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<b>let </b><font color="Maroon" title="c2">M</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="metric_1.html#NM1" title="METRIC_1:NM.1">MetrSpace</a>;<br/><b>let </b><font color="Maroon" title="c3">p</font>, <font color="Maroon" title="c4">r</font>, <font color="Maroon" title="c5">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">M</font>;<br/>



<a NAME="E1:35"/><i><font color="Green" title="E19">A1</font></i>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="sub_metr.html#K3" title="SUB_METR:func.3">open_dist_Segment</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">r</font> implies <font color="Maroon" title="c5">x</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="c4">r</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:35_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="sub_metr.html#K3" title="SUB_METR:func.3">open_dist_Segment</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">r</font>
 ;<br/>

<a NAME="E2:35_1"/><b>then </b>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">q</font> where <font color="Olive" title="b1">q</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">M</font> : <font color="Olive" title="b1">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="c4">r</font> </span>}</span> 
 
;<br/>
<a NAME="E3:35_1"/><b>then </b>
 ex <font color="Olive" title="b1">q</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">M</font> st <br/>( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">q</font> &amp; <font color="Olive" title="b1">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="c4">r</font> )
 
;<br/>
<b>hence </b><a NAME="E4:35_1"/>
<font color="Maroon" title="c5">x</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="c4">r</font>
 ;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E2:35"/>
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="sub_metr.html#K3" title="SUB_METR:func.3">open_dist_Segment</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">r</font> iff <font color="Maroon" title="c5">x</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="c4">r</font> )
 <b>by </b><i><a class="txt" href="sub_metr.html#E1:35"><i><font color="Green" title="E19">A1</font></i></a></i>;<br/>


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