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<div class="add">

<b>set </b><font color="Maroon" title="c4">X</font> =  <a href="geomtrap.html#K8" title="GEOMTRAP:func.8">Af</a> <font color="Maroon" title="c3">MOS</font>;<br/>
<a NAME="E1:101_1_1"/>
 <a href="geomtrap.html#K8" title="GEOMTRAP:func.8">Af</a> <font color="Maroon" title="c3">MOS</font> is <a href="geomtrap.html#V3" title="GEOMTRAP:attr.3">OrdTrapSpace-like</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><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#K8" title="GEOMTRAP:func.8">Af</a> <font color="Maroon" title="c3">MOS</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="geomtrap.html#D14" target="_self" title="GEOMTRAP:def.14">GEOMTRAP:def 14</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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#K8" title="GEOMTRAP:func.8">Af</a> <font color="Maroon" title="c3">MOS</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/>

<div><i><font color="Green" title="E84">A1</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="101_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c15">a</font>, <font color="Maroon" title="c16">b</font>, <font color="Maroon" title="c17">c</font>, <font color="Maroon" title="c18">d</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#K8" title="GEOMTRAP:func.8">Af</a> <font color="Maroon" title="c3">MOS</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a'</font>, <font color="Olive" title="b2">b'</font>, <font color="Olive" title="b3">c'</font>, <font color="Olive" title="b4">d'</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">MOS</font>  st <font color="Maroon" title="c15">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a'</font> &amp; <font color="Maroon" title="c16">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b'</font> &amp; <font color="Maroon" title="c17">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">c'</font> &amp; <font color="Maroon" title="c18">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">d'</font> holds <br/>( <font color="Maroon" title="c15">a</font>,<font color="Maroon" title="c16">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c17">c</font>,<font color="Maroon" title="c18">d</font> iff <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> )</span><br/><b>let </b><font color="Maroon" title="c19">a'</font>, <font color="Maroon" title="c20">b'</font>, <font color="Maroon" title="c21">c'</font>, <font color="Maroon" title="c22">d'</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">MOS</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c15">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">a'</font> &amp; <font color="Maroon" title="c16">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">b'</font> &amp; <font color="Maroon" title="c17">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">c'</font> &amp; <font color="Maroon" title="c18">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">d'</font> implies ( <font color="Maroon" title="c15">a</font>,<font color="Maroon" title="c16">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c17">c</font>,<font color="Maroon" title="c18">d</font> iff <font color="Maroon" title="c19">a'</font>,<font color="Maroon" title="c20">b'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c21">c'</font>,<font color="Maroon" title="c22">d'</font> ) )</span><br/><b>assume </b><a NAME="E1:101_1_1_1_1"/><i><font color="Green" title="E85">A2</font></i>: 
( <font color="Maroon" title="c15">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">a'</font> &amp; <font color="Maroon" title="c16">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">b'</font> &amp; <font color="Maroon" title="c17">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">c'</font> &amp; <font color="Maroon" title="c18">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">d'</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c15">a</font>,<font color="Maroon" title="c16">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c17">c</font>,<font color="Maroon" title="c18">d</font> iff <font color="Maroon" title="c19">a'</font>,<font color="Maroon" title="c20">b'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c21">c'</font>,<font color="Maroon" title="c22">d'</font> )</span><br/><div><i><font color="Green" title="E86">A3</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="101_1_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:101_1_1_1_1_1"/>
<font color="Maroon" title="c15">a</font>,<font color="Maroon" title="c16">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c17">c</font>,<font color="Maroon" title="c18">d</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c19">a'</font>,<font color="Maroon" title="c20">b'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c21">c'</font>,<font color="Maroon" title="c22">d'</font></span><br/><a NAME="E2:101_1_1_1_1_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c15">a</font>,<font color="Maroon" title="c16">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c17">c</font>,<font color="Maroon" title="c18">d</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="analoaf.html#U1" title="ANALOAF:sel.1">CONGR</a> of <font color="Maroon" title="c3">MOS</font>
 <b>by </b><i><a class="ref" href="analoaf.html#D2" title="ANALOAF:def.2">ANALOAF:def 2</a></i>;<br/><b>hence </b><a NAME="E3:101_1_1_1_1_1"/>
<font color="Maroon" title="c19">a'</font>,<font color="Maroon" title="c20">b'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c21">c'</font>,<font color="Maroon" title="c22">d'</font>
 <b>by </b><i><a class="txt" href="#E1:101_1_1_1_1"><i><font color="Green" title="E85">A2</font></i></a>, <a class="ref" href="analoaf.html#D2" title="ANALOAF:def.2">ANALOAF:def 2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_1_1_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:101_1_1_1_1_2"/>
<font color="Maroon" title="c19">a'</font>,<font color="Maroon" title="c20">b'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c21">c'</font>,<font color="Maroon" title="c22">d'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c15">a</font>,<font color="Maroon" title="c16">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c17">c</font>,<font color="Maroon" title="c18">d</font></span><br/><a NAME="E2:101_1_1_1_1_2"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c15">a</font>,<font color="Maroon" title="c16">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>,<span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c17">c</font>,<font color="Maroon" title="c18">d</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="analoaf.html#U1" title="ANALOAF:sel.1">CONGR</a> of <font color="Maroon" title="c3">MOS</font>
 <b>by </b><i><a class="txt" href="#E1:101_1_1_1_1"><i><font color="Green" title="E85">A2</font></i></a>, <a class="ref" href="analoaf.html#D2" title="ANALOAF:def.2">ANALOAF:def 2</a></i>;<br/><b>hence </b><a NAME="E3:101_1_1_1_1_2"/>
<font color="Maroon" title="c15">a</font>,<font color="Maroon" title="c16">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c17">c</font>,<font color="Maroon" title="c18">d</font>
 <b>by </b><i><a class="ref" href="analoaf.html#D2" title="ANALOAF:def.2">ANALOAF:def 2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E4:101_1_1_1_1"/>
( <font color="Maroon" title="c15">a</font>,<font color="Maroon" title="c16">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c17">c</font>,<font color="Maroon" title="c18">d</font> iff <font color="Maroon" title="c19">a'</font>,<font color="Maroon" title="c20">b'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c21">c'</font>,<font color="Maroon" title="c22">d'</font> )
 <b>by </b><i><a class="txt" href="#E2:101_1_1_1_1"><i><font color="Green" title="E86">A3</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>reconsider </b><font color="Maroon" title="c15">a'</font> = <font color="Maroon" title="c5">a</font>, <font color="Maroon" title="c16">b'</font> = <font color="Maroon" title="c6">b</font>, <font color="Maroon" title="c17">c'</font> = <font color="Maroon" title="c7">c</font>, <font color="Maroon" title="c18">d'</font> = <font color="Maroon" title="c8">d</font>, <font color="Maroon" title="c19">a1'</font> = <font color="Maroon" title="c9">a1</font>, <font color="Maroon" title="c20">b1'</font> = <font color="Maroon" title="c10">b1</font>, <font color="Maroon" title="c21">c1'</font> = <font color="Maroon" title="c11">c1</font>, <font color="Maroon" title="c22">d1'</font> = <font color="Maroon" title="c12">d1</font>, <font color="Maroon" title="c23">p'</font> = <font color="Maroon" title="c13">p</font>, <font color="Maroon" title="c24">q'</font> = <font color="Maroon" title="c14">q</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">MOS</font> ;<br/>
<div><i><font color="Green" title="E85">A4</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="101_1_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:101_1_1_1_2"/>
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <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><br/><a NAME="E2:101_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c15">a'</font>,<font color="Maroon" title="c16">b'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c16">b'</font>,<font color="Maroon" title="c17">c'</font>
 <b>by </b><i><a class="txt" href="#E1:101_1_1_1"><i><font color="Green" title="E84">A1</font></i></a></i>;<br/><b>hence </b><a NAME="E3:101_1_1_1_2"/>
( <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> )
 <b>by </b><i><a class="ref" href="geomtrap.html#D13" target="_self" title="GEOMTRAP:def.13">Def13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<div><i><font color="Green" title="E86">A5</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="101_1_1_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:101_1_1_1_3"/>
( <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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <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><br/><a NAME="E2:101_1_1_1_3"/><b>then </b>
( <font color="Maroon" title="c15">a'</font>,<font color="Maroon" title="c16">b'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c19">a1'</font>,<font color="Maroon" title="c20">b1'</font> &amp; <font color="Maroon" title="c15">a'</font>,<font color="Maroon" title="c16">b'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c21">c1'</font>,<font color="Maroon" title="c22">d1'</font> &amp; <font color="Maroon" title="c15">a'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">b'</font> )
 <b>by </b><i><a class="txt" href="#E1:101_1_1_1"><i><font color="Green" title="E84">A1</font></i></a></i>;<br/><a NAME="E3:101_1_1_1_3"/><b>then </b>
<font color="Maroon" title="c19">a1'</font>,<font color="Maroon" title="c20">b1'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c21">c1'</font>,<font color="Maroon" title="c22">d1'</font>
 <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="E4:101_1_1_1_3"/>
<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>
 <b>by </b><i><a class="txt" href="#E1:101_1_1_1"><i><font color="Green" title="E84">A1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<div><i><font color="Green" title="E87">A6</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="101_1_1_1_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:101_1_1_1_4"/>
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <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><br/><a NAME="E2:101_1_1_1_4"/><b>then </b>
<font color="Maroon" title="c15">a'</font>,<font color="Maroon" title="c16">b'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c17">c'</font>,<font color="Maroon" title="c18">d'</font>
 <b>by </b><i><a class="txt" href="#E1:101_1_1_1"><i><font color="Green" title="E84">A1</font></i></a></i>;<br/><a NAME="E3:101_1_1_1_4"/><b>then </b>
( <font color="Maroon" title="c17">c'</font>,<font color="Maroon" title="c18">d'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c15">a'</font>,<font color="Maroon" title="c16">b'</font> &amp; <font color="Maroon" title="c16">b'</font>,<font color="Maroon" title="c15">a'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c18">d'</font>,<font color="Maroon" title="c17">c'</font> )
 <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="E4:101_1_1_1_4"/>
( <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> )
 <b>by </b><i><a class="txt" href="#E1:101_1_1_1"><i><font color="Green" title="E84">A1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E6:101_1_1_1"/><i><font color="Green" title="E88">A7</font></i>: 
 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#K8" title="GEOMTRAP:func.8">Af</a> <font color="Maroon" title="c3">MOS</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> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_1_1_1_5"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon" title="c25">x'</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">MOS</font><b> such that </b><br/><a NAME="E2:101_1_1_1_5"/><i><font color="Green" title="E89">A8</font></i>: 
( <font color="Maroon" title="c15">a'</font>,<font color="Maroon" title="c16">b'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c17">c'</font>,<font color="Maroon" title="c25">x'</font> or <font color="Maroon" title="c15">a'</font>,<font color="Maroon" title="c16">b'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c25">x'</font>,<font color="Maroon" title="c17">c'</font> )
 <b>by </b><i><a class="ref" href="geomtrap.html#D13" target="_self" title="GEOMTRAP:def.13">Def13</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c26">x</font> = <font color="Maroon" title="c25">x'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="geomtrap.html#K8" title="GEOMTRAP:func.8">Af</a> <font color="Maroon" title="c3">MOS</font></span>)</span> ;<br/>
<b>take </b>
<font color="Maroon" title="c26">x</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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="c26">x</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="c26">x</font>,<font color="Maroon" title="c7">c</font> )</span><br/>

<b>thus </b><a NAME="E4:101_1_1_1_5"/>
( <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="c26">x</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="c26">x</font>,<font color="Maroon" title="c7">c</font> )
 <b>by </b><i><a class="txt" href="#E1:101_1_1_1"><i><font color="Green" title="E84">A1</font></i></a>, <a class="txt" href="#E2:101_1_1_1_5"><i><font color="Green" title="E89">A8</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="101_1_1_1_6"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:101_1_1_1_6"/>
( <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> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> or <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/><a NAME="E2:101_1_1_1_6"/><b>then </b>
( <font color="Maroon" title="c15">a'</font>,<font color="Maroon" title="c16">b'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c17">c'</font>,<font color="Maroon" title="c23">p'</font> &amp; <font color="Maroon" title="c15">a'</font>,<font color="Maroon" title="c16">b'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c17">c'</font>,<font color="Maroon" title="c24">q'</font> )
 <b>by </b><i><a class="txt" href="#E1:101_1_1_1"><i><font color="Green" title="E84">A1</font></i></a></i>;<br/><b>hence </b><a NAME="E3:101_1_1_1_6"/>
( <font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b</font> or <font color="Maroon" title="c13">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">q</font> )
 <b>by </b><i><a class="ref" href="geomtrap.html#D13" target="_self" title="GEOMTRAP:def.13">Def13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:101_1_1_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> ) ) &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#K8" title="GEOMTRAP:func.8">Af</a> <font color="Maroon" title="c3">MOS</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> ) )
 <b>by </b><i><a class="txt" href="#E3:101_1_1_1"><i><font color="Green" title="E85">A4</font></i></a>, <a class="txt" href="#E4:101_1_1_1"><i><font color="Green" title="E86">A5</font></i></a>, <a class="txt" href="#E5:101_1_1_1"><i><font color="Green" title="E87">A6</font></i></a>, <a class="txt" href="#E6:101_1_1_1"><i><font color="Green" title="E88">A7</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:101_1_1"/>
 <a href="geomtrap.html#K8" title="GEOMTRAP:func.8">Af</a> <font color="Maroon" title="c3">MOS</font> is <a href="geomtrap.html#V3" title="GEOMTRAP:attr.3">OrdTrapSpace-like</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
