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<b>let </b><font color="Maroon">V</font> be   <a href="rlvect_1.html#NM2">RealLinearSpace</a>;<br/><b>let </b><font color="Maroon">w</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>;<br/><b>let </b><font color="Maroon">y</font> be   <a href="rlvect_1.html#NM1">VECTOR</a> of <font color="Maroon">V</font>;<br/>





<b>assume </b><a NAME="E1:95"/><i><font color="Green">E45</font></i>: 
 <a href="analmetr.html#R1">Gen</a> <font color="Maroon">w</font>,<font color="Maroon">y</font>
 ;<br/>

<b>set </b><font color="Maroon">X</font> =  <a href="geomtrap.html#K7" target="_self">DTrSpace</a> <font color="Maroon">V</font>,<font color="Maroon">w</font>,<font color="Maroon">y</font>;<br/>
<a NAME="E2:95"/>
for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font>, <font color="Olive">d</font>, <font color="Olive">a1</font>, <font color="Olive">b1</font>, <font color="Olive">c1</font>, <font color="Olive">d1</font>, <font color="Olive">p</font>, <font color="Olive">q</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="geomtrap.html#K7" target="_self">DTrSpace</a> <font color="Maroon">V</font>,<font color="Maroon">w</font>,<font color="Maroon">y</font></span>)</span> holds <br/> ( <font color="Olive">a</font> <a href="geomtrap.html#K9" target="_self">#</a> <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> <a href="geomtrap.html#K9" target="_self">#</a> <font color="Olive">a</font> &amp; <font color="Olive">a</font> <a href="geomtrap.html#K9" target="_self">#</a> <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> &amp; <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="geomtrap.html#K9" target="_self">#</a> <font color="Olive">b</font></span>)</span> <a href="geomtrap.html#K9" target="_self">#</a> <span class="p1">(<span class="default"><font color="Olive">c</font> <a href="geomtrap.html#K9" target="_self">#</a> <font color="Olive">d</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">a</font> <a href="geomtrap.html#K9" target="_self">#</a> <font color="Olive">c</font></span>)</span> <a href="geomtrap.html#K9" target="_self">#</a> <span class="p1">(<span class="default"><font color="Olive">b</font> <a href="geomtrap.html#K9" target="_self">#</a> <font color="Olive">d</font></span>)</span> &amp;  ex <font color="Olive">p</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="geomtrap.html#K7" target="_self">DTrSpace</a> <font color="Maroon">V</font>,<font color="Maroon">w</font>,<font color="Maroon">y</font></span>)</span> st <font color="Olive">p</font> <a href="geomtrap.html#K9" target="_self">#</a> <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> &amp; ( <font color="Olive">a</font> <a href="geomtrap.html#K9" target="_self">#</a> <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">a</font> <a href="geomtrap.html#K9" target="_self">#</a> <font color="Olive">c</font> implies <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">c</font> ) &amp; ( <font color="Olive">a</font>,<font color="Olive">b</font> <a href="analoaf.html#R2">//</a> <font color="Olive">c</font>,<font color="Olive">d</font> implies <font color="Olive">a</font>,<font color="Olive">b</font> <a href="analoaf.html#R2">//</a> <font color="Olive">a</font> <a href="geomtrap.html#K9" target="_self">#</a> <font color="Olive">c</font>,<font color="Olive">b</font> <a href="geomtrap.html#K9" target="_self">#</a> <font color="Olive">d</font> ) &amp; (  not <font color="Olive">a</font>,<font color="Olive">b</font> <a href="analoaf.html#R2">//</a> <font color="Olive">c</font>,<font color="Olive">d</font> or <font color="Olive">a</font>,<font color="Olive">b</font> <a href="analoaf.html#R2">//</a> <font color="Olive">a</font> <a href="geomtrap.html#K9" target="_self">#</a> <font color="Olive">d</font>,<font color="Olive">b</font> <a href="geomtrap.html#K9" target="_self">#</a> <font color="Olive">c</font> or <font color="Olive">a</font>,<font color="Olive">b</font> <a href="analoaf.html#R2">//</a> <font color="Olive">b</font> <a href="geomtrap.html#K9" target="_self">#</a> <font color="Olive">c</font>,<font color="Olive">a</font> <a href="geomtrap.html#K9" target="_self">#</a> <font color="Olive">d</font> ) &amp; ( <font color="Olive">a</font>,<font color="Olive">b</font> <a href="analoaf.html#R2">//</a> <font color="Olive">c</font>,<font color="Olive">d</font> &amp; <font color="Olive">a</font> <a href="geomtrap.html#K9" target="_self">#</a> <font color="Olive">a1</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> &amp; <font color="Olive">b</font> <a href="geomtrap.html#K9" target="_self">#</a> <font color="Olive">b1</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> &amp; <font color="Olive">c</font> <a href="geomtrap.html#K9" target="_self">#</a> <font color="Olive">c1</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> &amp; <font color="Olive">d</font> <a href="geomtrap.html#K9" target="_self">#</a> <font color="Olive">d1</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> implies <font color="Olive">a1</font>,<font color="Olive">b1</font> <a href="analoaf.html#R2">//</a> <font color="Olive">c1</font>,<font color="Olive">d1</font> ) &amp; ( <font color="Olive">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">q</font> &amp; <font color="Olive">p</font>,<font color="Olive">q</font> <a href="analoaf.html#R2">//</a> <font color="Olive">a</font>,<font color="Olive">a1</font> &amp; <font color="Olive">p</font>,<font color="Olive">q</font> <a href="analoaf.html#R2">//</a> <font color="Olive">b</font>,<font color="Olive">b1</font> &amp; <font color="Olive">p</font>,<font color="Olive">q</font> <a href="analoaf.html#R2">//</a> <font color="Olive">c</font>,<font color="Olive">c1</font> &amp; <font color="Olive">p</font>,<font color="Olive">q</font> <a href="analoaf.html#R2">//</a> <font color="Olive">d</font>,<font color="Olive">d1</font> &amp; <font color="Olive">a</font>,<font color="Olive">b</font> <a href="analoaf.html#R2">//</a> <font color="Olive">c</font>,<font color="Olive">d</font> implies <font color="Olive">a1</font>,<font color="Olive">b1</font> <a href="analoaf.html#R2">//</a> <font color="Olive">c1</font>,<font color="Olive">d1</font> ) &amp; ( <font color="Olive">a</font>,<font color="Olive">b</font> <a href="analoaf.html#R2">//</a> <font color="Olive">b</font>,<font color="Olive">c</font> implies ( <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <font color="Olive">c</font> ) ) &amp; ( <font color="Olive">a</font>,<font color="Olive">b</font> <a href="analoaf.html#R2">//</a> <font color="Olive">a1</font>,<font color="Olive">b1</font> &amp; <font color="Olive">a</font>,<font color="Olive">b</font> <a href="analoaf.html#R2">//</a> <font color="Olive">c1</font>,<font color="Olive">d1</font> &amp; <font color="Olive">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b</font> implies <font color="Olive">a1</font>,<font color="Olive">b1</font> <a href="analoaf.html#R2">//</a> <font color="Olive">c1</font>,<font color="Olive">d1</font> ) &amp; ( <font color="Olive">a</font>,<font color="Olive">b</font> <a href="analoaf.html#R2">//</a> <font color="Olive">c</font>,<font color="Olive">d</font> implies ( <font color="Olive">c</font>,<font color="Olive">d</font> <a href="analoaf.html#R2">//</a> <font color="Olive">a</font>,<font color="Olive">b</font> &amp; <font color="Olive">b</font>,<font color="Olive">a</font> <a href="analoaf.html#R2">//</a> <font color="Olive">d</font>,<font color="Olive">c</font> ) ) &amp;  ex <font color="Olive">d</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="geomtrap.html#K7" target="_self">DTrSpace</a> <font color="Maroon">V</font>,<font color="Maroon">w</font>,<font color="Maroon">y</font></span>)</span> st <br/>( <font color="Olive">a</font>,<font color="Olive">b</font> <a href="analoaf.html#R2">//</a> <font color="Olive">c</font>,<font color="Olive">d</font> or <font color="Olive">a</font>,<font color="Olive">b</font> <a href="analoaf.html#R2">//</a> <font color="Olive">d</font>,<font color="Olive">c</font> ) &amp; ( <font color="Olive">a</font>,<font color="Olive">b</font> <a href="analoaf.html#R2">//</a> <font color="Olive">c</font>,<font color="Olive">p</font> &amp; <font color="Olive">a</font>,<font color="Olive">b</font> <a href="analoaf.html#R2">//</a> <font color="Olive">c</font>,<font color="Olive">q</font> &amp; not <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> implies <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Olive">q</font> ) )
 
<b>by </b><i>, , <a class="ref" href="geomtrap.html#T16" target="_self">Th16</a>, <a class="txt" href="geomtrap.html#E20"><i><font color="Green">Lemma69</font></i></a>, , , , , <a class="txt" href="geomtrap.html#E21"><i><font color="Green">Lemma74</font></i></a>, <a class="txt" href="geomtrap.html#E22"><i><font color="Green">Lemma75</font></i></a>, <a class="txt" href="geomtrap.html#E23"><i><font color="Green">Lemma76</font></i></a>, <a class="txt" href="geomtrap.html#E24"><i><font color="Green">Lemma77</font></i></a>, <a class="txt" href="geomtrap.html#E25"><i><font color="Green">Lemma78</font></i></a>, <a class="txt" href="geomtrap.html#E26"><i><font color="Green">Lemma79</font></i></a>, <a class="txt" href="geomtrap.html#E27"><i><font color="Green">Lemma80</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:95"/>
 <a href="geomtrap.html#K7" target="_self">DTrSpace</a> <font color="Maroon">V</font>,<font color="Maroon">w</font>,<font color="Maroon">y</font> is   <a href="geomtrap.html#NM1">MidOrdTrapSpace</a>
 <b>by </b><i/>;<br/>


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