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<b>let </b><font color="Maroon">M</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="metric_1.html#L1">MetrStruct</a> ;<br/><b>let </b><font color="Maroon">p</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font>, <font color="Maroon">r</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font>, <font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font>;<br/>



<a NAME="E1:37"/><i><font color="Green">E24</font></i>: 
(  not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="sub_metr.html#K4" target="_self">close_dist_Segment</a> <font color="Maroon">p</font>,<font color="Maroon">r</font> or <font color="Maroon">x</font> <a href="sub_metr.html#R1" target="_self">is_between</a> <font color="Maroon">p</font>,<font color="Maroon">r</font> or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:37_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="sub_metr.html#K4" target="_self">close_dist_Segment</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>
 ;<br/>

<a NAME="E2:37_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">q</font> where <font color="Olive">q</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font> : <font color="Olive">q</font> <a href="sub_metr.html#R1" target="_self">is_between</a> <font color="Maroon">p</font>,<font color="Maroon">r</font> </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">r</font></span><a href="domain_1.html#K7">}</a></span>
 
;<br/>
<a NAME="E3:37_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">q</font> where <font color="Olive">q</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font> : <font color="Olive">q</font> <a href="sub_metr.html#R1" target="_self">is_between</a> <font color="Maroon">p</font>,<font color="Maroon">r</font> </span>}</span>  or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">r</font></span><a href="domain_1.html#K7">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E4:37_1"/><b>then </b>
(  ex <font color="Olive">q</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">q</font> &amp; <font color="Olive">q</font> <a href="sub_metr.html#R1" target="_self">is_between</a> <font color="Maroon">p</font>,<font color="Maroon">r</font> ) or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> )
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<b>hence </b><a NAME="E5:37_1"/>
( <font color="Maroon">x</font> <a href="sub_metr.html#R1" target="_self">is_between</a> <font color="Maroon">p</font>,<font color="Maroon">r</font> or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> )
 ;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:37_2"/>
( <font color="Maroon">x</font> <a href="sub_metr.html#R1" target="_self">is_between</a> <font color="Maroon">p</font>,<font color="Maroon">r</font> or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> )
 ;<br/><a NAME="E2:37_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">q</font> where <font color="Olive">q</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font> : <font color="Olive">q</font> <a href="sub_metr.html#R1" target="_self">is_between</a> <font color="Maroon">p</font>,<font color="Maroon">r</font> </span>}</span>  or <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">r</font></span><a href="domain_1.html#K7">}</a></span> )
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E3:37_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">q</font> where <font color="Olive">q</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">M</font> : <font color="Olive">q</font> <a href="sub_metr.html#R1" target="_self">is_between</a> <font color="Maroon">p</font>,<font color="Maroon">r</font> </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">r</font></span><a href="domain_1.html#K7">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E4:37_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="sub_metr.html#K4" target="_self">close_dist_Segment</a> <font color="Maroon">p</font>,<font color="Maroon">r</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:37"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="sub_metr.html#K4" target="_self">close_dist_Segment</a> <font color="Maroon">p</font>,<font color="Maroon">r</font> iff ( <font color="Maroon">x</font> <a href="sub_metr.html#R1" target="_self">is_between</a> <font color="Maroon">p</font>,<font color="Maroon">r</font> or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> or <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> ) )
 <b>by </b><i/>;<br/>


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