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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">M</font>, <a href="midsp_1.html#K17" title="MIDSP_1:func.17">vectgroup</a> <font color="Maroon" title="c1">M</font> <a href="midsp_2.html#R1" title="MIDSP_2:pred.1">are_associated_wrp</a>  <a href="midsp_2.html#K5" title="MIDSP_2:func.5">vect</a> <font color="Maroon" title="c1">M</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">M</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="midsp_2.html#D2" target="_self" title="MIDSP_2:def.2">MIDSP_2:def 2</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">q</font>, <font color="Olive" title="b2">r</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/> ( <font color="Maroon" title="c2">p</font> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">r</font> iff <span class="p1">(<span class="default"><a href="midsp_2.html#K5" title="MIDSP_2:func.5">vect</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c2">p</font>,<font color="Olive" title="b2">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="midsp_2.html#K5" title="MIDSP_2:func.5">vect</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Olive" title="b2">r</font>,<font color="Olive" title="b1">q</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">q</font>, <font color="Maroon" title="c4">r</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">p</font> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <font color="Maroon" title="c3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">r</font> iff <span class="p1">(<span class="default"><a href="midsp_2.html#K5" title="MIDSP_2:func.5">vect</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="midsp_2.html#K5" title="MIDSP_2:func.5">vect</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">r</font>,<font color="Maroon" title="c3">q</font> )</span><br/>







<span class="kw">set </span><font color="Maroon" title="c5">w</font> =  <a href="midsp_2.html#K5" title="MIDSP_2:func.5">vect</a> <font color="Maroon" title="c1">M</font>;<br/>
<a NAME="E1:39"/>
( <span class="p1">(<span class="default"><a href="midsp_2.html#K5" title="MIDSP_2:func.5">vect</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">r</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="c2">p</font>,<font color="Maroon" title="c4">r</font> &amp; <span class="p1">(<span class="default"><a href="midsp_2.html#K5" title="MIDSP_2:func.5">vect</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">r</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="c3">q</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="midsp_2.html#D8" target="_self" title="MIDSP_2:def.8">Def8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:39"/>
( <font color="Maroon" title="c2">p</font> <a href="midsp_1.html#K1" title="MIDSP_1:func.1">@</a> <font color="Maroon" title="c3">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">r</font> iff <span class="p1">(<span class="default"><a href="midsp_2.html#K5" title="MIDSP_2:func.5">vect</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="midsp_2.html#K5" title="MIDSP_2:func.5">vect</a> <font color="Maroon" title="c1">M</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c4">r</font>,<font color="Maroon" title="c3">q</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="midsp_2.html#T26" target="_self" title="MIDSP_2:th.26">Th26</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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