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

<b>let </b><font color="Maroon" title="c1">OAS</font> be   <a href="analoaf.html#NM1" title="ANALOAF:NM.1">OAffinSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">OAS</font> <a href="papdesaf.html#NR7" title="PAPDESAF:NR.7">satisfies_DES_1</a>  implies <font color="Maroon" title="c1">OAS</font> <a href="papdesaf.html#NR6" title="PAPDESAF:NR.6">satisfies_DES</a>  )</span><br/>

<b>assume </b><a NAME="E1:19"/><i><font color="Green" title="E13">A1</font></i>: 
<font color="Maroon" title="c1">OAS</font> <a href="papdesaf.html#NR7" title="PAPDESAF:NR.7">satisfies_DES_1</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">OAS</font> <a href="papdesaf.html#NR6" title="PAPDESAF:NR.6">satisfies_DES</a> </span><br/>

<a NAME="E2:19"/>
for <font color="Olive" title="b1">o</font>, <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font>, <font color="Olive" title="b4">c</font>, <font color="Olive" title="b5">a1</font>, <font color="Olive" title="b6">b1</font>, <font color="Olive" title="b7">c1</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">OAS</font>  st <font color="Olive" title="b1">o</font>,<font color="Olive" title="b2">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b1">o</font>,<font color="Olive" title="b5">a1</font> &amp; <font color="Olive" title="b1">o</font>,<font color="Olive" title="b3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b1">o</font>,<font color="Olive" title="b6">b1</font> &amp; <font color="Olive" title="b1">o</font>,<font color="Olive" title="b4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b1">o</font>,<font color="Olive" title="b7">c1</font> &amp; not  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Olive" title="b1">o</font>,<font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font> &amp; not  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Olive" title="b1">o</font>,<font color="Olive" title="b2">a</font>,<font color="Olive" title="b4">c</font> &amp; <font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b5">a1</font>,<font color="Olive" title="b6">b1</font> &amp; <font color="Olive" title="b2">a</font>,<font color="Olive" title="b4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b5">a1</font>,<font color="Olive" title="b7">c1</font> holds <br/><font color="Olive" title="b3">b</font>,<font color="Olive" title="b4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b6">b1</font>,<font color="Olive" title="b7">c1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">o</font>, <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">a1</font>, <font color="Maroon" title="c7">b1</font>, <font color="Maroon" title="c8">c1</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">OAS</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c6">a1</font> &amp; <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c4">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c7">b1</font> &amp; <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c5">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c8">c1</font> &amp; not  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font> &amp; not  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font> &amp; <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">a1</font>,<font color="Maroon" title="c7">b1</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">a1</font>,<font color="Maroon" title="c8">c1</font> implies <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">b1</font>,<font color="Maroon" title="c8">c1</font> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:19_1"/><i><font color="Green" title="E14">A2</font></i>: 
( <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c6">a1</font> &amp; <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c4">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c7">b1</font> &amp; <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c5">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c8">c1</font> )
 <b>and </b><br/><a NAME="E2:19_1"/><i><font color="Green" title="E15">A3</font></i>: 
( not  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font> &amp; not  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font> )
 <b>and </b><br/><a NAME="E3:19_1"/><i><font color="Green" title="E16">A4</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">a1</font>,<font color="Maroon" title="c7">b1</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">a1</font>,<font color="Maroon" title="c8">c1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <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">b1</font>,<font color="Maroon" title="c8">c1</font></span><br/>

<b>consider </b><font color="Maroon" title="c9">a2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">OAS</font><b> such that </b><br/><a NAME="E5:19_1"/><i><font color="Green" title="E17">A5</font></i>: 
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c9">a2</font>
 <b>and </b><br/><a NAME="E6:19_1"/><i><font color="Green" title="E18">A6</font></i>: 
<font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">a2</font>
 <b>by </b><i><a class="ref" href="diraf.html#T17" title="DIRAF:th.17">DIRAF:17</a></i>;<br/>
<a NAME="E7:19_1"/><i><font color="Green" title="E19">A7</font></i>: 
( <font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">a</font> &amp; <font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> &amp; <font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">c</font> &amp; <font color="Maroon" title="c3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> &amp; <font color="Maroon" title="c3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">c</font> )
 
<b>by </b><i><a class="txt" href="papdesaf.html#E2:19_1"><i><font color="Green" title="E15">A3</font></i></a>, <a class="ref" href="diraf.html#T37" title="DIRAF:th.37">DIRAF:37</a></i>;<br/>
<a NAME="E8:19_1"/><i><font color="Green" title="E20">A8</font></i>: 
<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c2">o</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c9">a2</font>
 
<b>by </b><i><a class="txt" href="papdesaf.html#E5:19_1"><i><font color="Green" title="E17">A5</font></i></a>, <a class="ref" href="diraf.html#D3" title="DIRAF:def.3">DIRAF:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c10">b2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">OAS</font><b> such that </b><br/><a NAME="E10:19_1"/><i><font color="Green" title="E21">A9</font></i>: 
<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c2">o</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c10">b2</font>
 <b>and </b><br/><a NAME="E11:19_1"/><i><font color="Green" title="E22">A10</font></i>: 
<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">a2</font>,<font color="Maroon" title="c10">b2</font>
 <b>by </b><i><a class="txt" href="papdesaf.html#E7:19_1"><i><font color="Green" title="E19">A7</font></i></a>, <a class="ref" href="analoaf.html#D5" title="ANALOAF:def.5">ANALOAF:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c11">c2</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">OAS</font><b> such that </b><br/><a NAME="E13:19_1"/><i><font color="Green" title="E23">A11</font></i>: 
<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c2">o</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c11">c2</font>
 <b>and </b><br/><a NAME="E14:19_1"/><i><font color="Green" title="E24">A12</font></i>: 
<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">a2</font>,<font color="Maroon" title="c11">c2</font>
 <b>by </b><i><a class="txt" href="papdesaf.html#E7:19_1"><i><font color="Green" title="E19">A7</font></i></a>, <a class="txt" href="papdesaf.html#E8:19_1"><i><font color="Green" title="E20">A8</font></i></a>, <a class="ref" href="analoaf.html#D5" title="ANALOAF:def.5">ANALOAF:def 5</a></i>;<br/>
<a NAME="E15:19_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="c10">b2</font>,<font color="Maroon" title="c9">a2</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="c11">c2</font>,<font color="Maroon" title="c9">a2</font> )
 
<b>by </b><i><a class="txt" href="papdesaf.html#E11:19_1"><i><font color="Green" title="E22">A10</font></i></a>, <a class="txt" href="papdesaf.html#E14:19_1"><i><font color="Green" title="E24">A12</font></i></a>, <a class="ref" href="analoaf.html#D5" title="ANALOAF:def.5">ANALOAF:def 5</a></i>;<br/>
<a NAME="E16:19_1"/><b>then </b><i><font color="Green" title="E25">A13</font></i>: 
<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="c11">c2</font>,<font color="Maroon" title="c10">b2</font>
 
<b>by </b><i><a class="txt" href="papdesaf.html#E1:19"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="papdesaf.html#E2:19_1"><i><font color="Green" title="E15">A3</font></i></a>, <a class="txt" href="papdesaf.html#E8:19_1"><i><font color="Green" title="E20">A8</font></i></a>, <a class="txt" href="papdesaf.html#E10:19_1"><i><font color="Green" title="E21">A9</font></i></a>, <a class="txt" href="papdesaf.html#E13:19_1"><i><font color="Green" title="E23">A11</font></i></a>, <a class="ref" href="papdesaf.html#D16" target="_self" title="PAPDESAF:def.16">Def16</a></i>;<br/>
<a NAME="E17:19_1"/><i><font color="Green" title="E26">A14</font></i>: 
( <font color="Maroon" title="c9">a2</font>,<font color="Maroon" title="c2">o</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a</font> &amp; <font color="Maroon" title="c10">b2</font>,<font color="Maroon" title="c2">o</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c4">b</font> &amp; <font color="Maroon" title="c11">c2</font>,<font color="Maroon" title="c2">o</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c5">c</font> )
 
<b>by </b><i><a class="txt" href="papdesaf.html#E8:19_1"><i><font color="Green" title="E20">A8</font></i></a>, <a class="txt" href="papdesaf.html#E10:19_1"><i><font color="Green" title="E21">A9</font></i></a>, <a class="txt" href="papdesaf.html#E13:19_1"><i><font color="Green" title="E23">A11</font></i></a>, <a class="ref" href="diraf.html#T5" title="DIRAF:th.5">DIRAF:5</a></i>;<br/>
<a NAME="E18:19_1"/><b>then </b><i><font color="Green" title="E27">A15</font></i>: 
( <font color="Maroon" title="c9">a2</font>,<font color="Maroon" title="c2">o</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c6">a1</font> &amp; <font color="Maroon" title="c10">b2</font>,<font color="Maroon" title="c2">o</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c7">b1</font> &amp; <font color="Maroon" title="c11">c2</font>,<font color="Maroon" title="c2">o</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c8">c1</font> )
 
<b>by </b><i><a class="txt" href="papdesaf.html#E1:19_1"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="papdesaf.html#E7:19_1"><i><font color="Green" title="E19">A7</font></i></a>, <a class="ref" href="diraf.html#T6" title="DIRAF:th.6">DIRAF:6</a></i>;<br/>
<a NAME="E19:19_1"/>
( <font color="Maroon" title="c10">b2</font>,<font color="Maroon" title="c9">a2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font> &amp; <font color="Maroon" title="c11">c2</font>,<font color="Maroon" title="c9">a2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font> )
 
<b>by </b><i><a class="txt" href="papdesaf.html#E11:19_1"><i><font color="Green" title="E22">A10</font></i></a>, <a class="txt" href="papdesaf.html#E14:19_1"><i><font color="Green" title="E24">A12</font></i></a>, <a class="ref" href="diraf.html#T5" title="DIRAF:th.5">DIRAF:5</a></i>;<br/>
<a NAME="E20:19_1"/><b>then </b>
( <font color="Maroon" title="c10">b2</font>,<font color="Maroon" title="c9">a2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">a1</font>,<font color="Maroon" title="c7">b1</font> &amp; <font color="Maroon" title="c11">c2</font>,<font color="Maroon" title="c9">a2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">a1</font>,<font color="Maroon" title="c8">c1</font> )
 
<b>by </b><i><a class="txt" href="papdesaf.html#E3:19_1"><i><font color="Green" title="E16">A4</font></i></a>, <a class="txt" href="papdesaf.html#E7:19_1"><i><font color="Green" title="E19">A7</font></i></a>, <a class="ref" href="diraf.html#T6" title="DIRAF:th.6">DIRAF:6</a></i>;<br/>
<a NAME="E21:19_1"/><b>then </b><i><font color="Green" title="E28">A16</font></i>: 
( <font color="Maroon" title="c9">a2</font>,<font color="Maroon" title="c11">c2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">c1</font>,<font color="Maroon" title="c6">a1</font> &amp; <font color="Maroon" title="c9">a2</font>,<font color="Maroon" title="c10">b2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">b1</font>,<font color="Maroon" title="c6">a1</font> )
 
<b>by </b><i><a class="ref" href="diraf.html#T5" title="DIRAF:th.5">DIRAF:5</a></i>;<br/>
<a NAME="E22:19_1"/>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c9">a2</font>
 
<b>by </b><i><a class="txt" href="papdesaf.html#E5:19_1"><i><font color="Green" title="E17">A5</font></i></a>, <a class="ref" href="diraf.html#T34" title="DIRAF:th.34">DIRAF:34</a></i>;<br/>
<a NAME="E23:19_1"/><b>then </b><i><font color="Green" title="E29">A17</font></i>: 
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c9">a2</font>,<font color="Maroon" title="c3">a</font>
 
<b>by </b><i><a class="ref" href="diraf.html#T36" title="DIRAF:th.36">DIRAF:36</a></i>;<br/>
<a NAME="E24:19_1"/>
(  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c10">b2</font>,<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c4">b</font> &amp;  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c11">c2</font>,<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c5">c</font> )
 
<b>by </b><i><a class="txt" href="papdesaf.html#E17:19_1"><i><font color="Green" title="E26">A14</font></i></a>, <a class="ref" href="diraf.html#D3" title="DIRAF:def.3">DIRAF:def 3</a></i>;<br/>
<a NAME="E25:19_1"/><b>then </b><i><font color="Green" title="E30">A18</font></i>: 
(  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c10">b2</font>,<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c4">b</font> &amp;  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c11">c2</font>,<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c5">c</font> )
 
<b>by </b><i><a class="ref" href="diraf.html#T34" title="DIRAF:th.34">DIRAF:34</a></i>;<br/>
<a NAME="E26:19_1"/><i><font color="Green" title="E31">A19</font></i>: 
<font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">b2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:19_1_1"/>
<font color="Maroon" title="c2">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:19_1_1"/><b>then </b>
<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c9">a2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>
 
<b>by </b><i><a class="txt" href="papdesaf.html#E11:19_1"><i><font color="Green" title="E22">A10</font></i></a>, <a class="ref" href="diraf.html#T5" title="DIRAF:th.5">DIRAF:5</a></i>;<br/>
<a NAME="E3:19_1_1"/><b>then </b>
<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c9">a2</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>
 
<b>by </b><i><a class="ref" href="diraf.html#D4" title="DIRAF:def.4">DIRAF:def 4</a></i>;<br/>
<a NAME="E4:19_1_1"/><b>then </b>
(  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c9">a2</font>,<font color="Maroon" title="c2">o</font> &amp;  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c9">a2</font>,<font color="Maroon" title="c4">b</font> )
 
<b>by </b><i><a class="txt" href="papdesaf.html#E6:19_1"><i><font color="Green" title="E18">A6</font></i></a>, <a class="txt" href="papdesaf.html#E23:19_1"><i><font color="Green" title="E29">A17</font></i></a>, <a class="ref" href="diraf.html#T37" title="DIRAF:th.37">DIRAF:37</a>, <a class="ref" href="diraf.html#T39" title="DIRAF:th.39">DIRAF:39</a></i>;<br/>
<b>hence </b><a NAME="E5:19_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="papdesaf.html#E2:19_1"><i><font color="Green" title="E15">A3</font></i></a>, <a class="txt" href="papdesaf.html#E6:19_1"><i><font color="Green" title="E18">A6</font></i></a>, <a class="txt" href="papdesaf.html#E23:19_1"><i><font color="Green" title="E29">A17</font></i></a>, <a class="ref" href="diraf.html#T38" title="DIRAF:th.38">DIRAF:38</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="E27:19_1"/><i><font color="Green" title="E32">A20</font></i>: 
not  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c9">a2</font>,<font color="Maroon" title="c10">b2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:19_1_2"/><i><font color="Green" title="E33">A21</font></i>: 
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c9">a2</font>,<font color="Maroon" title="c10">b2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:19_1_2"/>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c9">a2</font>,<font color="Maroon" title="c2">o</font>
 
<b>by </b><i><a class="ref" href="diraf.html#T37" title="DIRAF:th.37">DIRAF:37</a></i>;<br/>
<a NAME="E3:19_1_2"/><b>then </b><i><font color="Green" title="E34">A22</font></i>: 
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c10">b2</font>,<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a</font>
 
<b>by </b><i><a class="txt" href="papdesaf.html#E6:19_1"><i><font color="Green" title="E18">A6</font></i></a>, <a class="txt" href="papdesaf.html#E23:19_1"><i><font color="Green" title="E29">A17</font></i></a>, <a class="txt" href="papdesaf.html#E1:19_1_2"><i><font color="Green" title="E33">A21</font></i></a>, <a class="ref" href="diraf.html#T38" title="DIRAF:th.38">DIRAF:38</a></i>;<br/>
<a NAME="E4:19_1_2"/>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c10">b2</font>,<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c2">o</font>
 
<b>by </b><i><a class="ref" href="diraf.html#T37" title="DIRAF:th.37">DIRAF:37</a></i>;<br/>
<b>hence </b><a NAME="E5:19_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="papdesaf.html#E2:19_1"><i><font color="Green" title="E15">A3</font></i></a>, <a class="txt" href="papdesaf.html#E25:19_1"><i><font color="Green" title="E30">A18</font></i></a>, <a class="txt" href="papdesaf.html#E26:19_1"><i><font color="Green" title="E31">A19</font></i></a>, <a class="txt" href="papdesaf.html#E3:19_1_2"><i><font color="Green" title="E34">A22</font></i></a>, <a class="ref" href="diraf.html#T38" title="DIRAF:th.38">DIRAF:38</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="E28:19_1"/><i><font color="Green" title="E33">A23</font></i>: 
<font color="Maroon" title="c2">o</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">c2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:19_1_3"/>
<font color="Maroon" title="c2">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">c2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:19_1_3"/><b>then </b>
<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c9">a2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font>
 
<b>by </b><i><a class="txt" href="papdesaf.html#E14:19_1"><i><font color="Green" title="E24">A12</font></i></a>, <a class="ref" href="diraf.html#T5" title="DIRAF:th.5">DIRAF:5</a></i>;<br/>
<a NAME="E3:19_1_3"/><b>then </b>
<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c9">a2</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font>
 
<b>by </b><i><a class="ref" href="diraf.html#D4" title="DIRAF:def.4">DIRAF:def 4</a></i>;<br/>
<a NAME="E4:19_1_3"/><b>then </b>
(  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c9">a2</font>,<font color="Maroon" title="c2">o</font> &amp;  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c9">a2</font>,<font color="Maroon" title="c5">c</font> )
 
<b>by </b><i><a class="txt" href="papdesaf.html#E6:19_1"><i><font color="Green" title="E18">A6</font></i></a>, <a class="txt" href="papdesaf.html#E23:19_1"><i><font color="Green" title="E29">A17</font></i></a>, <a class="ref" href="diraf.html#T37" title="DIRAF:th.37">DIRAF:37</a>, <a class="ref" href="diraf.html#T39" title="DIRAF:th.39">DIRAF:39</a></i>;<br/>
<b>hence </b><a NAME="E5:19_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="papdesaf.html#E2:19_1"><i><font color="Green" title="E15">A3</font></i></a>, <a class="txt" href="papdesaf.html#E6:19_1"><i><font color="Green" title="E18">A6</font></i></a>, <a class="txt" href="papdesaf.html#E23:19_1"><i><font color="Green" title="E29">A17</font></i></a>, <a class="ref" href="diraf.html#T38" title="DIRAF:th.38">DIRAF:38</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="E29:19_1"/>
not  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c9">a2</font>,<font color="Maroon" title="c11">c2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:19_1_4"/><i><font color="Green" title="E34">A24</font></i>: 
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c9">a2</font>,<font color="Maroon" title="c11">c2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:19_1_4"/>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c9">a2</font>,<font color="Maroon" title="c2">o</font>
 
<b>by </b><i><a class="ref" href="diraf.html#T37" title="DIRAF:th.37">DIRAF:37</a></i>;<br/>
<a NAME="E3:19_1_4"/><b>then </b><i><font color="Green" title="E35">A25</font></i>: 
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c11">c2</font>,<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c3">a</font>
 
<b>by </b><i><a class="txt" href="papdesaf.html#E6:19_1"><i><font color="Green" title="E18">A6</font></i></a>, <a class="txt" href="papdesaf.html#E23:19_1"><i><font color="Green" title="E29">A17</font></i></a>, <a class="txt" href="papdesaf.html#E1:19_1_4"><i><font color="Green" title="E34">A24</font></i></a>, <a class="ref" href="diraf.html#T38" title="DIRAF:th.38">DIRAF:38</a></i>;<br/>
<a NAME="E4:19_1_4"/>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c11">c2</font>,<font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c2">o</font>
 
<b>by </b><i><a class="ref" href="diraf.html#T37" title="DIRAF:th.37">DIRAF:37</a></i>;<br/>
<b>hence </b><a NAME="E5:19_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="papdesaf.html#E2:19_1"><i><font color="Green" title="E15">A3</font></i></a>, <a class="txt" href="papdesaf.html#E25:19_1"><i><font color="Green" title="E30">A18</font></i></a>, <a class="txt" href="papdesaf.html#E28:19_1"><i><font color="Green" title="E33">A23</font></i></a>, <a class="txt" href="papdesaf.html#E3:19_1_4"><i><font color="Green" title="E35">A25</font></i></a>, <a class="ref" href="diraf.html#T38" title="DIRAF:th.38">DIRAF:38</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="E30:19_1"/><b>then </b><i><font color="Green" title="E34">A26</font></i>: 
<font color="Maroon" title="c11">c2</font>,<font color="Maroon" title="c10">b2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">b1</font>,<font color="Maroon" title="c8">c1</font>
 
<b>by </b><i><a class="txt" href="papdesaf.html#E1:19"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="papdesaf.html#E18:19_1"><i><font color="Green" title="E27">A15</font></i></a>, <a class="txt" href="papdesaf.html#E21:19_1"><i><font color="Green" title="E28">A16</font></i></a>, <a class="txt" href="papdesaf.html#E27:19_1"><i><font color="Green" title="E32">A20</font></i></a>, <a class="ref" href="papdesaf.html#D16" target="_self" title="PAPDESAF:def.16">Def16</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:19_1_5"/><i><font color="Green" title="E35">A27</font></i>: 
<font color="Maroon" title="c11">c2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">b2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <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">b1</font>,<font color="Maroon" title="c8">c1</font></span><br/><a NAME="E2:19_1_5"/><i><font color="Green" title="E36">A28</font></i>: 
not  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c10">b2</font>,<font color="Maroon" title="c9">a2</font>
 <b>by </b><i><a class="txt" href="papdesaf.html#E27:19_1"><i><font color="Green" title="E32">A20</font></i></a>, <a class="ref" href="diraf.html#T36" title="DIRAF:th.36">DIRAF:36</a></i>;<br/><a NAME="E3:19_1_5"/><i><font color="Green" title="E37">A29</font></i>: 
(  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c10">b2</font>,<font color="Maroon" title="c4">b</font> &amp;  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">o</font>,<font color="Maroon" title="c10">b2</font>,<font color="Maroon" title="c5">c</font> )
 <b>by </b><i><a class="txt" href="papdesaf.html#E25:19_1"><i><font color="Green" title="E30">A18</font></i></a>, <a class="txt" href="papdesaf.html#E1:19_1_5"><i><font color="Green" title="E35">A27</font></i></a>, <a class="ref" href="diraf.html#T36" title="DIRAF:th.36">DIRAF:36</a></i>;<br/><a NAME="E4:19_1_5"/>
( <font color="Maroon" title="c10">b2</font>,<font color="Maroon" title="c9">a2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font> &amp; <font color="Maroon" title="c11">c2</font>,<font color="Maroon" title="c9">a2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font> )
 <b>by </b><i><a class="txt" href="papdesaf.html#E11:19_1"><i><font color="Green" title="E22">A10</font></i></a>, <a class="txt" href="papdesaf.html#E14:19_1"><i><font color="Green" title="E24">A12</font></i></a>, <a class="ref" href="diraf.html#T5" title="DIRAF:th.5">DIRAF:5</a></i>;<br/><a NAME="E5:19_1_5"/><b>then </b>
( <font color="Maroon" title="c10">b2</font>,<font color="Maroon" title="c9">a2</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font> &amp; <font color="Maroon" title="c10">b2</font>,<font color="Maroon" title="c9">a2</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font> )
 <b>by </b><i><a class="txt" href="papdesaf.html#E1:19_1_5"><i><font color="Green" title="E35">A27</font></i></a>, <a class="ref" href="diraf.html#D4" title="DIRAF:def.4">DIRAF:def 4</a></i>;<br/><a NAME="E6:19_1_5"/><b>then </b>
<font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font>
 <b>by </b><i><a class="txt" href="papdesaf.html#E23:19_1"><i><font color="Green" title="E29">A17</font></i></a>, <a class="txt" href="papdesaf.html#E2:19_1_5"><i><font color="Green" title="E36">A28</font></i></a>, <a class="txt" href="papdesaf.html#E3:19_1_5"><i><font color="Green" title="E37">A29</font></i></a>, <a class="ref" href="pasch.html#T11" title="PASCH:th.11">PASCH:11</a></i>;<br/><b>hence </b><a NAME="E7:19_1_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">b1</font>,<font color="Maroon" title="c8">c1</font>
 <b>by </b><i><a class="ref" href="diraf.html#T7" title="DIRAF:th.7">DIRAF:7</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="E32:19_1"/>
<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">b1</font>,<font color="Maroon" title="c8">c1</font>
 <b>by </b><i><a class="txt" href="papdesaf.html#E16:19_1"><i><font color="Green" title="E25">A13</font></i></a>, <a class="txt" href="papdesaf.html#E30:19_1"><i><font color="Green" title="E34">A26</font></i></a>, <a class="ref" href="diraf.html#T6" title="DIRAF:th.6">DIRAF:6</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="E3:19"/>
<font color="Maroon" title="c1">OAS</font> <a href="papdesaf.html#NR6" title="PAPDESAF:NR.6">satisfies_DES</a> 
 <b>by </b><i><a class="ref" href="papdesaf.html#D15" target="_self" title="PAPDESAF:def.15">Def15</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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