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<b>let </b><font color="Maroon" title="c1">M</font> be   <a href="midsp_1.html#NM1" title="MIDSP_1:NM.1">MidSp</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font> holds <br/> (  <a href="midsp_1.html#K10" title="MIDSP_1:func.10">vect</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="midsp_1.html#K10" title="MIDSP_1:func.10">vect</a> <font color="Olive" title="b3">r</font>,<font color="Olive" title="b4">s</font> iff <font color="Olive" title="b1">p</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Olive" title="b4">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">q</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Olive" title="b3">r</font> )</span><br/><b>let </b><font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c3">q</font>, <font color="Maroon" title="c4">r</font>, <font color="Maroon" title="c5">s</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="midsp_1.html#K10" title="MIDSP_1:func.10">vect</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="midsp_1.html#K10" title="MIDSP_1:func.10">vect</a> <font color="Maroon" title="c4">r</font>,<font color="Maroon" title="c5">s</font> iff <font color="Maroon" title="c2">p</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c5">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c4">r</font> )</span><br/>









<b>thus </b><a NAME="E1:34"/>
(  <a href="midsp_1.html#K10" title="MIDSP_1:func.10">vect</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="midsp_1.html#K10" title="MIDSP_1:func.10">vect</a> <font color="Maroon" title="c4">r</font>,<font color="Maroon" title="c5">s</font> implies <font color="Maroon" title="c2">p</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c5">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c4">r</font> )
 <b>by </b><i><a class="ref" href="midsp_1.html#T57" title="MIDSP_1:th.57">MIDSP_1:57</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">p</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c5">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c4">r</font> implies  <a href="midsp_1.html#K10" title="MIDSP_1:func.10">vect</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="midsp_1.html#K10" title="MIDSP_1:func.10">vect</a> <font color="Maroon" title="c4">r</font>,<font color="Maroon" title="c5">s</font> )</span><br/>

<b>thus </b><a NAME="E2:34"/>
( <font color="Maroon" title="c2">p</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c5">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c4">r</font> implies  <a href="midsp_1.html#K10" title="MIDSP_1:func.10">vect</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="midsp_1.html#K10" title="MIDSP_1:func.10">vect</a> <font color="Maroon" title="c4">r</font>,<font color="Maroon" title="c5">s</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><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" title="c2">p</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c5">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q</font> <a href="midsp_1.html#K3" title="MIDSP_1:func.3">@</a> <font color="Maroon" title="c4">r</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="midsp_1.html#K10" title="MIDSP_1:func.10">vect</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="midsp_1.html#K10" title="MIDSP_1:func.10">vect</a> <font color="Maroon" title="c4">r</font>,<font color="Maroon" title="c5">s</font></span><br/>

<a NAME="E2:34_1"/><b>then </b>
<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font> <a href="midsp_1.html#R1" title="MIDSP_1:pred.1">@@</a> <font color="Maroon" title="c4">r</font>,<font color="Maroon" title="c5">s</font>
 
<b>by </b><i><a class="ref" href="midsp_1.html#D5" title="MIDSP_1:def.5">MIDSP_1:def 5</a></i>;<br/>
<a NAME="E3:34_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="midsp_1.html#R2" title="MIDSP_1:pred.2">##</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">r</font>,<font color="Maroon" title="c5">s</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 
<b>by </b><i><a class="ref" href="midsp_1.html#T31" title="MIDSP_1:th.31">MIDSP_1:31</a></i>;<br/>
<b>hence </b><a NAME="E4:34_1"/>
 <a href="midsp_1.html#K10" title="MIDSP_1:func.10">vect</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="midsp_1.html#K10" title="MIDSP_1:func.10">vect</a> <font color="Maroon" title="c4">r</font>,<font color="Maroon" title="c5">s</font>
 <b>by </b><i><a class="ref" href="midsp_1.html#T56" title="MIDSP_1:th.56">MIDSP_1:56</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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


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