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<b>consider </b><font color="Maroon" title="c1">V</font> being   <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</a><b> such that </b><br/><a NAME="E2:94_1_1"/><i><font color="Green" title="E82">A1</font></i>: 
 ex <font color="Olive" title="b1">w</font>, <font color="Olive" title="b2">y</font> being  <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font> st  <a href="analmetr.html#R1" title="ANALMETR:pred.1">Gen</a> <font color="Olive" title="b1">w</font>,<font color="Olive" title="b2">y</font>
 <b>by </b><i><a class="ref" href="analmetr.html#T7" title="ANALMETR:th.7">ANALMETR:7</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c2">w</font>, <font color="Maroon" title="c3">y</font> being   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font><b> such that </b><br/><a NAME="E4:94_1_1"/><i><font color="Green" title="E83">A2</font></i>: 
 <a href="analmetr.html#R1" title="ANALMETR:pred.1">Gen</a> <font color="Maroon" title="c2">w</font>,<font color="Maroon" title="c3">y</font>
 <b>by </b><i><a class="txt" href="#E2:94_1_1"><i><font color="Green" title="E82">A1</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c4">X</font> =  <a href="geomtrap.html#K7" title="GEOMTRAP:func.7">DTrSpace</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">w</font>,<font color="Maroon" title="c3">y</font>;<br/>
<a NAME="E5:94_1_1"/>
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font>, <font color="Olive" title="b4">d</font>, <font color="Olive" title="b5">a1</font>, <font color="Olive" title="b6">b1</font>, <font color="Olive" title="b7">c1</font>, <font color="Olive" title="b8">d1</font>, <font color="Olive" title="b9">p</font>, <font color="Olive" title="b10">q</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="geomtrap.html#K7" title="GEOMTRAP:func.7">DTrSpace</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">w</font>,<font color="Maroon" title="c3">y</font></span>)</span> holds <br/> ( <font color="Olive" title="b1">a</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Olive" title="b1">a</font> &amp; <font color="Olive" title="b1">a</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> &amp; <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">c</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Olive" title="b4">d</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Olive" title="b3">c</font></span>)</span> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">b</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Olive" title="b4">d</font></span>)</span> &amp;  ex <font color="Olive" title="b11">p</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="geomtrap.html#K7" title="GEOMTRAP:func.7">DTrSpace</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">w</font>,<font color="Maroon" title="c3">y</font></span>)</span> st <font color="Olive" title="b11">p</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> &amp; ( <font color="Olive" title="b1">a</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Olive" title="b3">c</font> implies <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">c</font> ) &amp; ( <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b3">c</font>,<font color="Olive" title="b4">d</font> implies <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b1">a</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Olive" title="b3">c</font>,<font color="Olive" title="b2">b</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Olive" title="b4">d</font> ) &amp; (  not <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b3">c</font>,<font color="Olive" title="b4">d</font> or <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b1">a</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Olive" title="b4">d</font>,<font color="Olive" title="b2">b</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Olive" title="b3">c</font> or <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b2">b</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Olive" title="b3">c</font>,<font color="Olive" title="b1">a</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Olive" title="b4">d</font> ) &amp; ( <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b3">c</font>,<font color="Olive" title="b4">d</font> &amp; <font color="Olive" title="b1">a</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Olive" title="b5">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b9">p</font> &amp; <font color="Olive" title="b2">b</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Olive" title="b6">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b9">p</font> &amp; <font color="Olive" title="b3">c</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Olive" title="b7">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b9">p</font> &amp; <font color="Olive" title="b4">d</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Olive" title="b8">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b9">p</font> implies <font color="Olive" title="b5">a1</font>,<font color="Olive" title="b6">b1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b7">c1</font>,<font color="Olive" title="b8">d1</font> ) &amp; ( <font color="Olive" title="b9">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b10">q</font> &amp; <font color="Olive" title="b9">p</font>,<font color="Olive" title="b10">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b5">a1</font> &amp; <font color="Olive" title="b9">p</font>,<font color="Olive" title="b10">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b2">b</font>,<font color="Olive" title="b6">b1</font> &amp; <font color="Olive" title="b9">p</font>,<font color="Olive" title="b10">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b3">c</font>,<font color="Olive" title="b7">c1</font> &amp; <font color="Olive" title="b9">p</font>,<font color="Olive" title="b10">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b4">d</font>,<font color="Olive" title="b8">d1</font> &amp; <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b3">c</font>,<font color="Olive" title="b4">d</font> implies <font color="Olive" title="b5">a1</font>,<font color="Olive" title="b6">b1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b7">c1</font>,<font color="Olive" title="b8">d1</font> ) &amp; ( <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font> implies ( <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">c</font> ) ) &amp; ( <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b5">a1</font>,<font color="Olive" title="b6">b1</font> &amp; <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b7">c1</font>,<font color="Olive" title="b8">d1</font> &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">b</font> implies <font color="Olive" title="b5">a1</font>,<font color="Olive" title="b6">b1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b7">c1</font>,<font color="Olive" title="b8">d1</font> ) &amp; ( <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b3">c</font>,<font color="Olive" title="b4">d</font> implies ( <font color="Olive" title="b3">c</font>,<font color="Olive" title="b4">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> &amp; <font color="Olive" title="b2">b</font>,<font color="Olive" title="b1">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b4">d</font>,<font color="Olive" title="b3">c</font> ) ) &amp;  ex <font color="Olive" title="b11">d</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="geomtrap.html#K7" title="GEOMTRAP:func.7">DTrSpace</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">w</font>,<font color="Maroon" title="c3">y</font></span>)</span> st <br/>( <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b3">c</font>,<font color="Olive" title="b11">d</font> or <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b11">d</font>,<font color="Olive" title="b3">c</font> ) &amp; ( <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b3">c</font>,<font color="Olive" title="b9">p</font> &amp; <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b3">c</font>,<font color="Olive" title="b10">q</font> &amp; not <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font> implies <font color="Olive" title="b9">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b10">q</font> ) )
 
<b>by </b><i><a class="txt" href="#E4:94_1_1"><i><font color="Green" title="E83">A2</font></i></a>, <a class="txt" href="geomtrap.html#E82"><i><font color="Green" title="E67">Lm21</font></i></a>, <a class="txt" href="geomtrap.html#E83"><i><font color="Green" title="E68">Lm22</font></i></a>, <a class="txt" href="geomtrap.html#E84"><i><font color="Green" title="E69">Lm23</font></i></a>, <a class="txt" href="geomtrap.html#E85"><i><font color="Green" title="E70">Lm24</font></i></a>, <a class="txt" href="geomtrap.html#E86"><i><font color="Green" title="E71">Lm25</font></i></a>, <a class="txt" href="geomtrap.html#E87"><i><font color="Green" title="E72">Lm26</font></i></a>, <a class="txt" href="geomtrap.html#E88"><i><font color="Green" title="E73">Lm27</font></i></a>, <a class="txt" href="geomtrap.html#E89"><i><font color="Green" title="E74">Lm28</font></i></a>, <a class="txt" href="geomtrap.html#E90"><i><font color="Green" title="E75">Lm29</font></i></a>, <a class="txt" href="geomtrap.html#E91"><i><font color="Green" title="E76">Lm30</font></i></a>, <a class="txt" href="geomtrap.html#E92"><i><font color="Green" title="E77">Lm31</font></i></a>, <a class="txt" href="geomtrap.html#E93"><i><font color="Green" title="E78">Lm32</font></i></a>, <a class="txt" href="geomtrap.html#E94"><i><font color="Green" title="E79">Lm33</font></i></a>, <a class="txt" href="geomtrap.html#E95"><i><font color="Green" title="E80">Lm34</font></i></a></i>;<br/>
<a NAME="E6:94_1_1"/><b>then </b>
 <a href="geomtrap.html#K7" title="GEOMTRAP:func.7">DTrSpace</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c2">w</font>,<font color="Maroon" title="c3">y</font> is <a href="geomtrap.html#V2" title="GEOMTRAP:attr.2">MidOrdTrapSpace-like</a>
 
<b>by </b><i><a class="ref" href="geomtrap.html#D13" target="_self" title="GEOMTRAP:def.13">Def13</a></i>;<br/>
<b>hence </b><a NAME="E7:94_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  <a href="geomtrap.html#L1" title="GEOMTRAP:struct.1">AfMidStruct</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="geomtrap.html#V1" title="GEOMTRAP:attr.1">strict</a> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="geomtrap.html#V2" title="GEOMTRAP:attr.2">MidOrdTrapSpace-like</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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