<?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>;<br/><b>let </b><font color="Maroon" title="c2">p</font>, <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 <font color="Maroon" title="c1">OAS</font>;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:33"/><i><font color="Green" title="E16">A1</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="c3">a</font>,<font color="Maroon" title="c4">b</font>
 <b>and </b><br/><a NAME="E2:33"/><i><font color="Green" title="E17">A2</font></i>: 
 <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="c4">b</font>
 ;<br/>

<a NAME="E3:33"/><i><font color="Green" title="E18">A3</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">a</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="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="c4">b</font> )
 
<b>by </b><i><a class="txt" href="pasch.html#E1:33"><i><font color="Green" title="E16">A1</font></i></a>, <a class="ref" href="diraf.html#T37" title="DIRAF:th.37">DIRAF:37</a></i>;<br/>
<a NAME="E4:33"/><i><font color="Green" title="E19">A4</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="c4">b</font>
 
<b>by </b><i><a class="txt" href="pasch.html#E2:33"><i><font color="Green" title="E17">A2</font></i></a>, <a class="ref" href="diraf.html#T34" title="DIRAF:th.34">DIRAF:34</a></i>;<br/>
<a NAME="E5:33"/><i><font color="Green" title="E20">A5</font></i>: 
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c2">p</font>
 
<b>by </b><i><a class="txt" href="pasch.html#E2:33"><i><font color="Green" title="E17">A2</font></i></a>, <a class="ref" href="diraf.html#T13" title="DIRAF:th.13">DIRAF:13</a></i>;<br/>
<a NAME="E6:33"/><b>then </b><i><font color="Green" title="E21">A6</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="c5">c</font>,<font color="Maroon" title="c2">p</font>
 
<b>by </b><i><a class="ref" href="diraf.html#D3" title="DIRAF:def.3">DIRAF:def 3</a></i>;<br/>
<div><i><font color="Green" title="E22">A7</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_1"/><i><font color="Green" title="E23">A8</font></i>: 
( <font color="Maroon" title="c4">b</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="c5">c</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> )
 ;<br/><b>consider </b><font color="Maroon" title="c6">e1</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="E3:33_1"/><i><font color="Green" title="E24">A9</font></i>: 
(  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c6">e1</font>,<font color="Maroon" title="c3">a</font> &amp; <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">e1</font> )
 <b>by </b><i><a class="txt" href="pasch.html#E5:33"><i><font color="Green" title="E20">A5</font></i></a>, <a class="ref" href="pasch.html#T26" target="_self" title="PASCH:th.26">Th26</a></i>;<br/><b>consider </b><font color="Maroon" title="c7">e2</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="E5:33_1"/><i><font color="Green" title="E25">A10</font></i>: 
( <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="c5">c</font>,<font color="Maroon" title="c7">e2</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="c2">p</font>,<font color="Maroon" title="c7">e2</font> )
 <b>by </b><i><a class="txt" href="pasch.html#E6:33"><i><font color="Green" title="E21">A6</font></i></a>, <a class="txt" href="pasch.html#E1:33_1"><i><font color="Green" title="E23">A8</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="c8">e3</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="E7:33_1"/><i><font color="Green" title="E26">A11</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="c5">c</font>,<font color="Maroon" title="c8">e3</font> &amp; <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">e2</font>,<font color="Maroon" title="c8">e3</font> )
 <b>by </b><i><a class="txt" href="pasch.html#E1:33_1"><i><font color="Green" title="E23">A8</font></i></a>, <a class="txt" href="pasch.html#E5:33_1"><i><font color="Green" title="E25">A10</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="c9">e4</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="E9:33_1"/><i><font color="Green" title="E27">A12</font></i>: 
( <font color="Maroon" title="c6">e1</font>,<font color="Maroon" title="c5">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c9">e4</font> &amp; <font color="Maroon" title="c6">e1</font>,<font color="Maroon" title="c4">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c9">e4</font> )
 <b>by </b><i><a class="txt" href="pasch.html#E6:33"><i><font color="Green" title="E21">A6</font></i></a>, <a class="txt" href="pasch.html#E1:33_1"><i><font color="Green" title="E23">A8</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="c10">e5</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="E11:33_1"/><i><font color="Green" title="E28">A13</font></i>: 
( <font color="Maroon" title="c9">e4</font>,<font color="Maroon" title="c7">e2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c10">e5</font> &amp; <font color="Maroon" title="c9">e4</font>,<font color="Maroon" title="c5">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">e2</font>,<font color="Maroon" title="c10">e5</font> &amp; <font color="Maroon" title="c7">e2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">e5</font> )
 <b>by </b><i><a class="ref" href="analoaf.html#D5" title="ANALOAF:def.5">ANALOAF:def 5</a></i>;<br/><a NAME="E12:33_1"/><i><font color="Green" title="E29">A14</font></i>: 
<font color="Maroon" title="c8">e3</font>,<font color="Maroon" title="c5">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c2">p</font>
 <b>by </b><i><a class="txt" href="pasch.html#E7:33_1"><i><font color="Green" title="E26">A11</font></i></a>, <a class="ref" href="diraf.html#T5" title="DIRAF:th.5">DIRAF:5</a></i>;<br/><a NAME="E13:33_1"/><i><font color="Green" title="E30">A15</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="c5">c</font>,<font color="Maroon" title="c3">a</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:33_1_1"/>
 <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">a</font>
 ;<br/>

<a NAME="E2:33_1_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">a</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#T37" title="DIRAF:th.37">DIRAF:37</a></i>;<br/>
<b>hence </b><a NAME="E3:33_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="pasch.html#E1:33"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="pasch.html#E4:33"><i><font color="Green" title="E19">A4</font></i></a>, <a class="txt" href="pasch.html#E1:33_1"><i><font color="Green" title="E23">A8</font></i></a>, <a class="ref" href="diraf.html#T38" title="DIRAF:th.38">DIRAF:38</a></i>;<br/>


</div><b>end;</b></div><a NAME="E14:33_1"/><i><font color="Green" title="E31">A16</font></i>: 
not  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:33_1_2"/>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font>
 ;<br/>

<a NAME="E2:33_1_2"/><b>then </b>
(  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">a</font> &amp;  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c4">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="c4">b</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c4">b</font> )
 
<b>by </b><i><a class="txt" href="pasch.html#E4:33"><i><font color="Green" title="E19">A4</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/>
<b>hence </b><a NAME="E3:33_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="pasch.html#E1:33"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="pasch.html#E1:33_1"><i><font color="Green" title="E23">A8</font></i></a>, <a class="ref" href="diraf.html#T38" title="DIRAF:th.38">DIRAF:38</a></i>;<br/>


</div><b>end;</b></div><a NAME="E15:33_1"/><i><font color="Green" title="E32">A17</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="c7">e2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:33_1_3"/>
<font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">e2</font>
 ;<br/>

<a NAME="E2:33_1_3"/><b>then </b>
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c2">p</font>
 
<b>by </b><i><a class="txt" href="pasch.html#E5:33_1"><i><font color="Green" title="E25">A10</font></i></a>, <a class="ref" href="diraf.html#D3" title="DIRAF:def.3">DIRAF:def 3</a></i>;<br/>
<a NAME="E3:33_1_3"/><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="c5">c</font>,<font color="Maroon" title="c3">a</font>
 
<b>by </b><i><a class="ref" href="diraf.html#T13" title="DIRAF:th.13">DIRAF:13</a></i>;<br/>
<b>hence </b><a NAME="E4:33_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="pasch.html#E13:33_1"><i><font color="Green" title="E30">A15</font></i></a>, <a class="ref" href="diraf.html#T34" title="DIRAF:th.34">DIRAF:34</a></i>;<br/>


</div><b>end;</b></div><a NAME="E16:33_1"/><i><font color="Green" title="E33">A18</font></i>: 
<font color="Maroon" title="c5">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">e2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:33_1_4"/><i><font color="Green" title="E34">A19</font></i>: 
<font color="Maroon" title="c5">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">e2</font>
 ;<br/>

<a NAME="E2:33_1_4"/>
<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="c5">c</font>,<font color="Maroon" title="c4">b</font>
 
<b>by </b><i><a class="txt" href="pasch.html#E6:33"><i><font color="Green" title="E21">A6</font></i></a>, <a class="ref" href="diraf.html#T5" title="DIRAF:th.5">DIRAF:5</a></i>;<br/>
<a NAME="E3:33_1_4"/><b>then </b>
<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="c4">b</font>
 
<b>by </b><i><a class="txt" href="pasch.html#E1:33_1"><i><font color="Green" title="E23">A8</font></i></a>, <a class="txt" href="pasch.html#E5:33_1"><i><font color="Green" title="E25">A10</font></i></a>, <a class="txt" href="pasch.html#E1:33_1_4"><i><font color="Green" title="E34">A19</font></i></a>, <a class="ref" href="diraf.html#T6" title="DIRAF:th.6">DIRAF:6</a></i>;<br/>
<a NAME="E4:33_1_4"/><b>then </b>
<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="c4">b</font>,<font color="Maroon" title="c5">c</font>
 
<b>by </b><i><a class="ref" href="diraf.html#T5" title="DIRAF:th.5">DIRAF:5</a></i>;<br/>
<a NAME="E5:33_1_4"/><b>then </b>
(  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font> or  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">a</font> )
 
<b>by </b><i><a class="ref" href="diraf.html#T11" title="DIRAF:th.11">DIRAF:11</a></i>;<br/>
<a NAME="E6:33_1_4"/><b>then </b>
(  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font> or  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">a</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="E7:33_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="pasch.html#E14:33_1"><i><font color="Green" title="E31">A16</font></i></a>, <a class="ref" href="diraf.html#T36" title="DIRAF:th.36">DIRAF:36</a></i>;<br/>


</div><b>end;</b></div><a NAME="E17:33_1"/><i><font color="Green" title="E34">A20</font></i>: 
<font color="Maroon" title="c5">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">e3</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_5"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:33_1_5"/>
<font color="Maroon" title="c5">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">e3</font>
 ;<br/>

<a NAME="E2:33_1_5"/><b>then </b>
<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c7">e2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c2">p</font>
 
<b>by </b><i><a class="txt" href="pasch.html#E7:33_1"><i><font color="Green" title="E26">A11</font></i></a>, <a class="ref" href="diraf.html#T5" title="DIRAF:th.5">DIRAF:5</a></i>;<br/>
<a NAME="E3:33_1_5"/><b>then </b>
<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="c3">a</font>,<font color="Maroon" title="c2">p</font>
 
<b>by </b><i><a class="txt" href="pasch.html#E5:33_1"><i><font color="Green" title="E25">A10</font></i></a>, <a class="txt" href="pasch.html#E16:33_1"><i><font color="Green" title="E33">A18</font></i></a>, <a class="ref" href="diraf.html#T6" title="DIRAF:th.6">DIRAF:6</a></i>;<br/>
<a NAME="E4:33_1_5"/><b>then </b>
(  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c2">p</font> or  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c2">p</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="E5:33_1_5"/><b>then </b>
(  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c2">p</font> or  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c2">p</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/>
<b>hence </b><a NAME="E6:33_1_5"/>
contradiction
 <b>by </b><i><a class="txt" href="pasch.html#E13:33_1"><i><font color="Green" title="E30">A15</font></i></a>, <a class="ref" href="diraf.html#T36" title="DIRAF:th.36">DIRAF:36</a></i>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon" title="c11">x</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="E19:33_1"/><i><font color="Green" title="E35">A21</font></i>: 
( <font color="Maroon" title="c10">e5</font>,<font color="Maroon" title="c5">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c11">x</font> &amp; <font color="Maroon" title="c10">e5</font>,<font color="Maroon" title="c8">e3</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c11">x</font> )
 <b>by </b><i><a class="txt" href="pasch.html#E12:33_1"><i><font color="Green" title="E29">A14</font></i></a>, <a class="ref" href="analoaf.html#D5" title="ANALOAF:def.5">ANALOAF:def 5</a></i>;<br/><a NAME="E20:33_1"/><i><font color="Green" title="E36">A22</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="c8">e3</font>
 <b>by </b><i><a class="txt" href="pasch.html#E1:33_1"><i><font color="Green" title="E23">A8</font></i></a>, <a class="txt" href="pasch.html#E7:33_1"><i><font color="Green" title="E26">A11</font></i></a>, <a class="ref" href="analoaf.html#D5" title="ANALOAF:def.5">ANALOAF:def 5</a></i>;<br/><a NAME="E21:33_1"/><i><font color="Green" title="E37">A23</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="c8">e3</font>,<font color="Maroon" title="c7">e2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_6"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:33_1_6"/><i><font color="Green" title="E38">A24</font></i>: 
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c8">e3</font>,<font color="Maroon" title="c7">e2</font>
 ;<br/>

<a NAME="E2:33_1_6"/>
<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="c8">e3</font>
 
<b>by </b><i><a class="txt" href="pasch.html#E7:33_1"><i><font color="Green" title="E26">A11</font></i></a>, <a class="ref" href="analoaf.html#D5" title="ANALOAF:def.5">ANALOAF:def 5</a></i>;<br/>
<a NAME="E3:33_1_6"/><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="c5">c</font>,<font color="Maroon" title="c8">e3</font> or  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c8">e3</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="E4:33_1_6"/><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="c8">e3</font> or  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c8">e3</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="E5:33_1_6"/><b>then </b><i><font color="Green" title="E39">A25</font></i>: 
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c8">e3</font>,<font color="Maroon" title="c5">c</font>
 
<b>by </b><i><a class="ref" href="diraf.html#T36" title="DIRAF:th.36">DIRAF:36</a></i>;<br/>
<a NAME="E6:33_1_6"/>
<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="c3">a</font>,<font color="Maroon" title="c7">e2</font>
 
<b>by </b><i><a class="txt" href="pasch.html#E5:33_1"><i><font color="Green" title="E25">A10</font></i></a>, <a class="ref" href="analoaf.html#D5" title="ANALOAF:def.5">ANALOAF:def 5</a></i>;<br/>
<a NAME="E7:33_1_6"/><b>then </b>
(  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c7">e2</font> or  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c7">e2</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="E8:33_1_6"/><b>then </b>
(  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c7">e2</font> or  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c7">e2</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="E9:33_1_6"/><b>then </b>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c7">e2</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="E10:33_1_6"/><b>then </b><i><font color="Green" title="E40">A26</font></i>: 
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c8">e3</font>,<font color="Maroon" title="c3">a</font>
 
<b>by </b><i><a class="txt" href="pasch.html#E16:33_1"><i><font color="Green" title="E33">A18</font></i></a>, <a class="txt" href="pasch.html#E1:33_1_6"><i><font color="Green" title="E38">A24</font></i></a>, <a class="txt" href="pasch.html#E5:33_1_6"><i><font color="Green" title="E39">A25</font></i></a>, <a class="ref" href="diraf.html#T41" title="DIRAF:th.41">DIRAF:41</a></i>;<br/>
<a NAME="E11:33_1_6"/>
<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="c5">c</font>,<font color="Maroon" title="c4">b</font>
 
<b>by </b><i><a class="txt" href="pasch.html#E2:33"><i><font color="Green" title="E17">A2</font></i></a>, <a class="ref" href="diraf.html#D3" title="DIRAF:def.3">DIRAF:def 3</a></i>;<br/>
<a NAME="E12:33_1_6"/><b>then </b>
( <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="c4">b</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="c8">e3</font> )
 
<b>by </b><i><a class="txt" href="pasch.html#E7:33_1"><i><font color="Green" title="E26">A11</font></i></a>, <a class="ref" href="analoaf.html#D5" title="ANALOAF:def.5">ANALOAF:def 5</a></i>;<br/>
<a NAME="E13:33_1_6"/><b>then </b>
<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c8">e3</font>
 
<b>by </b><i><a class="txt" href="pasch.html#E1:33_1"><i><font color="Green" title="E23">A8</font></i></a>, <a class="ref" href="analoaf.html#D5" title="ANALOAF:def.5">ANALOAF:def 5</a></i>;<br/>
<a NAME="E14:33_1_6"/><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">b</font>,<font color="Maroon" title="c8">e3</font> or  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c8">e3</font>,<font color="Maroon" title="c4">b</font> )
 
<b>by </b><i><a class="ref" href="diraf.html#T11" title="DIRAF:th.11">DIRAF:11</a></i>;<br/>
<a NAME="E15:33_1_6"/><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="c4">b</font>,<font color="Maroon" title="c8">e3</font> or  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c8">e3</font>,<font color="Maroon" title="c4">b</font> )
 
<b>by </b><i><a class="ref" href="diraf.html#T34" title="DIRAF:th.34">DIRAF:34</a></i>;<br/>
<a NAME="E16:33_1_6"/><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="c8">e3</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">p</font>,<font color="Maroon" title="c8">e3</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/>
<b>hence </b><a NAME="E17:33_1_6"/>
contradiction
 <b>by </b><i><a class="txt" href="pasch.html#E1:33"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="pasch.html#E20:33_1"><i><font color="Green" title="E36">A22</font></i></a>, <a class="txt" href="pasch.html#E10:33_1_6"><i><font color="Green" title="E40">A26</font></i></a>, <a class="ref" href="diraf.html#T38" title="DIRAF:th.38">DIRAF:38</a></i>;<br/>


</div><b>end;</b></div><a NAME="E22:33_1"/><i><font color="Green" title="E38">A27</font></i>: 
<font color="Maroon" title="c5">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">e1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_7"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:33_1_7"/>
<font color="Maroon" title="c5">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">e1</font>
 ;<br/>

<a NAME="E2:33_1_7"/><b>then </b>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">a</font>
 
<b>by </b><i><a class="txt" href="pasch.html#E3:33_1"><i><font color="Green" title="E24">A9</font></i></a>, <a class="ref" href="diraf.html#T34" title="DIRAF:th.34">DIRAF:34</a></i>;<br/>
<b>hence </b><a NAME="E3:33_1_7"/>
contradiction
 <b>by </b><i><a class="txt" href="pasch.html#E14:33_1"><i><font color="Green" title="E31">A16</font></i></a>, <a class="ref" href="diraf.html#T36" title="DIRAF:th.36">DIRAF:36</a></i>;<br/>


</div><b>end;</b></div><a NAME="E23:33_1"/><i><font color="Green" title="E39">A28</font></i>: 
<font color="Maroon" title="c9">e4</font>,<font color="Maroon" title="c5">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">e2</font>,<font color="Maroon" title="c8">e3</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_8"><b>proof </b></a><div class="add">

<a NAME="E1:33_1_8"/><i><font color="Green" title="E40">A29</font></i>: 
<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">e1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">e2</font>,<font color="Maroon" title="c8">e3</font>
 
<b>by </b><i><a class="txt" href="pasch.html#E3:33"><i><font color="Green" title="E18">A3</font></i></a>, <a class="txt" href="pasch.html#E3:33_1"><i><font color="Green" title="E24">A9</font></i></a>, <a class="txt" href="pasch.html#E7:33_1"><i><font color="Green" title="E26">A11</font></i></a>, <a class="ref" href="analoaf.html#D5" title="ANALOAF:def.5">ANALOAF:def 5</a></i>;<br/>
<a NAME="E2:33_1_8"/>
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c6">e1</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c9">e4</font>
 
<b>by </b><i><a class="txt" href="pasch.html#E9:33_1"><i><font color="Green" title="E27">A12</font></i></a>, <a class="ref" href="diraf.html#D3" title="DIRAF:def.3">DIRAF:def 3</a></i>;<br/>
<a NAME="E3:33_1_8"/><b>then </b>
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c9">e4</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">e1</font>
 
<b>by </b><i><a class="ref" href="diraf.html#T13" title="DIRAF:th.13">DIRAF:13</a></i>;<br/>
<a NAME="E4:33_1_8"/><b>then </b>
<font color="Maroon" title="c9">e4</font>,<font color="Maroon" title="c5">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">e1</font>
 
<b>by </b><i><a class="ref" href="diraf.html#D3" title="DIRAF:def.3">DIRAF:def 3</a></i>;<br/>
<b>hence </b><a NAME="E5:33_1_8"/>
<font color="Maroon" title="c9">e4</font>,<font color="Maroon" title="c5">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">e2</font>,<font color="Maroon" title="c8">e3</font>
 <b>by </b><i><a class="txt" href="pasch.html#E22:33_1"><i><font color="Green" title="E38">A27</font></i></a>, <a class="txt" href="pasch.html#E1:33_1_8"><i><font color="Green" title="E40">A29</font></i></a>, <a class="ref" href="diraf.html#T6" title="DIRAF:th.6">DIRAF:6</a></i>;<br/>


</div><b>end;</b></div><a NAME="E24:33_1"/><i><font color="Green" title="E40">A30</font></i>: 
<font color="Maroon" title="c4">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">e1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_9"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:33_1_9"/>
<font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">e1</font>
 ;<br/>

<a NAME="E2:33_1_9"/><b>then </b>
( <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c4">b</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="c5">c</font>,<font color="Maroon" title="c4">b</font> )
 
<b>by </b><i><a class="txt" href="pasch.html#E2:33"><i><font color="Green" title="E17">A2</font></i></a>, <a class="txt" href="pasch.html#E3:33_1"><i><font color="Green" title="E24">A9</font></i></a>, <a class="ref" href="diraf.html#D3" title="DIRAF:def.3">DIRAF:def 3</a></i>;<br/>
<a NAME="E3:33_1_9"/><b>then </b>
( <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c4">b</font> &amp; <font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c4">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c5">c</font> )
 
<b>by </b><i><a class="ref" href="diraf.html#T5" title="DIRAF:th.5">DIRAF:5</a></i>;<br/>
<a NAME="E4:33_1_9"/><b>then </b>
<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c5">c</font>
 
<b>by </b><i><a class="txt" href="pasch.html#E1:33_1"><i><font color="Green" title="E23">A8</font></i></a>, <a class="ref" href="diraf.html#T6" title="DIRAF:th.6">DIRAF:6</a></i>;<br/>
<a NAME="E5:33_1_9"/><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="c3">a</font>,<font color="Maroon" title="c5">c</font> or  <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="c3">a</font> )
 
<b>by </b><i><a class="ref" href="diraf.html#T11" title="DIRAF:th.11">DIRAF:11</a></i>;<br/>
<a NAME="E6:33_1_9"/><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">a</font>,<font color="Maroon" title="c5">c</font> or  <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">a</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="E7:33_1_9"/>
contradiction
 <b>by </b><i><a class="txt" href="pasch.html#E13:33_1"><i><font color="Green" title="E30">A15</font></i></a>, <a class="ref" href="diraf.html#T36" title="DIRAF:th.36">DIRAF:36</a></i>;<br/>


</div><b>end;</b></div><a NAME="E25:33_1"/><i><font color="Green" title="E41">A31</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="c9">e4</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_10"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:33_1_10"/>
<font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">e4</font>
 ;<br/>

<a NAME="E2:33_1_10"/><b>then </b>
<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">e1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c5">c</font>
 
<b>by </b><i><a class="txt" href="pasch.html#E9:33_1"><i><font color="Green" title="E27">A12</font></i></a>, <a class="ref" href="diraf.html#T5" title="DIRAF:th.5">DIRAF:5</a></i>;<br/>
<a NAME="E3:33_1_10"/><b>then </b>
<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c5">c</font>
 
<b>by </b><i><a class="txt" href="pasch.html#E3:33_1"><i><font color="Green" title="E24">A9</font></i></a>, <a class="txt" href="pasch.html#E22:33_1"><i><font color="Green" title="E38">A27</font></i></a>, <a class="ref" href="diraf.html#T6" title="DIRAF:th.6">DIRAF:6</a></i>;<br/>
<a NAME="E4:33_1_10"/><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="c3">a</font>,<font color="Maroon" title="c5">c</font> or  <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="c3">a</font> )
 
<b>by </b><i><a class="ref" href="diraf.html#T11" title="DIRAF:th.11">DIRAF:11</a></i>;<br/>
<a NAME="E5:33_1_10"/><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">a</font>,<font color="Maroon" title="c5">c</font> or  <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">a</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="E6:33_1_10"/>
contradiction
 <b>by </b><i><a class="txt" href="pasch.html#E13:33_1"><i><font color="Green" title="E30">A15</font></i></a>, <a class="ref" href="diraf.html#T36" title="DIRAF:th.36">DIRAF:36</a></i>;<br/>


</div><b>end;</b></div><a NAME="E26:33_1"/><i><font color="Green" title="E42">A32</font></i>: 
<font color="Maroon" title="c7">e2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">e4</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_11"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:33_1_11"/>
<font color="Maroon" title="c7">e2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">e4</font>
 ;<br/>

<a NAME="E2:33_1_11"/><b>then </b>
<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c7">e2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">e1</font>,<font color="Maroon" title="c5">c</font>
 
<b>by </b><i><a class="txt" href="pasch.html#E9:33_1"><i><font color="Green" title="E27">A12</font></i></a>, <a class="ref" href="diraf.html#T5" title="DIRAF:th.5">DIRAF:5</a></i>;<br/>
<a NAME="E3:33_1_11"/><b>then </b>
<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">e1</font>,<font color="Maroon" title="c5">c</font>
 
<b>by </b><i><a class="txt" href="pasch.html#E5:33_1"><i><font color="Green" title="E25">A10</font></i></a>, <a class="txt" href="pasch.html#E16:33_1"><i><font color="Green" title="E33">A18</font></i></a>, <a class="ref" href="diraf.html#T6" title="DIRAF:th.6">DIRAF:6</a></i>;<br/>
<a NAME="E4:33_1_11"/><b>then </b>
<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">e1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">a</font>
 
<b>by </b><i><a class="ref" href="diraf.html#T5" title="DIRAF:th.5">DIRAF:5</a></i>;<br/>
<a NAME="E5:33_1_11"/><b>then </b>
<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">a</font>
 
<b>by </b><i><a class="txt" href="pasch.html#E3:33_1"><i><font color="Green" title="E24">A9</font></i></a>, <a class="txt" href="pasch.html#E22:33_1"><i><font color="Green" title="E38">A27</font></i></a>, <a class="ref" href="diraf.html#T6" title="DIRAF:th.6">DIRAF:6</a></i>;<br/>
<a NAME="E6:33_1_11"/><b>then </b>
<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c2">p</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="ref" href="diraf.html#T5" title="DIRAF:th.5">DIRAF:5</a></i>;<br/>
<a NAME="E7:33_1_11"/><b>then </b>
(  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c5">c</font> or  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c3">a</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#T11" title="DIRAF:th.11">DIRAF:11</a></i>;<br/>
<a NAME="E8:33_1_11"/><b>then </b>
(  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c5">c</font> or  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c3">a</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#T34" title="DIRAF:th.34">DIRAF:34</a></i>;<br/>
<b>hence </b><a NAME="E9:33_1_11"/>
contradiction
 <b>by </b><i><a class="txt" href="pasch.html#E13:33_1"><i><font color="Green" title="E30">A15</font></i></a>, <a class="ref" href="diraf.html#T36" title="DIRAF:th.36">DIRAF:36</a></i>;<br/>


</div><b>end;</b></div><a NAME="E27:33_1"/><i><font color="Green" title="E43">A33</font></i>: 
<font color="Maroon" title="c5">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">e4</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_12"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:33_1_12"/>
<font color="Maroon" title="c5">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">e4</font>
 ;<br/>

<a NAME="E2:33_1_12"/><b>then </b>
( <font color="Maroon" title="c6">e1</font>,<font color="Maroon" title="c4">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c5">c</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="c5">c</font>,<font color="Maroon" title="c4">b</font> )
 
<b>by </b><i><a class="txt" href="pasch.html#E2:33"><i><font color="Green" title="E17">A2</font></i></a>, <a class="txt" href="pasch.html#E9:33_1"><i><font color="Green" title="E27">A12</font></i></a>, <a class="ref" href="diraf.html#D3" title="DIRAF:def.3">DIRAF:def 3</a></i>;<br/>
<a NAME="E3:33_1_12"/><b>then </b>
<font color="Maroon" title="c6">e1</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="c4">b</font>
 
<b>by </b><i><a class="txt" href="pasch.html#E1:33_1"><i><font color="Green" title="E23">A8</font></i></a>, <a class="ref" href="diraf.html#T6" title="DIRAF:th.6">DIRAF:6</a></i>;<br/>
<a NAME="E4:33_1_12"/><b>then </b>
<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c6">e1</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>
 
<b>by </b><i><a class="ref" href="diraf.html#T5" title="DIRAF:th.5">DIRAF:5</a></i>;<br/>
<a NAME="E5:33_1_12"/><b>then </b>
(  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c6">e1</font>,<font color="Maroon" title="c5">c</font> or  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">e1</font> )
 
<b>by </b><i><a class="ref" href="diraf.html#T11" title="DIRAF:th.11">DIRAF:11</a></i>;<br/>
<a NAME="E6:33_1_12"/><b>then </b>
(  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c6">e1</font>,<font color="Maroon" title="c5">c</font> or  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">e1</font> )
 
<b>by </b><i><a class="ref" href="diraf.html#T34" title="DIRAF:th.34">DIRAF:34</a></i>;<br/>
<a NAME="E7:33_1_12"/><b>then </b>
(  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c6">e1</font>,<font color="Maroon" title="c5">c</font> &amp;  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c6">e1</font>,<font color="Maroon" title="c3">a</font> &amp;  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c6">e1</font>,<font color="Maroon" title="c4">b</font> )
 
<b>by </b><i><a class="txt" href="pasch.html#E3:33_1"><i><font color="Green" title="E24">A9</font></i></a>, <a class="ref" href="diraf.html#T34" title="DIRAF:th.34">DIRAF:34</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/>
<b>hence </b><a NAME="E8:33_1_12"/>
contradiction
 <b>by </b><i><a class="txt" href="pasch.html#E14:33_1"><i><font color="Green" title="E31">A16</font></i></a>, <a class="txt" href="pasch.html#E24:33_1"><i><font color="Green" title="E40">A30</font></i></a>, <a class="ref" href="diraf.html#T38" title="DIRAF:th.38">DIRAF:38</a></i>;<br/>


</div><b>end;</b></div><a NAME="E28:33_1"/><i><font color="Green" title="E44">A34</font></i>: 
<font color="Maroon" title="c5">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">e5</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_13"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:33_1_13"/>
<font color="Maroon" title="c5">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">e5</font>
 ;<br/>

<a NAME="E2:33_1_13"/><b>then </b>
<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c9">e4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c7">e2</font>
 
<b>by </b><i><a class="txt" href="pasch.html#E11:33_1"><i><font color="Green" title="E28">A13</font></i></a>, <a class="ref" href="diraf.html#T5" title="DIRAF:th.5">DIRAF:5</a></i>;<br/>
<a NAME="E3:33_1_13"/><b>then </b>
<font color="Maroon" title="c6">e1</font>,<font color="Maroon" title="c5">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c7">e2</font>
 
<b>by </b><i><a class="txt" href="pasch.html#E9:33_1"><i><font color="Green" title="E27">A12</font></i></a>, <a class="txt" href="pasch.html#E27:33_1"><i><font color="Green" title="E43">A33</font></i></a>, <a class="ref" href="diraf.html#T6" title="DIRAF:th.6">DIRAF:6</a></i>;<br/>
<a NAME="E4:33_1_13"/><b>then </b>
<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c7">e2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">e1</font>,<font color="Maroon" title="c5">c</font>
 
<b>by </b><i><a class="ref" href="diraf.html#T5" title="DIRAF:th.5">DIRAF:5</a></i>;<br/>
<a NAME="E5:33_1_13"/><b>then </b>
<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">e1</font>,<font color="Maroon" title="c5">c</font>
 
<b>by </b><i><a class="txt" href="pasch.html#E5:33_1"><i><font color="Green" title="E25">A10</font></i></a>, <a class="txt" href="pasch.html#E16:33_1"><i><font color="Green" title="E33">A18</font></i></a>, <a class="ref" href="diraf.html#T6" title="DIRAF:th.6">DIRAF:6</a></i>;<br/>
<a NAME="E6:33_1_13"/><b>then </b>
<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">e1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">a</font>
 
<b>by </b><i><a class="ref" href="diraf.html#T5" title="DIRAF:th.5">DIRAF:5</a></i>;<br/>
<a NAME="E7:33_1_13"/><b>then </b>
<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">a</font>
 
<b>by </b><i><a class="txt" href="pasch.html#E3:33_1"><i><font color="Green" title="E24">A9</font></i></a>, <a class="txt" href="pasch.html#E22:33_1"><i><font color="Green" title="E38">A27</font></i></a>, <a class="ref" href="diraf.html#T6" title="DIRAF:th.6">DIRAF:6</a></i>;<br/>
<a NAME="E8:33_1_13"/><b>then </b>
<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c2">p</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="ref" href="diraf.html#T5" title="DIRAF:th.5">DIRAF:5</a></i>;<br/>
<a NAME="E9:33_1_13"/><b>then </b>
(  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c5">c</font> or  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c3">a</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#T11" title="DIRAF:th.11">DIRAF:11</a></i>;<br/>
<a NAME="E10:33_1_13"/><b>then </b>
(  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c5">c</font> or  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c3">a</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#T34" title="DIRAF:th.34">DIRAF:34</a></i>;<br/>
<b>hence </b><a NAME="E11:33_1_13"/>
contradiction
 <b>by </b><i><a class="txt" href="pasch.html#E13:33_1"><i><font color="Green" title="E30">A15</font></i></a>, <a class="ref" href="diraf.html#T36" title="DIRAF:th.36">DIRAF:36</a></i>;<br/>


</div><b>end;</b></div><a NAME="E29:33_1"/><i><font color="Green" title="E45">A35</font></i>: 
<font color="Maroon" title="c3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">e1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_14"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:33_1_14"/>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">e1</font>
 ;<br/>

<a NAME="E2:33_1_14"/><b>then </b>
<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c2">p</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="pasch.html#E3:33_1"><i><font color="Green" title="E24">A9</font></i></a>, <a class="ref" href="diraf.html#T5" title="DIRAF:th.5">DIRAF:5</a></i>;<br/>
<a NAME="E3:33_1_14"/><b>then </b>
(  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c5">c</font> or  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c3">a</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#T11" title="DIRAF:th.11">DIRAF:11</a></i>;<br/>
<a NAME="E4:33_1_14"/><b>then </b>
(  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c5">c</font> or  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c3">a</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#T34" title="DIRAF:th.34">DIRAF:34</a></i>;<br/>
<b>hence </b><a NAME="E5:33_1_14"/>
contradiction
 <b>by </b><i><a class="txt" href="pasch.html#E13:33_1"><i><font color="Green" title="E30">A15</font></i></a>, <a class="ref" href="diraf.html#T36" title="DIRAF:th.36">DIRAF:36</a></i>;<br/>


</div><b>end;</b></div><a NAME="E30:33_1"/>
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c6">e1</font>,<font color="Maroon" title="c4">b</font>
 <b>by </b><i><a class="txt" href="pasch.html#E3:33_1"><i><font color="Green" title="E24">A9</font></i></a>, <a class="ref" href="diraf.html#T13" title="DIRAF:th.13">DIRAF:13</a></i>;<br/><a NAME="E31:33_1"/><b>then </b>
<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c6">e1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">e1</font>,<font color="Maroon" title="c4">b</font>
 <b>by </b><i><a class="ref" href="diraf.html#D3" title="DIRAF:def.3">DIRAF:def 3</a></i>;<br/><a NAME="E32:33_1"/><b>then </b>
( <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c6">e1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">e1</font>,<font color="Maroon" title="c4">b</font> &amp; <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c6">e1</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="ref" href="analoaf.html#D5" title="ANALOAF:def.5">ANALOAF:def 5</a></i>;<br/><a NAME="E33:33_1"/><b>then </b>
<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">e1</font>,<font color="Maroon" title="c4">b</font>
 <b>by </b><i><a class="txt" href="pasch.html#E29:33_1"><i><font color="Green" title="E45">A35</font></i></a>, <a class="ref" href="analoaf.html#D5" title="ANALOAF:def.5">ANALOAF:def 5</a></i>;<br/><a NAME="E34:33_1"/><b>then </b>
<font color="Maroon" title="c6">e1</font>,<font color="Maroon" title="c4">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c7">e2</font>
 <b>by </b><i><a class="txt" href="pasch.html#E3:33"><i><font color="Green" title="E18">A3</font></i></a>, <a class="txt" href="pasch.html#E5:33_1"><i><font color="Green" title="E25">A10</font></i></a>, <a class="ref" href="analoaf.html#D5" title="ANALOAF:def.5">ANALOAF:def 5</a></i>;<br/><a NAME="E35:33_1"/><b>then </b><i><font color="Green" title="E46">A36</font></i>: 
<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c9">e4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c7">e2</font>
 <b>by </b><i><a class="txt" href="pasch.html#E9:33_1"><i><font color="Green" title="E27">A12</font></i></a>, <a class="txt" href="pasch.html#E24:33_1"><i><font color="Green" title="E40">A30</font></i></a>, <a class="ref" href="analoaf.html#D5" title="ANALOAF:def.5">ANALOAF:def 5</a></i>;<br/><a NAME="E36:33_1"/>
 <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="c8">e3</font>
 <b>by </b><i><a class="txt" href="pasch.html#E7:33_1"><i><font color="Green" title="E26">A11</font></i></a>, <a class="ref" href="diraf.html#D3" title="DIRAF:def.3">DIRAF:def 3</a></i>;<br/><a NAME="E37:33_1"/><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="c9">e4</font>,<font color="Maroon" title="c7">e2</font>
 <b>by </b><i><a class="txt" href="pasch.html#E21:33_1"><i><font color="Green" title="E37">A23</font></i></a>, <a class="txt" href="pasch.html#E23:33_1"><i><font color="Green" title="E39">A28</font></i></a>, <a class="txt" href="pasch.html#E25:33_1"><i><font color="Green" title="E41">A31</font></i></a>, <a class="txt" href="pasch.html#E35:33_1"><i><font color="Green" title="E46">A36</font></i></a>, <a class="ref" href="pasch.html#T23" target="_self" title="PASCH:th.23">Th23</a></i>;<br/><a NAME="E38:33_1"/><b>then </b>
<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c9">e4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">e4</font>,<font color="Maroon" title="c7">e2</font>
 <b>by </b><i><a class="ref" href="diraf.html#D3" title="DIRAF:def.3">DIRAF:def 3</a></i>;<br/><a NAME="E39:33_1"/><b>then </b>
( <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c9">e4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">e4</font>,<font color="Maroon" title="c7">e2</font> &amp; <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c9">e4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c7">e2</font> )
 <b>by </b><i><a class="ref" href="analoaf.html#D5" title="ANALOAF:def.5">ANALOAF:def 5</a></i>;<br/><a NAME="E40:33_1"/><b>then </b><i><font color="Green" title="E47">A37</font></i>: 
<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c7">e2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">e4</font>,<font color="Maroon" title="c7">e2</font>
 <b>by </b><i><a class="txt" href="pasch.html#E25:33_1"><i><font color="Green" title="E41">A31</font></i></a>, <a class="ref" href="analoaf.html#D5" title="ANALOAF:def.5">ANALOAF:def 5</a></i>;<br/><a NAME="E41:33_1"/><b>then </b><i><font color="Green" title="E48">A38</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="c9">e4</font>,<font color="Maroon" title="c7">e2</font>
 <b>by </b><i><a class="txt" href="pasch.html#E5:33_1"><i><font color="Green" title="E25">A10</font></i></a>, <a class="txt" href="pasch.html#E15:33_1"><i><font color="Green" title="E32">A17</font></i></a>, <a class="ref" href="diraf.html#T6" title="DIRAF:th.6">DIRAF:6</a></i>;<br/><a NAME="E42:33_1"/><b>then </b><i><font color="Green" title="E49">A39</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="c10">e5</font>
 <b>by </b><i><a class="txt" href="pasch.html#E11:33_1"><i><font color="Green" title="E28">A13</font></i></a>, <a class="txt" href="pasch.html#E26:33_1"><i><font color="Green" title="E42">A32</font></i></a>, <a class="ref" href="diraf.html#T6" title="DIRAF:th.6">DIRAF:6</a></i>;<br/><a NAME="E43:33_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="c10">e5</font> &amp; <font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c10">e5</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c5">c</font> )
 <b>by </b><i><a class="txt" href="pasch.html#E11:33_1"><i><font color="Green" title="E28">A13</font></i></a>, <a class="txt" href="pasch.html#E19:33_1"><i><font color="Green" title="E35">A21</font></i></a>, <a class="txt" href="pasch.html#E26:33_1"><i><font color="Green" title="E42">A32</font></i></a>, <a class="txt" href="pasch.html#E41:33_1"><i><font color="Green" title="E48">A38</font></i></a>, <a class="ref" href="diraf.html#T5" title="DIRAF:th.5">DIRAF:5</a>, <a class="ref" href="diraf.html#T6" title="DIRAF:th.6">DIRAF:6</a></i>;<br/><a NAME="E44:33_1"/><b>then </b><i><font color="Green" title="E50">A40</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="c11">x</font>,<font color="Maroon" title="c5">c</font>
 <b>by </b><i><a class="txt" href="pasch.html#E28:33_1"><i><font color="Green" title="E44">A34</font></i></a>, <a class="ref" href="diraf.html#T6" title="DIRAF:th.6">DIRAF:6</a></i>;<br/><a NAME="E45:33_1"/><i><font color="Green" title="E51">A41</font></i>: 
<font color="Maroon" title="c7">e2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">e3</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_15"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:33_1_15"/>
<font color="Maroon" title="c7">e2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">e3</font>
 ;<br/>

<a NAME="E2:33_1_15"/><b>then </b>
<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c7">e2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c5">c</font>
 
<b>by </b><i><a class="txt" href="pasch.html#E7:33_1"><i><font color="Green" title="E26">A11</font></i></a>, <a class="ref" href="diraf.html#T5" title="DIRAF:th.5">DIRAF:5</a></i>;<br/>
<a NAME="E3:33_1_15"/><b>then </b>
<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="c2">p</font>,<font color="Maroon" title="c5">c</font>
 
<b>by </b><i><a class="txt" href="pasch.html#E5:33_1"><i><font color="Green" title="E25">A10</font></i></a>, <a class="txt" href="pasch.html#E16:33_1"><i><font color="Green" title="E33">A18</font></i></a>, <a class="ref" href="diraf.html#T6" title="DIRAF:th.6">DIRAF:6</a></i>;<br/>
<a NAME="E4:33_1_15"/><b>then </b>
<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="c5">c</font>,<font color="Maroon" title="c2">p</font>
 
<b>by </b><i><a class="ref" href="diraf.html#T5" title="DIRAF:th.5">DIRAF:5</a></i>;<br/>
<a NAME="E5:33_1_15"/><b>then </b>
(  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c2">p</font> or  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">a</font> )
 
<b>by </b><i><a class="ref" href="diraf.html#T11" title="DIRAF:th.11">DIRAF:11</a></i>;<br/>
<a NAME="E6:33_1_15"/><b>then </b>
(  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c2">p</font> or  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">a</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="E7:33_1_15"/>
contradiction
 <b>by </b><i><a class="txt" href="pasch.html#E13:33_1"><i><font color="Green" title="E30">A15</font></i></a>, <a class="ref" href="diraf.html#T36" title="DIRAF:th.36">DIRAF:36</a></i>;<br/>


</div><b>end;</b></div><a NAME="E46:33_1"/><i><font color="Green" title="E52">A42</font></i>: 
<font color="Maroon" title="c10">e5</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">e3</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_16"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:33_1_16"/>
<font color="Maroon" title="c10">e5</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">e3</font>
 ;<br/>

<a NAME="E2:33_1_16"/><b>then </b>
<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c8">e3</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="pasch.html#E42:33_1"><i><font color="Green" title="E49">A39</font></i></a>, <a class="ref" href="diraf.html#T5" title="DIRAF:th.5">DIRAF:5</a></i>;<br/>
<a NAME="E3:33_1_16"/><b>then </b>
<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="c3">a</font>,<font color="Maroon" title="c4">b</font>
 
<b>by </b><i><a class="txt" href="pasch.html#E7:33_1"><i><font color="Green" title="E26">A11</font></i></a>, <a class="txt" href="pasch.html#E17:33_1"><i><font color="Green" title="E34">A20</font></i></a>, <a class="ref" href="diraf.html#T6" title="DIRAF:th.6">DIRAF:6</a></i>;<br/>
<a NAME="E4:33_1_16"/><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="c4">b</font>,<font color="Maroon" title="c3">a</font>
 
<b>by </b><i><a class="ref" href="diraf.html#T5" title="DIRAF:th.5">DIRAF:5</a></i>;<br/>
<a NAME="E5:33_1_16"/><b>then </b>
<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="c4">b</font>,<font color="Maroon" title="c3">a</font>
 
<b>by </b><i><a class="txt" href="pasch.html#E6:33"><i><font color="Green" title="E21">A6</font></i></a>, <a class="txt" href="pasch.html#E1:33_1"><i><font color="Green" title="E23">A8</font></i></a>, <a class="ref" href="diraf.html#T6" title="DIRAF:th.6">DIRAF:6</a></i>;<br/>
<a NAME="E6:33_1_16"/><b>then </b>
(  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">a</font> or  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">a</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="E7:33_1_16"/><b>then </b>
(  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c3">a</font> or  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">a</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/>
<b>hence </b><a NAME="E8:33_1_16"/>
contradiction
 <b>by </b><i><a class="txt" href="pasch.html#E14:33_1"><i><font color="Green" title="E31">A16</font></i></a>, <a class="ref" href="diraf.html#T36" title="DIRAF:th.36">DIRAF:36</a></i>;<br/>


</div><b>end;</b></div><a NAME="E47:33_1"/><i><font color="Green" title="E53">A43</font></i>: 
<font color="Maroon" title="c8">e3</font>,<font color="Maroon" title="c2">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">e3</font>,<font color="Maroon" title="c5">c</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_17"><b>proof </b></a><div class="add">

<a NAME="E1:33_1_17"/>
 <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="c8">e3</font>
 
<b>by </b><i><a class="txt" href="pasch.html#E7:33_1"><i><font color="Green" title="E26">A11</font></i></a>, <a class="ref" href="diraf.html#D3" title="DIRAF:def.3">DIRAF:def 3</a></i>;<br/>
<a NAME="E2:33_1_17"/><b>then </b>
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c8">e3</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#T13" title="DIRAF:th.13">DIRAF:13</a></i>;<br/>
<a NAME="E3:33_1_17"/><b>then </b>
<font color="Maroon" title="c8">e3</font>,<font color="Maroon" title="c5">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c2">p</font>
 
<b>by </b><i><a class="ref" href="diraf.html#D3" title="DIRAF:def.3">DIRAF:def 3</a></i>;<br/>
<a NAME="E4:33_1_17"/><b>then </b>
<font color="Maroon" title="c8">e3</font>,<font color="Maroon" title="c5">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">e3</font>,<font color="Maroon" title="c2">p</font>
 
<b>by </b><i><a class="ref" href="analoaf.html#D5" title="ANALOAF:def.5">ANALOAF:def 5</a></i>;<br/>
<b>hence </b><a NAME="E5:33_1_17"/>
<font color="Maroon" title="c8">e3</font>,<font color="Maroon" title="c2">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">e3</font>,<font color="Maroon" title="c5">c</font>
 <b>by </b><i><a class="ref" href="diraf.html#T5" title="DIRAF:th.5">DIRAF:5</a></i>;<br/>


</div><b>end;</b></div><a NAME="E48:33_1"/><i><font color="Green" title="E54">A44</font></i>: 
<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c7">e2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c10">e5</font>
 <b>by </b><i><a class="txt" href="pasch.html#E11:33_1"><i><font color="Green" title="E28">A13</font></i></a>, <a class="txt" href="pasch.html#E26:33_1"><i><font color="Green" title="E42">A32</font></i></a>, <a class="txt" href="pasch.html#E40:33_1"><i><font color="Green" title="E47">A37</font></i></a>, <a class="ref" href="diraf.html#T6" title="DIRAF:th.6">DIRAF:6</a></i>;<br/><a NAME="E49:33_1"/><i><font color="Green" title="E55">A45</font></i>: 
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c7">e2</font>,<font color="Maroon" title="c8">e3</font>,<font color="Maroon" title="c10">e5</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_18"><b>proof </b></a><div class="add">

<a NAME="E1:33_1_18"/>
<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">e1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">e4</font>,<font color="Maroon" title="c5">c</font>
 
<b>by </b><i><a class="txt" href="pasch.html#E9:33_1"><i><font color="Green" title="E27">A12</font></i></a>, <a class="ref" href="diraf.html#T5" title="DIRAF:th.5">DIRAF:5</a></i>;<br/>
<a NAME="E2:33_1_18"/><b>then </b>
<font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">e1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">e2</font>,<font color="Maroon" title="c10">e5</font>
 
<b>by </b><i><a class="txt" href="pasch.html#E11:33_1"><i><font color="Green" title="E28">A13</font></i></a>, <a class="txt" href="pasch.html#E27:33_1"><i><font color="Green" title="E43">A33</font></i></a>, <a class="ref" href="diraf.html#T6" title="DIRAF:th.6">DIRAF:6</a></i>;<br/>
<a NAME="E3:33_1_18"/><b>then </b>
<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">e2</font>,<font color="Maroon" title="c10">e5</font>
 
<b>by </b><i><a class="txt" href="pasch.html#E3:33_1"><i><font color="Green" title="E24">A9</font></i></a>, <a class="txt" href="pasch.html#E22:33_1"><i><font color="Green" title="E38">A27</font></i></a>, <a class="ref" href="diraf.html#T6" title="DIRAF:th.6">DIRAF:6</a></i>;<br/>
<a NAME="E4:33_1_18"/><b>then </b>
<font color="Maroon" title="c7">e2</font>,<font color="Maroon" title="c8">e3</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">e2</font>,<font color="Maroon" title="c10">e5</font>
 
<b>by </b><i><a class="txt" href="pasch.html#E3:33"><i><font color="Green" title="E18">A3</font></i></a>, <a class="txt" href="pasch.html#E7:33_1"><i><font color="Green" title="E26">A11</font></i></a>, <a class="ref" href="analoaf.html#D5" title="ANALOAF:def.5">ANALOAF:def 5</a></i>;<br/>
<a NAME="E5:33_1_18"/><b>then </b>
(  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c7">e2</font>,<font color="Maroon" title="c8">e3</font>,<font color="Maroon" title="c10">e5</font> or  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c7">e2</font>,<font color="Maroon" title="c10">e5</font>,<font color="Maroon" title="c8">e3</font> )
 
<b>by </b><i><a class="ref" href="diraf.html#T11" title="DIRAF:th.11">DIRAF:11</a></i>;<br/>
<a NAME="E6:33_1_18"/><b>then </b>
(  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c7">e2</font>,<font color="Maroon" title="c8">e3</font>,<font color="Maroon" title="c10">e5</font> or  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c7">e2</font>,<font color="Maroon" title="c10">e5</font>,<font color="Maroon" title="c8">e3</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="E7:33_1_18"/>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c7">e2</font>,<font color="Maroon" title="c8">e3</font>,<font color="Maroon" title="c10">e5</font>
 <b>by </b><i><a class="ref" href="diraf.html#T36" title="DIRAF:th.36">DIRAF:36</a></i>;<br/>


</div><b>end;</b></div><a NAME="E50:33_1"/><i><font color="Green" title="E56">A46</font></i>: 
not  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c8">e3</font>,<font color="Maroon" title="c7">e2</font>,<font color="Maroon" title="c2">p</font>
 <b>by </b><i><a class="txt" href="pasch.html#E21:33_1"><i><font color="Green" title="E37">A23</font></i></a>, <a class="ref" href="diraf.html#T36" title="DIRAF:th.36">DIRAF:36</a></i>;<br/><a NAME="E51:33_1"/><b>then </b>
not  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c7">e2</font>,<font color="Maroon" title="c8">e3</font>,<font color="Maroon" title="c10">e5</font>
 <b>by </b><i><a class="txt" href="pasch.html#E46:33_1"><i><font color="Green" title="E52">A42</font></i></a>, <a class="txt" href="pasch.html#E47:33_1"><i><font color="Green" title="E53">A43</font></i></a>, <a class="txt" href="pasch.html#E48:33_1"><i><font color="Green" title="E54">A44</font></i></a>, <a class="txt" href="pasch.html#E49:33_1"><i><font color="Green" title="E55">A45</font></i></a>, <a class="ref" href="pasch.html#T24" target="_self" title="PASCH:th.24">Th24</a></i>;<br/><a NAME="E52:33_1"/><b>then </b>
(  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c8">e3</font>,<font color="Maroon" title="c7">e2</font>,<font color="Maroon" title="c10">e5</font> or  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c7">e2</font>,<font color="Maroon" title="c10">e5</font>,<font color="Maroon" title="c8">e3</font> )
 <b>by </b><i><a class="txt" href="pasch.html#E49:33_1"><i><font color="Green" title="E55">A45</font></i></a>, <a class="ref" href="diraf.html#T35" title="DIRAF:th.35">DIRAF:35</a></i>;<br/><a NAME="E53:33_1"/><b>then </b>
( <font color="Maroon" title="c8">e3</font>,<font color="Maroon" title="c7">e2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">e2</font>,<font color="Maroon" title="c10">e5</font> or  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c8">e3</font>,<font color="Maroon" title="c10">e5</font>,<font color="Maroon" title="c7">e2</font> )
 <b>by </b><i><a class="ref" href="diraf.html#T13" title="DIRAF:th.13">DIRAF:13</a>, <a class="ref" href="diraf.html#D3" title="DIRAF:def.3">DIRAF:def 3</a></i>;<br/><a NAME="E54:33_1"/><b>then </b>
( <font color="Maroon" title="c8">e3</font>,<font color="Maroon" title="c7">e2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">e3</font>,<font color="Maroon" title="c10">e5</font> or <font color="Maroon" title="c8">e3</font>,<font color="Maroon" title="c10">e5</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">e5</font>,<font color="Maroon" title="c7">e2</font> )
 <b>by </b><i><a class="ref" href="analoaf.html#D5" title="ANALOAF:def.5">ANALOAF:def 5</a>, <a class="ref" href="diraf.html#D3" title="DIRAF:def.3">DIRAF:def 3</a></i>;<br/><a NAME="E55:33_1"/><b>then </b>
( <font color="Maroon" title="c8">e3</font>,<font color="Maroon" title="c10">e5</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">e3</font>,<font color="Maroon" title="c7">e2</font> &amp; <font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c10">e5</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c7">e2</font> &amp;  <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="c8">e3</font> )
 <b>by </b><i><a class="txt" href="pasch.html#E7:33_1"><i><font color="Green" title="E26">A11</font></i></a>, <a class="txt" href="pasch.html#E48:33_1"><i><font color="Green" title="E54">A44</font></i></a>, <a class="ref" href="analoaf.html#D5" title="ANALOAF:def.5">ANALOAF:def 5</a>, <a class="ref" href="diraf.html#T5" title="DIRAF:th.5">DIRAF:5</a>, <a class="ref" href="diraf.html#D3" title="DIRAF:def.3">DIRAF:def 3</a></i>;<br/><a NAME="E56:33_1"/><b>then </b>
( <font color="Maroon" title="c8">e3</font>,<font color="Maroon" title="c10">e5</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">e3</font>,<font color="Maroon" title="c7">e2</font> &amp; <font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c10">e5</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c7">e2</font> &amp;  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c8">e3</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#T13" title="DIRAF:th.13">DIRAF:13</a></i>;<br/><a NAME="E57:33_1"/><b>then </b>
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c8">e3</font>,<font color="Maroon" title="c10">e5</font>,<font color="Maroon" title="c7">e2</font>
 <b>by </b><i><a class="txt" href="pasch.html#E17:33_1"><i><font color="Green" title="E34">A20</font></i></a>, <a class="txt" href="pasch.html#E50:33_1"><i><font color="Green" title="E56">A46</font></i></a>, <a class="ref" href="pasch.html#T22" target="_self" title="PASCH:th.22">Th22</a></i>;<br/><a NAME="E58:33_1"/><b>then </b>
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c7">e2</font>,<font color="Maroon" title="c10">e5</font>,<font color="Maroon" title="c8">e3</font>
 <b>by </b><i><a class="ref" href="diraf.html#T13" title="DIRAF:th.13">DIRAF:13</a></i>;<br/><a NAME="E59:33_1"/><b>then </b>
<font color="Maroon" title="c7">e2</font>,<font color="Maroon" title="c10">e5</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">e5</font>,<font color="Maroon" title="c8">e3</font>
 <b>by </b><i><a class="ref" href="diraf.html#D3" title="DIRAF:def.3">DIRAF:def 3</a></i>;<br/><a NAME="E60:33_1"/><b>then </b>
( <font color="Maroon" title="c7">e2</font>,<font color="Maroon" title="c10">e5</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">e5</font>,<font color="Maroon" title="c8">e3</font> &amp; <font color="Maroon" title="c7">e2</font>,<font color="Maroon" title="c10">e5</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">e2</font>,<font color="Maroon" title="c8">e3</font> )
 <b>by </b><i><a class="ref" href="analoaf.html#D5" title="ANALOAF:def.5">ANALOAF:def 5</a></i>;<br/><a NAME="E61:33_1"/><b>then </b>
<font color="Maroon" title="c7">e2</font>,<font color="Maroon" title="c8">e3</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">e5</font>,<font color="Maroon" title="c8">e3</font>
 <b>by </b><i><a class="txt" href="pasch.html#E11:33_1"><i><font color="Green" title="E28">A13</font></i></a>, <a class="ref" href="analoaf.html#D5" title="ANALOAF:def.5">ANALOAF:def 5</a></i>;<br/><a NAME="E62:33_1"/><b>then </b>
<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">e5</font>,<font color="Maroon" title="c8">e3</font>
 <b>by </b><i><a class="txt" href="pasch.html#E7:33_1"><i><font color="Green" title="E26">A11</font></i></a>, <a class="txt" href="pasch.html#E45:33_1"><i><font color="Green" title="E51">A41</font></i></a>, <a class="ref" href="diraf.html#T6" title="DIRAF:th.6">DIRAF:6</a></i>;<br/><a NAME="E63:33_1"/><b>then </b>
<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c11">x</font>
 <b>by </b><i><a class="txt" href="pasch.html#E19:33_1"><i><font color="Green" title="E35">A21</font></i></a>, <a class="txt" href="pasch.html#E46:33_1"><i><font color="Green" title="E52">A42</font></i></a>, <a class="ref" href="diraf.html#T6" title="DIRAF:th.6">DIRAF:6</a></i>;<br/><a NAME="E64:33_1"/><b>then </b>
( <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c11">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">a</font> &amp; <font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c11">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">a</font> )
 <b>by </b><i><a class="txt" href="pasch.html#E44:33_1"><i><font color="Green" title="E50">A40</font></i></a>, <a class="ref" href="diraf.html#T5" title="DIRAF:th.5">DIRAF:5</a></i>;<br/><a NAME="E65:33_1"/><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="c11">x</font>,<font color="Maroon" title="c3">a</font>
 <b>by </b><i><a class="txt" href="pasch.html#E1:33"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="pasch.html#E2:33"><i><font color="Green" title="E17">A2</font></i></a>, <a class="txt" href="pasch.html#E1:33_1"><i><font color="Green" title="E23">A8</font></i></a>, <a class="ref" href="pasch.html#T22" target="_self" title="PASCH:th.22">Th22</a></i>;<br/><b>hence </b><a NAME="E66:33_1"/>
 ex <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">OAS</font> st <br/>(  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c2">p</font>,<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c3">a</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="Olive" title="b1">x</font>,<font color="Maroon" title="c5">c</font> )
 <b>by </b><i><a class="txt" href="pasch.html#E44:33_1"><i><font color="Green" title="E50">A40</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E8:33"/><i><font color="Green" title="E23">A47</font></i>: 
( <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> implies ( <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="c3">a</font>,<font color="Maroon" title="c5">c</font> &amp;  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c3">a</font> ) )
 
<b>by </b><i><a class="ref" href="diraf.html#T4" title="DIRAF:th.4">DIRAF:4</a>, <a class="ref" href="diraf.html#T14" title="DIRAF:th.14">DIRAF:14</a></i>;<br/>
<a NAME="E9:33"/><i><font color="Green" title="E24">A48</font></i>: 
( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> implies ( <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="c3">a</font>,<font color="Maroon" title="c5">c</font> &amp;  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c3">a</font> ) )
 
<b>by </b><i><a class="ref" href="diraf.html#T7" title="DIRAF:th.7">DIRAF:7</a>, <a class="ref" href="diraf.html#T14" title="DIRAF:th.14">DIRAF:14</a></i>;<br/>
<a NAME="E10:33"/>
( <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">c</font> implies ( <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> &amp;  <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="c3">a</font> ) )
 
<b>by </b><i><a class="ref" href="diraf.html#T7" title="DIRAF:th.7">DIRAF:7</a>, <a class="ref" href="diraf.html#T14" title="DIRAF:th.14">DIRAF:14</a></i>;<br/>
<b>hence </b><a NAME="E11:33"/>
 ex <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">OAS</font> st <br/>(  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c2">p</font>,<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c3">a</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="Olive" title="b1">x</font>,<font color="Maroon" title="c5">c</font> )
 <b>by </b><i><a class="txt" href="pasch.html#E7:33"><i><font color="Green" title="E22">A7</font></i></a>, <a class="txt" href="pasch.html#E8:33"><i><font color="Green" title="E23">A47</font></i></a>, <a class="txt" href="pasch.html#E9:33"><i><font color="Green" title="E24">A48</font></i></a></i>;<br/>


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