<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">ADG</font> be   <a href="tdgroup.html#NM2" title="TDGROUP:NM.2">Uniquely_Two_Divisible_Group</a>;<br/>

<b>assume </b><a NAME="E1:23"/><i><font color="Green" title="E17">A1</font></i>: 
 ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">ADG</font> st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">b</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c2">A</font> =  <a href="tdgroup.html#K2" title="TDGROUP:func.2">AV</a> <font color="Maroon" title="c1">ADG</font>;<br/>
<a NAME="E2:23"/><i><font color="Green" title="E18">A2</font></i>: 
 <a href="tdgroup.html#K2" title="TDGROUP:func.2">AV</a> <font color="Maroon" title="c1">ADG</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="analoaf.html#G1" title="ANALOAF:aggr.1">AffinStruct</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">ADG</font>,<span class="p1">(<span class="default"><a href="tdgroup.html#K1" title="TDGROUP:func.1">CONGRD</a> <font color="Maroon" title="c1">ADG</font></span>)</span> #)
 
;<br/>
<a NAME="E3:23"/><i><font color="Green" title="E19">A3</font></i>: 
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="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="tdgroup.html#K2" title="TDGROUP:func.2">AV</a> <font color="Maroon" title="c1">ADG</font></span>)</span><br/> for <font color="Olive" title="b5">a</font>, <font color="Olive" title="b6">b</font>, <font color="Olive" title="b7">c</font>, <font color="Olive" title="b8">d</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">ADG</font>  st <font color="Olive" title="b5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a'</font> &amp; <font color="Olive" title="b6">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b'</font> &amp; <font color="Olive" title="b7">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">c'</font> &amp; <font color="Olive" title="b8">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">d'</font> holds <br/>( <font color="Olive" title="b5">a</font>,<font color="Olive" title="b6">b</font> <a href="tdgroup.html#R1" title="TDGROUP:pred.1">==&gt;</a> <font color="Olive" title="b7">c</font>,<font color="Olive" title="b8">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> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">a'</font>, <font color="Maroon" title="c4">b'</font>, <font color="Maroon" title="c5">c'</font>, <font color="Maroon" title="c6">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="tdgroup.html#K2" title="TDGROUP:func.2">AV</a> <font color="Maroon" title="c1">ADG</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c7">a</font>, <font color="Maroon" title="c8">b</font>, <font color="Maroon" title="c9">c</font>, <font color="Maroon" title="c10">d</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">ADG</font>;<br/>









<b>assume </b><a NAME="E1:23_1"/><i><font color="Green" title="E20">A4</font></i>: 
( <font color="Maroon" title="c7">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a'</font> &amp; <font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b'</font> &amp; <font color="Maroon" title="c9">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c'</font> &amp; <font color="Maroon" title="c10">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">d'</font> )
 ;<br/>

<div><i><font color="Green" title="E21">A5</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:23_1_1"/>
<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font> <a href="tdgroup.html#R1" title="TDGROUP:pred.1">==&gt;</a> <font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">d</font>
 ;<br/><a NAME="E2:23_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">d</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tdgroup.html#K1" title="TDGROUP:func.1">CONGRD</a> <font color="Maroon" title="c1">ADG</font>
 <b>by </b><i><a class="txt" href="tdgroup.html#E2:23"><i><font color="Green" title="E18">A2</font></i></a>, <a class="ref" href="tdgroup.html#D6" target="_self" title="TDGROUP:def.6">Def6</a></i>;<br/><b>hence </b><a NAME="E3:23_1_1"/>
<font color="Maroon" title="c3">a'</font>,<font color="Maroon" title="c4">b'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">c'</font>,<font color="Maroon" title="c6">d'</font>
 <b>by </b><i><a class="txt" href="tdgroup.html#E1:23_1"><i><font color="Green" title="E20">A4</font></i></a>, <a class="ref" href="analoaf.html#D2" title="ANALOAF:def.2">ANALOAF:def 2</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:23_1_2"/>
<font color="Maroon" title="c3">a'</font>,<font color="Maroon" title="c4">b'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">c'</font>,<font color="Maroon" title="c6">d'</font>
 ;<br/><a NAME="E2:23_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">d</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tdgroup.html#K1" title="TDGROUP:func.1">CONGRD</a> <font color="Maroon" title="c1">ADG</font>
 <b>by </b><i><a class="txt" href="tdgroup.html#E1:23_1"><i><font color="Green" title="E20">A4</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="E3:23_1_2"/>
<font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font> <a href="tdgroup.html#R1" title="TDGROUP:pred.1">==&gt;</a> <font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">d</font>
 <b>by </b><i><a class="txt" href="tdgroup.html#E2:23"><i><font color="Green" title="E18">A2</font></i></a>, <a class="ref" href="tdgroup.html#D6" target="_self" title="TDGROUP:def.6">Def6</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:23_1"/>
( <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font> <a href="tdgroup.html#R1" title="TDGROUP:pred.1">==&gt;</a> <font color="Maroon" title="c9">c</font>,<font color="Maroon" title="c10">d</font> iff <font color="Maroon" title="c3">a'</font>,<font color="Maroon" title="c4">b'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">c'</font>,<font color="Maroon" title="c6">d'</font> )
 <b>by </b><i><a class="txt" href="tdgroup.html#E2:23_1"><i><font color="Green" title="E21">A5</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E4:23"/>
 ex <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="tdgroup.html#K2" title="TDGROUP:func.2">AV</a> <font color="Maroon" title="c1">ADG</font></span>)</span> st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">b</font>
 <b>by </b><i><a class="txt" href="tdgroup.html#E1:23"><i><font color="Green" title="E17">A1</font></i></a></i>;<br/>

<b>thus </b><a NAME="E5:23"/>
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="tdgroup.html#K2" title="TDGROUP:func.2">AV</a> <font color="Maroon" title="c1">ADG</font></span>)</span>  st <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="b3">c</font> holds <br/><font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font>, <font color="Maroon" title="c5">c</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="tdgroup.html#K2" title="TDGROUP:func.2">AV</a> <font color="Maroon" title="c1">ADG</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:23_2"/><i><font color="Green" title="E20">A6</font></i>: 
<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c5">c</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c6">a'</font> = <font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c7">b'</font> = <font color="Maroon" title="c4">b</font>, <font color="Maroon" title="c8">c'</font> = <font color="Maroon" title="c5">c</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">ADG</font> ;<br/>
<a NAME="E3:23_2"/>
<font color="Maroon" title="c6">a'</font>,<font color="Maroon" title="c7">b'</font> <a href="tdgroup.html#R1" title="TDGROUP:pred.1">==&gt;</a> <font color="Maroon" title="c8">c'</font>,<font color="Maroon" title="c8">c'</font>
 
<b>by </b><i><a class="txt" href="tdgroup.html#E3:23"><i><font color="Green" title="E19">A3</font></i></a>, <a class="txt" href="tdgroup.html#E1:23_2"><i><font color="Green" title="E20">A6</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:23_2"/>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font>
 <b>by </b><i><a class="ref" href="tdgroup.html#T13" target="_self" title="TDGROUP:th.13">Th13</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E6:23"/>
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font>, <font color="Olive" title="b4">d</font>, <font color="Olive" title="b5">p</font>, <font color="Olive" title="b6">q</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="tdgroup.html#K2" title="TDGROUP:func.2">AV</a> <font color="Maroon" title="c1">ADG</font></span>)</span>  st <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b5">p</font>,<font color="Olive" title="b6">q</font> &amp; <font color="Olive" title="b3">c</font>,<font color="Olive" title="b4">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b5">p</font>,<font color="Olive" title="b6">q</font> holds <br/><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b3">c</font>,<font color="Olive" title="b4">d</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font>, <font color="Maroon" title="c5">c</font>, <font color="Maroon" title="c6">d</font>, <font color="Maroon" title="c7">p</font>, <font color="Maroon" title="c8">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="tdgroup.html#K2" title="TDGROUP:func.2">AV</a> <font color="Maroon" title="c1">ADG</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:23_3"/><i><font color="Green" title="E20">A7</font></i>: 
( <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">p</font>,<font color="Maroon" title="c8">q</font> &amp; <font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">p</font>,<font color="Maroon" title="c8">q</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c9">a'</font> = <font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c10">b'</font> = <font color="Maroon" title="c4">b</font>, <font color="Maroon" title="c11">c'</font> = <font color="Maroon" title="c5">c</font>, <font color="Maroon" title="c12">d'</font> = <font color="Maroon" title="c6">d</font>, <font color="Maroon" title="c13">p'</font> = <font color="Maroon" title="c7">p</font>, <font color="Maroon" title="c14">q'</font> = <font color="Maroon" title="c8">q</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">ADG</font> ;<br/>
<a NAME="E3:23_3"/>
( <font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c10">b'</font> <a href="tdgroup.html#R1" title="TDGROUP:pred.1">==&gt;</a> <font color="Maroon" title="c13">p'</font>,<font color="Maroon" title="c14">q'</font> &amp; <font color="Maroon" title="c11">c'</font>,<font color="Maroon" title="c12">d'</font> <a href="tdgroup.html#R1" title="TDGROUP:pred.1">==&gt;</a> <font color="Maroon" title="c13">p'</font>,<font color="Maroon" title="c14">q'</font> )
 
<b>by </b><i><a class="txt" href="tdgroup.html#E3:23"><i><font color="Green" title="E19">A3</font></i></a>, <a class="txt" href="tdgroup.html#E1:23_3"><i><font color="Green" title="E20">A7</font></i></a></i>;<br/>
<a NAME="E4:23_3"/><b>then </b>
<font color="Maroon" title="c9">a'</font>,<font color="Maroon" title="c10">b'</font> <a href="tdgroup.html#R1" title="TDGROUP:pred.1">==&gt;</a> <font color="Maroon" title="c11">c'</font>,<font color="Maroon" title="c12">d'</font>
 
<b>by </b><i><a class="ref" href="tdgroup.html#T14" target="_self" title="TDGROUP:th.14">Th14</a></i>;<br/>
<b>hence </b><a NAME="E5:23_3"/>
<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">d</font>
 <b>by </b><i><a class="txt" href="tdgroup.html#E3:23"><i><font color="Green" title="E19">A3</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E7:23"/>
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="tdgroup.html#K2" title="TDGROUP:func.2">AV</a> <font color="Maroon" title="c1">ADG</font></span>)</span>  ex <font color="Olive" title="b4">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="tdgroup.html#K2" title="TDGROUP:func.2">AV</a> <font color="Maroon" title="c1">ADG</font></span>)</span> st <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>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font>, <font color="Maroon" title="c5">c</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="tdgroup.html#K2" title="TDGROUP:func.2">AV</a> <font color="Maroon" title="c1">ADG</font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon" title="c6">a'</font> = <font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c7">b'</font> = <font color="Maroon" title="c4">b</font>, <font color="Maroon" title="c8">c'</font> = <font color="Maroon" title="c5">c</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">ADG</font> ;<br/>
<b>consider </b><font color="Maroon" title="c9">d'</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">ADG</font><b> such that </b><br/><a NAME="E3:23_4"/><i><font color="Green" title="E20">A8</font></i>: 
<font color="Maroon" title="c6">a'</font>,<font color="Maroon" title="c7">b'</font> <a href="tdgroup.html#R1" title="TDGROUP:pred.1">==&gt;</a> <font color="Maroon" title="c8">c'</font>,<font color="Maroon" title="c9">d'</font>
 <b>by </b><i><a class="ref" href="tdgroup.html#T15" target="_self" title="TDGROUP:th.15">Th15</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">d</font> = <font color="Maroon" title="c9">d'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="tdgroup.html#K2" title="TDGROUP:func.2">AV</a> <font color="Maroon" title="c1">ADG</font></span>)</span> ;<br/>
<b>take </b>
<font color="Maroon" title="c10">d</font>
;<br/>

<b>thus </b><a NAME="E5:23_4"/>
<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c10">d</font>
 <b>by </b><i><a class="txt" href="tdgroup.html#E3:23"><i><font color="Green" title="E19">A3</font></i></a>, <a class="txt" href="tdgroup.html#E3:23_4"><i><font color="Green" title="E20">A8</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E8:23"/>
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">a'</font>, <font color="Olive" title="b5">b'</font>, <font color="Olive" title="b6">c'</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="tdgroup.html#K2" title="TDGROUP:func.2">AV</a> <font color="Maroon" title="c1">ADG</font></span>)</span>  st <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="b4">a'</font>,<font color="Olive" title="b5">b'</font> &amp; <font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b4">a'</font>,<font color="Olive" title="b6">c'</font> holds <br/><font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b5">b'</font>,<font color="Olive" title="b6">c'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font>, <font color="Maroon" title="c5">c</font>, <font color="Maroon" title="c6">a'</font>, <font color="Maroon" title="c7">b'</font>, <font color="Maroon" title="c8">c'</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="tdgroup.html#K2" title="TDGROUP:func.2">AV</a> <font color="Maroon" title="c1">ADG</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:23_5"/><i><font color="Green" title="E20">A9</font></i>: 
( <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">a'</font>,<font color="Maroon" title="c7">b'</font> &amp; <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">a'</font>,<font color="Maroon" title="c8">c'</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c9">p</font> = <font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c10">q</font> = <font color="Maroon" title="c4">b</font>, <font color="Maroon" title="c11">r</font> = <font color="Maroon" title="c5">c</font>, <font color="Maroon" title="c12">p'</font> = <font color="Maroon" title="c6">a'</font>, <font color="Maroon" title="c13">q'</font> = <font color="Maroon" title="c7">b'</font>, <font color="Maroon" title="c14">r'</font> = <font color="Maroon" title="c8">c'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">ADG</font> ;<br/>
<a NAME="E3:23_5"/>
( <font color="Maroon" title="c9">p</font>,<font color="Maroon" title="c10">q</font> <a href="tdgroup.html#R1" title="TDGROUP:pred.1">==&gt;</a> <font color="Maroon" title="c12">p'</font>,<font color="Maroon" title="c13">q'</font> &amp; <font color="Maroon" title="c9">p</font>,<font color="Maroon" title="c11">r</font> <a href="tdgroup.html#R1" title="TDGROUP:pred.1">==&gt;</a> <font color="Maroon" title="c12">p'</font>,<font color="Maroon" title="c14">r'</font> )
 
<b>by </b><i><a class="txt" href="tdgroup.html#E3:23"><i><font color="Green" title="E19">A3</font></i></a>, <a class="txt" href="tdgroup.html#E1:23_5"><i><font color="Green" title="E20">A9</font></i></a></i>;<br/>
<a NAME="E4:23_5"/><b>then </b>
<font color="Maroon" title="c10">q</font>,<font color="Maroon" title="c11">r</font> <a href="tdgroup.html#R1" title="TDGROUP:pred.1">==&gt;</a> <font color="Maroon" title="c13">q'</font>,<font color="Maroon" title="c14">r'</font>
 
<b>by </b><i><a class="ref" href="tdgroup.html#T16" target="_self" title="TDGROUP:th.16">Th16</a></i>;<br/>
<b>hence </b><a NAME="E5:23_5"/>
<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">b'</font>,<font color="Maroon" title="c8">c'</font>
 <b>by </b><i><a class="txt" href="tdgroup.html#E3:23"><i><font color="Green" title="E19">A3</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E9:23"/>
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">c</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="tdgroup.html#K2" title="TDGROUP:func.2">AV</a> <font color="Maroon" title="c1">ADG</font></span>)</span>  ex <font color="Olive" title="b3">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="tdgroup.html#K2" title="TDGROUP:func.2">AV</a> <font color="Maroon" title="c1">ADG</font></span>)</span> st <font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b3">b</font>,<font color="Olive" title="b2">c</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">c</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="tdgroup.html#K2" title="TDGROUP:func.2">AV</a> <font color="Maroon" title="c1">ADG</font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon" title="c5">a'</font> = <font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c6">c'</font> = <font color="Maroon" title="c4">c</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">ADG</font> ;<br/>
<b>consider </b><font color="Maroon" title="c7">b'</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="c1">ADG</font><b> such that </b><br/><a NAME="E3:23_6"/><i><font color="Green" title="E20">A10</font></i>: 
<font color="Maroon" title="c5">a'</font>,<font color="Maroon" title="c7">b'</font> <a href="tdgroup.html#R1" title="TDGROUP:pred.1">==&gt;</a> <font color="Maroon" title="c7">b'</font>,<font color="Maroon" title="c6">c'</font>
 <b>by </b><i><a class="ref" href="tdgroup.html#T17" target="_self" title="TDGROUP:th.17">Th17</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">b</font> = <font color="Maroon" title="c7">b'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="tdgroup.html#K2" title="TDGROUP:func.2">AV</a> <font color="Maroon" title="c1">ADG</font></span>)</span> ;<br/>
<b>take </b>
<font color="Maroon" title="c8">b</font>
;<br/>

<b>thus </b><a NAME="E5:23_6"/>
<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c8">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c4">c</font>
 <b>by </b><i><a class="txt" href="tdgroup.html#E3:23"><i><font color="Green" title="E19">A3</font></i></a>, <a class="txt" href="tdgroup.html#E3:23_6"><i><font color="Green" title="E20">A10</font></i></a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E10:23"/>
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">b'</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="tdgroup.html#K2" title="TDGROUP:func.2">AV</a> <font color="Maroon" title="c1">ADG</font></span>)</span>  st <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">c</font> &amp; <font color="Olive" title="b1">a</font>,<font color="Olive" title="b4">b'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b4">b'</font>,<font color="Olive" title="b3">c</font> holds <br/><font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">b'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font>, <font color="Maroon" title="c5">c</font>, <font color="Maroon" title="c6">b'</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="tdgroup.html#K2" title="TDGROUP:func.2">AV</a> <font color="Maroon" title="c1">ADG</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:23_7"/><i><font color="Green" title="E20">A11</font></i>: 
( <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font> &amp; <font color="Maroon" title="c3">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="c5">c</font> )
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c7">a'</font> = <font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c8">p</font> = <font color="Maroon" title="c4">b</font>, <font color="Maroon" title="c9">c'</font> = <font color="Maroon" title="c5">c</font>, <font color="Maroon" title="c10">p'</font> = <font color="Maroon" title="c6">b'</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">ADG</font> ;<br/>
<a NAME="E3:23_7"/>
( <font color="Maroon" title="c7">a'</font>,<font color="Maroon" title="c8">p</font> <a href="tdgroup.html#R1" title="TDGROUP:pred.1">==&gt;</a> <font color="Maroon" title="c8">p</font>,<font color="Maroon" title="c9">c'</font> &amp; <font color="Maroon" title="c7">a'</font>,<font color="Maroon" title="c10">p'</font> <a href="tdgroup.html#R1" title="TDGROUP:pred.1">==&gt;</a> <font color="Maroon" title="c10">p'</font>,<font color="Maroon" title="c9">c'</font> )
 
<b>by </b><i><a class="txt" href="tdgroup.html#E3:23"><i><font color="Green" title="E19">A3</font></i></a>, <a class="txt" href="tdgroup.html#E1:23_7"><i><font color="Green" title="E20">A11</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:23_7"/>
<font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b'</font>
 <b>by </b><i><a class="ref" href="tdgroup.html#T18" target="_self" title="TDGROUP:th.18">Th18</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E11:23"/>
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="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="tdgroup.html#K2" title="TDGROUP:func.2">AV</a> <font color="Maroon" title="c1">ADG</font></span>)</span>  st <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> holds <br/><font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b2">b</font>,<font color="Olive" title="b4">d</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font>, <font color="Maroon" title="c5">c</font>, <font color="Maroon" title="c6">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="tdgroup.html#K2" title="TDGROUP:func.2">AV</a> <font color="Maroon" title="c1">ADG</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:23_8"/><i><font color="Green" title="E20">A12</font></i>: 
<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">d</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c7">a'</font> = <font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c8">b'</font> = <font color="Maroon" title="c4">b</font>, <font color="Maroon" title="c9">c'</font> = <font color="Maroon" title="c5">c</font>, <font color="Maroon" title="c10">d'</font> = <font color="Maroon" title="c6">d</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">ADG</font> ;<br/>
<a NAME="E3:23_8"/>
<font color="Maroon" title="c7">a'</font>,<font color="Maroon" title="c8">b'</font> <a href="tdgroup.html#R1" title="TDGROUP:pred.1">==&gt;</a> <font color="Maroon" title="c9">c'</font>,<font color="Maroon" title="c10">d'</font>
 
<b>by </b><i><a class="txt" href="tdgroup.html#E3:23"><i><font color="Green" title="E19">A3</font></i></a>, <a class="txt" href="tdgroup.html#E1:23_8"><i><font color="Green" title="E20">A12</font></i></a></i>;<br/>
<a NAME="E4:23_8"/><b>then </b>
<font color="Maroon" title="c7">a'</font>,<font color="Maroon" title="c9">c'</font> <a href="tdgroup.html#R1" title="TDGROUP:pred.1">==&gt;</a> <font color="Maroon" title="c8">b'</font>,<font color="Maroon" title="c10">d'</font>
 
<b>by </b><i><a class="ref" href="tdgroup.html#T19" target="_self" title="TDGROUP:th.19">Th19</a></i>;<br/>
<b>hence </b><a NAME="E5:23_8"/>
<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c6">d</font>
 <b>by </b><i><a class="txt" href="tdgroup.html#E3:23"><i><font color="Green" title="E19">A3</font></i></a></i>;<br/>


</div><b>end;</b></div>


</div>
