric_1.html#R1" title="METRIC_1:pred.1">is_between</a> <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c5">r</font> &amp;  not <font color="Maroon" title="c5">r</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="c4">q</font> )</span><br/>

<a NAME="E2:71"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A2</font></span>: 
 <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="ref" href="metric_1.html#D22" target="_self" title="METRIC_1:def.22">Def23</a></span>;<br/>
<a NAME="E3:71"/><span class="lab"><font color="Green" title="E52">A3</font></span>: 
<font color="Maroon" title="c3">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">q</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:71"><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">thus </span><a NAME="E4:71"/>
 not <font color="Maroon" title="c3">p</font> <a href="metric_1.html#R1" title="METRIC_1:pred.1">is_between</a> <font color="Maroon" title="c4">q</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">  not <font color="Maroon" title="c5">r</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="c4">q</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:71_1"/>
<font color="Maroon" title="c3">p</font> <a href="metric_1.html#R1" title="METRIC_1:pred.1">is_between</a> <font color="Maroon" title="c4">q</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"> contradiction</span><br/>

<a NAME="E2:71_1"/><span class="kw">then </span>
 <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"><span class="p2">(<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="c3">p</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="c3">p</font>,<font color="Maroon" title="c5">r</font>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:71"><span class="lab"><font color="Green" title="E51">A2</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="E3:71_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:71"><span class="lab"><font color="Green" title="E52">A3</font></span></a>, <a class="txt" href="metric_1.html#E43"><span class="lab"><font color="Green" title="E36">Lm8</font></span></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">assume </span><a NAME="E5:71"/>
<font color="Maroon" title="c5">r</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="c4">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E6:71"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A4</font></span>: 
 <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>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="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="p2">(<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>)</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="c5">r</font>,<font color="Maroon" title="c4">q</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:71"><span class="lab"><font color="Green" title="E51">A2</font></span></a>, <a class="ref" href="metric_1.html#D22" target="_self" title="METRIC_1:def.22">Def23</a></span>;<br/>
<a NAME="E7:71"/>
<font color="Maroon" title="c4">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:71"><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="E8:71"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:71"><span class="lab"><font color="Green" title="E53">A4</font></span></a>, <a class="txt" href="metric_1.html#E43"><span class="lab"><font color="Green" title="E36">Lm8</font></span></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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