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<b>let </b><font color="Maroon">M</font> be   <a href="midsp_1.html#NM1">MidSp</a>;<br/><b>let </b><font color="Maroon">p</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font>;<br/><b>let </b><font color="Maroon">q</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font>;<br/><b>let </b><font color="Maroon">r</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font>;<br/><b>let </b><font color="Maroon">s</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">M</font>;<br/>









<b>thus </b><a NAME="E1:34"/>
(  <a href="midsp_1.html#K11">vect</a> <font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="midsp_1.html#K11">vect</a> <font color="Maroon">r</font>,<font color="Maroon">s</font> implies <font color="Maroon">p</font> <a href="midsp_1.html#K4">@</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="midsp_1.html#K4">@</a> <font color="Maroon">r</font> )
 <b>by </b><i><a class="ref" href="midsp_1.html#T57">MIDSP_1:57</a></i>;<br/>

<b>thus </b><a NAME="E2:34"/>
( <font color="Maroon">p</font> <a href="midsp_1.html#K4">@</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="midsp_1.html#K4">@</a> <font color="Maroon">r</font> implies  <a href="midsp_1.html#K11">vect</a> <font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="midsp_1.html#K11">vect</a> <font color="Maroon">r</font>,<font color="Maroon">s</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:34_1"/>
<font color="Maroon">p</font> <a href="midsp_1.html#K4">@</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="midsp_1.html#K4">@</a> <font color="Maroon">r</font>
 ;<br/>

<a NAME="E2:34_1"/><b>then </b>
<font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="midsp_1.html#R1">@@</a> <font color="Maroon">r</font>,<font color="Maroon">s</font>
 
<b>by </b><i><a class="ref" href="midsp_1.html#D5">MIDSP_1:def 5</a></i>;<br/>
<a NAME="E3:34_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">p</font>,<font color="Maroon">q</font></span><a href="domain_1.html#K1">]</a></span> <a href="midsp_1.html#R2">##</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">s</font></span><a href="domain_1.html#K1">]</a></span>
 
<b>by </b><i><a class="ref" href="midsp_1.html#T31">MIDSP_1:31</a></i>;<br/>
<b>hence </b><a NAME="E4:34_1"/>
 <a href="midsp_1.html#K11">vect</a> <font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="midsp_1.html#K11">vect</a> <font color="Maroon">r</font>,<font color="Maroon">s</font>
 <b>by </b><i><a class="ref" href="midsp_1.html#T56">MIDSP_1:56</a></i>;<br/>


</div><b>end;</b></div>


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