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<span class="kw">let </span><font color="Maroon" title="c1">V</font> be   <a href="rlvect_1.html#NM2" title="RLVECT_1:NM.2">RealLinearSpace</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">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> holds <br/> <a href="geomtrap.html#K7" title="GEOMTRAP:func.7">DTrSpace</a> <font color="Maroon" title="c1">V</font>,<font color="Olive" title="b1">w</font>,<font color="Olive" title="b2">y</font> is   <a href="geomtrap.html#NM1" title="GEOMTRAP:NM.1">MidOrdTrapSpace</a></span><br/><span class="kw">let </span><font color="Maroon" title="c2">w</font>, <font color="Maroon" title="c3">y</font> be   <a href="rlvect_1.html#NM1" title="RLVECT_1:NM.1">VECTOR</a> of <font color="Maroon" title="c1">V</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <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> implies  <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#NM1" title="GEOMTRAP:NM.1">MidOrdTrapSpace</a> )</span><br/>





<span class="kw">set </span><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/>
<span class="kw">assume </span><a NAME="E1:99"/><span class="lab"><font color="Green" title="E82">A1</font></span>: 
 <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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <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#NM1" title="GEOMTRAP:NM.1">MidOrdTrapSpace</a></span><br/>

<a NAME="E2:99"/>
 <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> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="99_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c6">b</font>, <font color="Maroon" title="c7">c</font>, <font color="Maroon" title="c8">d</font>, <font color="Maroon" title="c9">a1</font>, <font color="Maroon" title="c10">b1</font>, <font color="Maroon" title="c11">c1</font>, <font color="Maroon" title="c12">d1</font>, <font color="Maroon" title="c13">p</font>, <font color="Maroon" title="c14">q</font> be   <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>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="geomtrap.html#D13" target="_self" title="GEOMTRAP:def.13">GEOMTRAP:def 13</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="c5">a</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c5">a</font> &amp; <font color="Maroon" title="c5">a</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">c</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c8">d</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c7">c</font></span>)</span> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c8">d</font></span>)</span> &amp;  ex <font color="Olive" title="b1">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="b1">p</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> &amp; ( <font color="Maroon" title="c5">a</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c7">c</font> implies <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">c</font> ) &amp; ( <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">d</font> implies <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c6">b</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c8">d</font> ) &amp; (  not <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">d</font> or <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c8">d</font>,<font color="Maroon" title="c6">b</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c7">c</font> or <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c5">a</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c8">d</font> ) &amp; ( <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">d</font> &amp; <font color="Maroon" title="c5">a</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c9">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">p</font> &amp; <font color="Maroon" title="c6">b</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c10">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">p</font> &amp; <font color="Maroon" title="c7">c</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c11">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">p</font> &amp; <font color="Maroon" title="c8">d</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c12">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">p</font> implies <font color="Maroon" title="c9">a1</font>,<font color="Maroon" title="c10">b1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">c1</font>,<font color="Maroon" title="c12">d1</font> ) &amp; ( <font color="Maroon" title="c13">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">q</font> &amp; <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">a1</font> &amp; <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c10">b1</font> &amp; <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c11">c1</font> &amp; <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">d</font>,<font color="Maroon" title="c12">d1</font> &amp; <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">d</font> implies <font color="Maroon" title="c9">a1</font>,<font color="Maroon" title="c10">b1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">c1</font>,<font color="Maroon" title="c12">d1</font> ) &amp; ( <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font> implies ( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> &amp; <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">c</font> ) ) &amp; ( <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">a1</font>,<font color="Maroon" title="c10">b1</font> &amp; <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">c1</font>,<font color="Maroon" title="c12">d1</font> &amp; <font color="Maroon" title="c5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font> implies <font color="Maroon" title="c9">a1</font>,<font color="Maroon" title="c10">b1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">c1</font>,<font color="Maroon" title="c12">d1</font> ) &amp; ( <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">d</font> implies ( <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> &amp; <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c5">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">d</font>,<font color="Maroon" title="c7">c</font> ) ) &amp;  ex <font color="Olive" title="b1">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="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Olive" title="b1">d</font> or <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b1">d</font>,<font color="Maroon" title="c7">c</font> ) &amp; ( <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c13">p</font> &amp; <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c14">q</font> &amp; not <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> implies <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">q</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E1:99_1"/>
( <font color="Maroon" title="c5">a</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c5">a</font> &amp; <font color="Maroon" title="c5">a</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c6">b</font></span>)</span> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">c</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c8">d</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">a</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c7">c</font></span>)</span> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">b</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c8">d</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="geomtrap.html#E87"><span class="lab"><font color="Green" title="E72">Lm27</font></span></a>, <a class="txt" href="geomtrap.html#E88"><span class="lab"><font color="Green" title="E73">Lm28</font></span></a>, <a class="txt" href="geomtrap.html#E89"><span class="lab"><font color="Green" title="E74">Lm29</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">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="b1">p</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> &amp; ( <font color="Maroon" title="c5">a</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c7">c</font> implies <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">c</font> ) &amp; ( <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">d</font> implies <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c6">b</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c8">d</font> ) &amp; (  not <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">d</font> or <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c8">d</font>,<font color="Maroon" title="c6">b</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c7">c</font> or <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c5">a</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c8">d</font> ) &amp; ( <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">d</font> &amp; <font color="Maroon" title="c5">a</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c9">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">p</font> &amp; <font color="Maroon" title="c6">b</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c10">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">p</font> &amp; <font color="Maroon" title="c7">c</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c11">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">p</font> &amp; <font color="Maroon" title="c8">d</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c12">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">p</font> implies <font color="Maroon" title="c9">a1</font>,<font color="Maroon" title="c10">b1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">c1</font>,<font color="Maroon" title="c12">d1</font> ) &amp; ( <font color="Maroon" title="c13">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">q</font> &amp; <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">a1</font> &amp; <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c10">b1</font> &amp; <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c11">c1</font> &amp; <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">d</font>,<font color="Maroon" title="c12">d1</font> &amp; <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">d</font> implies <font color="Maroon" title="c9">a1</font>,<font color="Maroon" title="c10">b1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">c1</font>,<font color="Maroon" title="c12">d1</font> ) &amp; ( <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font> implies ( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> &amp; <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">c</font> ) ) &amp; ( <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">a1</font>,<font color="Maroon" title="c10">b1</font> &amp; <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">c1</font>,<font color="Maroon" title="c12">d1</font> &amp; <font color="Maroon" title="c5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font> implies <font color="Maroon" title="c9">a1</font>,<font color="Maroon" title="c10">b1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">c1</font>,<font color="Maroon" title="c12">d1</font> ) &amp; ( <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">d</font> implies ( <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> &amp; <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c5">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">d</font>,<font color="Maroon" title="c7">c</font> ) ) &amp;  ex <font color="Olive" title="b1">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="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Olive" title="b1">d</font> or <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b1">d</font>,<font color="Maroon" title="c7">c</font> ) &amp; ( <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c13">p</font> &amp; <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c14">q</font> &amp; not <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> implies <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">q</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E2:99_1"/>
 ex <font color="Olive" title="b1">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="b1">p</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="geomtrap.html#E90"><span class="lab"><font color="Green" title="E75">Lm30</font></span></a></span>; <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="c5">a</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c7">c</font> implies <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">c</font> ) &amp; ( <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">d</font> implies <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c6">b</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c8">d</font> ) &amp; (  not <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">d</font> or <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c8">d</font>,<font color="Maroon" title="c6">b</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c7">c</font> or <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c5">a</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c8">d</font> ) &amp; ( <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">d</font> &amp; <font color="Maroon" title="c5">a</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c9">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">p</font> &amp; <font color="Maroon" title="c6">b</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c10">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">p</font> &amp; <font color="Maroon" title="c7">c</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c11">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">p</font> &amp; <font color="Maroon" title="c8">d</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c12">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">p</font> implies <font color="Maroon" title="c9">a1</font>,<font color="Maroon" title="c10">b1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">c1</font>,<font color="Maroon" title="c12">d1</font> ) &amp; ( <font color="Maroon" title="c13">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">q</font> &amp; <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">a1</font> &amp; <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c10">b1</font> &amp; <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c11">c1</font> &amp; <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">d</font>,<font color="Maroon" title="c12">d1</font> &amp; <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">d</font> implies <font color="Maroon" title="c9">a1</font>,<font color="Maroon" title="c10">b1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">c1</font>,<font color="Maroon" title="c12">d1</font> ) &amp; ( <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font> implies ( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> &amp; <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">c</font> ) ) &amp; ( <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">a1</font>,<font color="Maroon" title="c10">b1</font> &amp; <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">c1</font>,<font color="Maroon" title="c12">d1</font> &amp; <font color="Maroon" title="c5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font> implies <font color="Maroon" title="c9">a1</font>,<font color="Maroon" title="c10">b1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">c1</font>,<font color="Maroon" title="c12">d1</font> ) &amp; ( <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">d</font> implies ( <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> &amp; <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c5">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">d</font>,<font color="Maroon" title="c7">c</font> ) ) &amp;  ex <font color="Olive" title="b1">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="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Olive" title="b1">d</font> or <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b1">d</font>,<font color="Maroon" title="c7">c</font> ) &amp; ( <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c13">p</font> &amp; <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c14">q</font> &amp; not <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> implies <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">q</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E3:99_1"/>
( <font color="Maroon" title="c5">a</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c7">c</font> implies <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">c</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="geomtrap.html#E91"><span class="lab"><font color="Green" title="E76">Lm31</font></span></a></span>; <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="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">d</font> implies <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c6">b</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c8">d</font> ) &amp; (  not <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">d</font> or <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c8">d</font>,<font color="Maroon" title="c6">b</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c7">c</font> or <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c5">a</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c8">d</font> ) &amp; ( <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">d</font> &amp; <font color="Maroon" title="c5">a</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c9">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">p</font> &amp; <font color="Maroon" title="c6">b</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c10">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">p</font> &amp; <font color="Maroon" title="c7">c</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c11">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">p</font> &amp; <font color="Maroon" title="c8">d</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c12">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">p</font> implies <font color="Maroon" title="c9">a1</font>,<font color="Maroon" title="c10">b1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">c1</font>,<font color="Maroon" title="c12">d1</font> ) &amp; ( <font color="Maroon" title="c13">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">q</font> &amp; <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">a1</font> &amp; <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c10">b1</font> &amp; <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c11">c1</font> &amp; <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">d</font>,<font color="Maroon" title="c12">d1</font> &amp; <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">d</font> implies <font color="Maroon" title="c9">a1</font>,<font color="Maroon" title="c10">b1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">c1</font>,<font color="Maroon" title="c12">d1</font> ) &amp; ( <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font> implies ( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> &amp; <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">c</font> ) ) &amp; ( <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">a1</font>,<font color="Maroon" title="c10">b1</font> &amp; <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">c1</font>,<font color="Maroon" title="c12">d1</font> &amp; <font color="Maroon" title="c5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font> implies <font color="Maroon" title="c9">a1</font>,<font color="Maroon" title="c10">b1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">c1</font>,<font color="Maroon" title="c12">d1</font> ) &amp; ( <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">d</font> implies ( <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> &amp; <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c5">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">d</font>,<font color="Maroon" title="c7">c</font> ) ) &amp;  ex <font color="Olive" title="b1">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="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Olive" title="b1">d</font> or <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b1">d</font>,<font color="Maroon" title="c7">c</font> ) &amp; ( <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c13">p</font> &amp; <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c14">q</font> &amp; not <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> implies <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">q</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E4:99_1"/>
( <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">d</font> implies <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c6">b</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c8">d</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:99"><span class="lab"><font color="Green" title="E82">A1</font></span></a>, <a class="txt" href="geomtrap.html#E92"><span class="lab"><font color="Green" title="E77">Lm32</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( (  not <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">d</font> or <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c8">d</font>,<font color="Maroon" title="c6">b</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c7">c</font> or <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c5">a</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c8">d</font> ) &amp; ( <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">d</font> &amp; <font color="Maroon" title="c5">a</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c9">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">p</font> &amp; <font color="Maroon" title="c6">b</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c10">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">p</font> &amp; <font color="Maroon" title="c7">c</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c11">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">p</font> &amp; <font color="Maroon" title="c8">d</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c12">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">p</font> implies <font color="Maroon" title="c9">a1</font>,<font color="Maroon" title="c10">b1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">c1</font>,<font color="Maroon" title="c12">d1</font> ) &amp; ( <font color="Maroon" title="c13">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">q</font> &amp; <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">a1</font> &amp; <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c10">b1</font> &amp; <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c11">c1</font> &amp; <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">d</font>,<font color="Maroon" title="c12">d1</font> &amp; <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">d</font> implies <font color="Maroon" title="c9">a1</font>,<font color="Maroon" title="c10">b1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">c1</font>,<font color="Maroon" title="c12">d1</font> ) &amp; ( <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font> implies ( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> &amp; <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">c</font> ) ) &amp; ( <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">a1</font>,<font color="Maroon" title="c10">b1</font> &amp; <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">c1</font>,<font color="Maroon" title="c12">d1</font> &amp; <font color="Maroon" title="c5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font> implies <font color="Maroon" title="c9">a1</font>,<font color="Maroon" title="c10">b1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">c1</font>,<font color="Maroon" title="c12">d1</font> ) &amp; ( <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">d</font> implies ( <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> &amp; <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c5">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">d</font>,<font color="Maroon" title="c7">c</font> ) ) &amp;  ex <font color="Olive" title="b1">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="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Olive" title="b1">d</font> or <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b1">d</font>,<font color="Maroon" title="c7">c</font> ) &amp; ( <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c13">p</font> &amp; <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c14">q</font> &amp; not <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> implies <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">q</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E5:99_1"/>
(  not <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">d</font> or <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c8">d</font>,<font color="Maroon" title="c6">b</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c7">c</font> or <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c5">a</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c8">d</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:99"><span class="lab"><font color="Green" title="E82">A1</font></span></a>, <a class="txt" href="geomtrap.html#E93"><span class="lab"><font color="Green" title="E78">Lm33</font></span></a></span>; <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="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">d</font> &amp; <font color="Maroon" title="c5">a</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c9">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">p</font> &amp; <font color="Maroon" title="c6">b</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c10">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">p</font> &amp; <font color="Maroon" title="c7">c</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c11">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">p</font> &amp; <font color="Maroon" title="c8">d</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c12">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">p</font> implies <font color="Maroon" title="c9">a1</font>,<font color="Maroon" title="c10">b1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">c1</font>,<font color="Maroon" title="c12">d1</font> ) &amp; ( <font color="Maroon" title="c13">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">q</font> &amp; <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">a1</font> &amp; <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c10">b1</font> &amp; <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c11">c1</font> &amp; <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">d</font>,<font color="Maroon" title="c12">d1</font> &amp; <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">d</font> implies <font color="Maroon" title="c9">a1</font>,<font color="Maroon" title="c10">b1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">c1</font>,<font color="Maroon" title="c12">d1</font> ) &amp; ( <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font> implies ( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> &amp; <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">c</font> ) ) &amp; ( <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">a1</font>,<font color="Maroon" title="c10">b1</font> &amp; <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">c1</font>,<font color="Maroon" title="c12">d1</font> &amp; <font color="Maroon" title="c5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font> implies <font color="Maroon" title="c9">a1</font>,<font color="Maroon" title="c10">b1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">c1</font>,<font color="Maroon" title="c12">d1</font> ) &amp; ( <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">d</font> implies ( <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> &amp; <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c5">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">d</font>,<font color="Maroon" title="c7">c</font> ) ) &amp;  ex <font color="Olive" title="b1">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="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Olive" title="b1">d</font> or <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b1">d</font>,<font color="Maroon" title="c7">c</font> ) &amp; ( <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c13">p</font> &amp; <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c14">q</font> &amp; not <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> implies <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">q</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E6:99_1"/>
( <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">d</font> &amp; <font color="Maroon" title="c5">a</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c9">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">p</font> &amp; <font color="Maroon" title="c6">b</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c10">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">p</font> &amp; <font color="Maroon" title="c7">c</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c11">c1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">p</font> &amp; <font color="Maroon" title="c8">d</font> <a href="geomtrap.html#K9" title="GEOMTRAP:func.9">#</a> <font color="Maroon" title="c12">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">p</font> implies <font color="Maroon" title="c9">a1</font>,<font color="Maroon" title="c10">b1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">c1</font>,<font color="Maroon" title="c12">d1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="geomtrap.html#E94"><span class="lab"><font color="Green" title="E79">Lm34</font></span></a></span>; <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="c13">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">q</font> &amp; <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">a1</font> &amp; <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c10">b1</font> &amp; <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c11">c1</font> &amp; <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">d</font>,<font color="Maroon" title="c12">d1</font> &amp; <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">d</font> implies <font color="Maroon" title="c9">a1</font>,<font color="Maroon" title="c10">b1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">c1</font>,<font color="Maroon" title="c12">d1</font> ) &amp; ( <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font> implies ( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> &amp; <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">c</font> ) ) &amp; ( <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">a1</font>,<font color="Maroon" title="c10">b1</font> &amp; <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">c1</font>,<font color="Maroon" title="c12">d1</font> &amp; <font color="Maroon" title="c5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font> implies <font color="Maroon" title="c9">a1</font>,<font color="Maroon" title="c10">b1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">c1</font>,<font color="Maroon" title="c12">d1</font> ) &amp; ( <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">d</font> implies ( <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> &amp; <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c5">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">d</font>,<font color="Maroon" title="c7">c</font> ) ) &amp;  ex <font color="Olive" title="b1">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="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Olive" title="b1">d</font> or <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b1">d</font>,<font color="Maroon" title="c7">c</font> ) &amp; ( <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c13">p</font> &amp; <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c14">q</font> &amp; not <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> implies <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">q</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E7:99_1"/>
( <font color="Maroon" title="c13">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">q</font> &amp; <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c9">a1</font> &amp; <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c10">b1</font> &amp; <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c11">c1</font> &amp; <font color="Maroon" title="c13">p</font>,<font color="Maroon" title="c14">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">d</font>,<font color="Maroon" title="c12">d1</font> &amp; <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">d</font> implies <font color="Maroon" title="c9">a1</font>,<font color="Maroon" title="c10">b1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">c1</font>,<font color="Maroon" title="c12">d1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:99"><span class="lab"><font color="Green" title="E82">A1</font></span></a>, <a class="txt" href="geomtrap.html#E95"><span class="lab"><font color="Green" title="E80">Lm35</font></span></a></span>; <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="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font> implies ( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> &amp; <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">c</font> ) ) &amp; ( <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">a1</font>,<font color="Maroon" title="c10">b1</font> &amp; <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">c1</font>,<font color="Maroon" title="c12">d1</font> &amp; <font color="Maroon" title="c5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font> implies <font color="Maroon" title="c9">a1</font>,<font color="Maroon" title="c10">b1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">c1</font>,<font color="Maroon" title="c12">d1</font> ) &amp; ( <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">d</font> implies ( <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> &amp; <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c5">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">d</font>,<font color="Maroon" title="c7">c</font> ) ) &amp;  ex <font color="Olive" title="b1">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="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Olive" title="b1">d</font> or <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b1">d</font>,<font color="Maroon" title="c7">c</font> ) &amp; ( <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c13">p</font> &amp; <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c14">q</font> &amp; not <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> implies <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">q</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E8:99_1"/>
( <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c7">c</font> implies ( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> &amp; <font color="Maroon" title="c6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">c</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:99"><span class="lab"><font color="Green" title="E82">A1</font></span></a>, <a class="txt" href="geomtrap.html#E82"><span class="lab"><font color="Green" title="E67">Lm22</font></span></a></span>; <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="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">a1</font>,<font color="Maroon" title="c10">b1</font> &amp; <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">c1</font>,<font color="Maroon" title="c12">d1</font> &amp; <font color="Maroon" title="c5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font> implies <font color="Maroon" title="c9">a1</font>,<font color="Maroon" title="c10">b1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">c1</font>,<font color="Maroon" title="c12">d1</font> ) &amp; ( <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">d</font> implies ( <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> &amp; <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c5">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">d</font>,<font color="Maroon" title="c7">c</font> ) ) &amp;  ex <font color="Olive" title="b1">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="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Olive" title="b1">d</font> or <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b1">d</font>,<font color="Maroon" title="c7">c</font> ) &amp; ( <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c13">p</font> &amp; <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c14">q</font> &amp; not <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> implies <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">q</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E9:99_1"/>
( <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">a1</font>,<font color="Maroon" title="c10">b1</font> &amp; <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">c1</font>,<font color="Maroon" title="c12">d1</font> &amp; <font color="Maroon" title="c5">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font> implies <font color="Maroon" title="c9">a1</font>,<font color="Maroon" title="c10">b1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">c1</font>,<font color="Maroon" title="c12">d1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:99"><span class="lab"><font color="Green" title="E82">A1</font></span></a>, <a class="txt" href="geomtrap.html#E83"><span class="lab"><font color="Green" title="E68">Lm23</font></span></a></span>; <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="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">d</font> implies ( <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> &amp; <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c5">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">d</font>,<font color="Maroon" title="c7">c</font> ) ) &amp;  ex <font color="Olive" title="b1">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="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Olive" title="b1">d</font> or <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b1">d</font>,<font color="Maroon" title="c7">c</font> ) &amp; ( <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c13">p</font> &amp; <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c14">q</font> &amp; not <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> implies <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">q</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E10:99_1"/>
( <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">d</font> implies ( <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c8">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> &amp; <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c5">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">d</font>,<font color="Maroon" title="c7">c</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="geomtrap.html#E84"><span class="lab"><font color="Green" title="E69">Lm24</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">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="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Olive" title="b1">d</font> or <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b1">d</font>,<font color="Maroon" title="c7">c</font> ) &amp; ( <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c13">p</font> &amp; <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c14">q</font> &amp; not <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> implies <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">q</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E11:99_1"/>
 ex <font color="Olive" title="b1">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="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Olive" title="b1">d</font> or <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b1">d</font>,<font color="Maroon" title="c7">c</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:99"><span class="lab"><font color="Green" title="E82">A1</font></span></a>, <a class="txt" href="geomtrap.html#E85"><span class="lab"><font color="Green" title="E70">Lm25</font></span></a></span>; <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="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c13">p</font> &amp; <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c14">q</font> &amp; not <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> implies <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">q</font> )</span><br/>

<span class="kw">thus </span><a NAME="E12:99_1"/>
( <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c13">p</font> &amp; <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">c</font>,<font color="Maroon" title="c14">q</font> &amp; not <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> implies <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">q</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:99"><span class="lab"><font color="Green" title="E82">A1</font></span></a>, <a class="txt" href="geomtrap.html#E86"><span class="lab"><font color="Green" title="E71">Lm26</font></span></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/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:99"/>
 <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#NM1" title="GEOMTRAP:NM.1">MidOrdTrapSpace</a>
 ; <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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