<?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"> for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">d</font>, <font color="Olive" title="b3">a</font>, <font color="Olive" title="b4">c</font>, <font color="Olive" title="b5">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">OAS</font>  st  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">d</font>,<font color="Olive" title="b3">a</font> &amp; <font color="Olive" title="b4">c</font>,<font color="Olive" title="b2">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b5">b</font>,<font color="Olive" title="b3">a</font> &amp; <font color="Olive" title="b1">p</font>,<font color="Olive" title="b4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b5">b</font> &amp; not  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b3">a</font>,<font color="Olive" title="b5">b</font> &amp; <font color="Olive" title="b1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">c</font> holds <br/> <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b4">c</font>,<font color="Olive" title="b5">b</font></span><br/><b>let </b><font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c3">d</font>, <font color="Maroon" title="c4">a</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 <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"> (  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">a</font> &amp; <font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c4">a</font> &amp; <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c5">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">b</font> &amp; not  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">b</font> &amp; <font color="Maroon" title="c2">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">c</font> implies  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">b</font> )</span><br/>











<b>assume </b><b>that </b><br/><a NAME="E1:22"/><i><font color="Green" title="E11">A1</font></i>: 
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">a</font>
 <b>and </b><br/><a NAME="E2:22"/><i><font color="Green" title="E12">A2</font></i>: 
<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c4">a</font>
 <b>and </b><br/><a NAME="E3:22"/><i><font color="Green" title="E13">A3</font></i>: 
<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c5">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">b</font>
 <b>and </b><br/><a NAME="E4:22"/><i><font color="Green" title="E14">A4</font></i>: 
not  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">b</font>
 <b>and </b><br/><a NAME="E5:22"/><i><font color="Green" title="E15">A5</font></i>: 
<font color="Maroon" title="c2">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">b</font></span><br/>

<a NAME="E6:22"/>
(  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">b</font> or  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c5">c</font> )
 
<b>by </b><i><a class="txt" href="#E3:22"><i><font color="Green" title="E13">A3</font></i></a>, <a class="ref" href="diraf.html#T11" title="DIRAF:th.11">DIRAF:11</a></i>;<br/>
<a NAME="E7:22"/><b>then </b><i><font color="Green" title="E16">A6</font></i>: 
(  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">b</font> or  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">b</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="E8:22"/><b>then </b><i><font color="Green" title="E17">A7</font></i>: 
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">b</font>
 
<b>by </b><i><a class="ref" href="diraf.html#T36" title="DIRAF:th.36">DIRAF:36</a></i>;<br/>
<a NAME="E9:22"/><i><font color="Green" title="E18">A8</font></i>: 
<font color="Maroon" title="c2">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">d</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:22_1"/>
<font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">d</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="E2:22_1"/><b>then </b>
( <font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c2">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c4">a</font> &amp; <font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c2">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c2">p</font> )
 
<b>by </b><i><a class="txt" href="#E2:22"><i><font color="Green" title="E12">A2</font></i></a>, <a class="txt" href="#E3:22"><i><font color="Green" title="E13">A3</font></i></a>, <a class="ref" href="diraf.html#T5" title="DIRAF:th.5">DIRAF:5</a></i>;<br/>
<a NAME="E3:22_1"/><b>then </b>
<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c2">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c4">a</font>
 
<b>by </b><i><a class="txt" href="#E5:22"><i><font color="Green" title="E15">A5</font></i></a>, <a class="ref" href="analoaf.html#D5" title="ANALOAF:def.5">ANALOAF:def 5</a></i>;<br/>
<a NAME="E4:22_1"/><b>then </b>
(  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">a</font> or  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">p</font> )
 
<b>by </b><i><a class="ref" href="diraf.html#T11" title="DIRAF:th.11">DIRAF:11</a></i>;<br/>
<a NAME="E5:22_1"/><b>then </b>
(  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">a</font> or  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">p</font> )
 
<b>by </b><i><a class="ref" href="diraf.html#T34" title="DIRAF:th.34">DIRAF:34</a></i>;<br/>
<b>hence </b><a NAME="E6:22_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E4:22"><i><font color="Green" title="E14">A4</font></i></a>, <a class="ref" href="diraf.html#T36" title="DIRAF:th.36">DIRAF:36</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="22_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:22_2"/><i><font color="Green" title="E19">A9</font></i>: 
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c5">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">b</font></span><br/><a NAME="E2:22_2"/>
<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">a</font>
 <b>by </b><i><a class="txt" href="#E1:22"><i><font color="Green" title="E11">A1</font></i></a>, <a class="ref" href="diraf.html#D3" title="DIRAF:def.3">DIRAF:def 3</a></i>;<br/><a NAME="E3:22_2"/><b>then </b>
<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">a</font>
 <b>by </b><i><a class="ref" href="analoaf.html#D5" title="ANALOAF:def.5">ANALOAF:def 5</a></i>;<br/><a NAME="E4:22_2"/><b>then </b><i><font color="Green" title="E20">A10</font></i>: 
( <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">d</font> &amp; <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c4">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">d</font> &amp; <font color="Maroon" title="c2">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">b</font> )
 <b>by </b><i><a class="txt" href="#E2:22"><i><font color="Green" title="E12">A2</font></i></a>, <a class="txt" href="#E4:22"><i><font color="Green" title="E14">A4</font></i></a>, <a class="ref" href="diraf.html#T5" title="DIRAF:th.5">DIRAF:5</a>, <a class="ref" href="diraf.html#T37" title="DIRAF:th.37">DIRAF:37</a></i>;<br/><a NAME="E5:22_2"/>
not  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">c</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:22_2_1"/>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c5">c</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="E2:22_2_1"/><b>then </b>
(  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">d</font> &amp;  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c2">p</font> )
 
<b>by </b><i><a class="ref" href="diraf.html#T36" title="DIRAF:th.36">DIRAF:36</a>, <a class="ref" href="diraf.html#T37" title="DIRAF:th.37">DIRAF:37</a></i>;<br/>
<a NAME="E3:22_2_1"/><b>then </b>
(  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c6">b</font> &amp;  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c2">p</font> &amp;  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">a</font> )
 
<b>by </b><i><a class="txt" href="#E1:22"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="#E5:22"><i><font color="Green" title="E15">A5</font></i></a>, <a class="txt" href="#E8:22"><i><font color="Green" title="E17">A7</font></i></a>, <a class="ref" href="diraf.html#T34" title="DIRAF:th.34">DIRAF:34</a>, <a class="ref" href="diraf.html#T38" title="DIRAF:th.38">DIRAF:38</a></i>;<br/>
<b>hence </b><a NAME="E4:22_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E4:22"><i><font color="Green" title="E14">A4</font></i></a>, <a class="txt" href="#E9:22"><i><font color="Green" title="E18">A8</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="E6:22_2"/><b>then </b>
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">d</font>
 <b>by </b><i><a class="txt" href="#E1:22_2"><i><font color="Green" title="E19">A9</font></i></a>, <a class="txt" href="#E4:22_2"><i><font color="Green" title="E20">A10</font></i></a>, <a class="ref" href="pasch.html#T22" target="_self" title="PASCH:th.22">Th22</a></i>;<br/><a NAME="E7:22_2"/><b>then </b>
(  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c2">p</font> &amp;  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c2">p</font> )
 <b>by </b><i><a class="txt" href="#E1:22"><i><font color="Green" title="E11">A1</font></i></a>, <a class="ref" href="diraf.html#T13" title="DIRAF:th.13">DIRAF:13</a></i>;<br/><a NAME="E8:22_2"/><b>then </b>
<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c4">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c4">a</font>
 <b>by </b><i><a class="txt" href="#E2:22"><i><font color="Green" title="E12">A2</font></i></a>, <a class="ref" href="diraf.html#T18" title="DIRAF:th.18">DIRAF:18</a></i>;<br/><a NAME="E9:22_2"/><b>then </b>
<font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">b</font>
 <b>by </b><i><a class="ref" href="diraf.html#T5" title="DIRAF:th.5">DIRAF:5</a></i>;<br/><a NAME="E10:22_2"/><b>then </b>
(  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">b</font> or  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c5">c</font> )
 <b>by </b><i><a class="ref" href="diraf.html#T11" title="DIRAF:th.11">DIRAF:11</a></i>;<br/><a NAME="E11:22_2"/><b>then </b>
(  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">b</font> or  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c4">a</font>,<font color="Maroon" title="c6">b</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="E12:22_2"/><b>then </b>
(  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c4">a</font> &amp;  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c2">p</font> &amp;  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c6">b</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">b</font> )
 <b>by </b><i><a class="txt" href="#E7:22"><i><font color="Green" title="E16">A6</font></i></a>, <a class="ref" href="diraf.html#T36" title="DIRAF:th.36">DIRAF:36</a>, <a class="ref" href="diraf.html#T37" title="DIRAF:th.37">DIRAF:37</a></i>;<br/><a NAME="E13:22_2"/><b>then </b>
<font color="Maroon" title="c6">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="#E4:22"><i><font color="Green" title="E14">A4</font></i></a>, <a class="ref" href="diraf.html#T38" title="DIRAF:th.38">DIRAF:38</a></i>;<br/><b>hence </b><a NAME="E14:22_2"/>
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">b</font>
 <b>by </b><i><a class="ref" href="diraf.html#T14" title="DIRAF:th.14">DIRAF:14</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="E11:22"/>
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">b</font>
 <b>by </b><i><a class="txt" href="#E3:22"><i><font color="Green" title="E13">A3</font></i></a>, <a class="ref" href="diraf.html#T11" title="DIRAF:th.11">DIRAF:11</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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