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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="Red">::  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>, <font color="Olive" title="b4">s</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="sub_metr.html#R1" title="SUB_METR:pred.1">is_between</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b3">r</font> &amp; <font color="Olive" title="b3">r</font> <a href="sub_metr.html#R1" title="SUB_METR:pred.1">is_between</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b4">s</font> holds <br/>( <font color="Olive" title="b2">q</font> <a href="sub_metr.html#R1" title="SUB_METR:pred.1">is_between</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b4">s</font> &amp; <font color="Olive" title="b3">r</font> <a href="sub_metr.html#R1" title="SUB_METR:pred.1">is_between</a> <font color="Olive" title="b2">q</font>,<font color="Olive" title="b4">s</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>, <font color="Maroon" title="c6">s</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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">q</font> <a href="sub_metr.html#R1" title="SUB_METR:pred.1">is_between</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c5">r</font> &amp; <font color="Maroon" title="c5">r</font> <a href="sub_metr.html#R1" title="SUB_METR:pred.1">is_between</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c6">s</font> implies ( <font color="Maroon" title="c4">q</font> <a href="sub_metr.html#R1" title="SUB_METR:pred.1">is_between</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c6">s</font> &amp; <font color="Maroon" title="c5">r</font> <a href="sub_metr.html#R1" title="SUB_METR:pred.1">is_between</a> <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c6">s</font> ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:66"/><span class="lab"><font color="Green" title="E17">A1</font></span>: 
<font color="Maroon" title="c4">q</font> <a href="sub_metr.html#R1" title="SUB_METR: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="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c5">r</font> <a href="sub_metr.html#R1" title="SUB_METR:pred.1">is_between</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c6">s</font> or ( <font color="Maroon" title="c4">q</font> <a href="sub_metr.html#R1" title="SUB_METR:pred.1">is_between</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c6">s</font> &amp; <font color="Maroon" title="c5">r</font> <a href="sub_metr.html#R1" title="SUB_METR:pred.1">is_between</a> <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c6">s</font> ) )</span><br/>

<a NAME="E2:66"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A2</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="ref" href="sub_metr.html#D7" target="_self" title="SUB_METR:def.7">Def7</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:66"/><span class="lab"><font color="Green" title="E19">A3</font></span>: 
<font color="Maroon" title="c5">r</font> <a href="sub_metr.html#R1" title="SUB_METR:pred.1">is_between</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c6">s</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">q</font> <a href="sub_metr.html#R1" title="SUB_METR:pred.1">is_between</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c6">s</font> &amp; <font color="Maroon" title="c5">r</font> <a href="sub_metr.html#R1" title="SUB_METR:pred.1">is_between</a> <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c6">s</font> )</span><br/>

<a NAME="E4:66"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A4</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="c6">s</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="c6">s</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="sub_metr.html#D7" target="_self" title="SUB_METR:def.7">Def7</a></span>;<br/>
<a NAME="E5:66"/>
 <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="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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="#E1:66"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="ref" href="sub_metr.html#D7" target="_self" title="SUB_METR:def.7">Def7</a></span>;<br/>
<a NAME="E6:66"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A5</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="c6">s</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="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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="c6">s</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:66"><span class="lab"><font color="Green" title="E19">A3</font></span></a>, <a class="ref" href="sub_metr.html#D7" target="_self" title="SUB_METR:def.7">Def7</a></span>;<br/>
<a NAME="E7:66"/>
 <a href="metric_1.html#K4" title="METRIC_1:func.4">dist</a> <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c6">s</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="c4">q</font>,<font color="Maroon" title="c5">r</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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="c6">s</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#T4" title="METRIC_1:th.4">METRIC_1:4</a></span>;<br/>
<a NAME="E8:66"/><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="c6">s</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="c3">p</font>,<font color="Maroon" title="c4">q</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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="c6">s</font></span>)</span> &amp; <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="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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="c6">s</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="c4">q</font>,<font color="Maroon" title="c5">r</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<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="c6">s</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="metric_1.html#T4" title="METRIC_1:th.4">METRIC_1:4</a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/>
<a NAME="E9:66"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A6</font></span>: 
<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="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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="c6">s</font></span>)</span> <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="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</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="c5">r</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<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="c6">s</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:66"><span class="lab"><font color="Green" title="E21">A5</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/>
<a NAME="E10:66"/><span class="lab"><font color="Green" title="E23">A7</font></span>: 
<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="#E1:66"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="ref" href="sub_metr.html#D7" target="_self" title="SUB_METR:def.7">Def7</a></span>;<br/>
<a NAME="E11:66"/><span class="kw">then </span>
<font color="Maroon" title="c4">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:66"><span class="lab"><font color="Green" title="E21">A5</font></span></a>, <a class="ref" href="metric_1.html#T2" title="METRIC_1:th.2">METRIC_1:2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:66"/>
( <font color="Maroon" title="c4">q</font> <a href="sub_metr.html#R1" title="SUB_METR:pred.1">is_between</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c6">s</font> &amp; <font color="Maroon" title="c5">r</font> <a href="sub_metr.html#R1" title="SUB_METR:pred.1">is_between</a> <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c6">s</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:66"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="txt" href="#E10:66"><span class="lab"><font color="Green" title="E23">A7</font></span></a>, <a class="txt" href="#E4:66"><span class="lab"><font color="Green" title="E20">A4</font></span></a>, <a class="txt" href="#E6:66"><span class="lab"><font color="Green" title="E21">A5</font></span></a>, <a class="txt" href="#E9:66"><span class="lab"><font color="Green" title="E22">A6</font></span></a>, <a class="ref" href="sub_metr.html#D7" target="_self" title="SUB_METR:def.7">Def7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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