<?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">b</font>, <font color="Maroon" title="c4">c</font>, <font color="Maroon" title="c5">a</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="E1:32"/><i><font color="Green" title="E15">A1</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">b</font>
 <b>and </b><br/><a NAME="E2:32"/><i><font color="Green" title="E16">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="c3">b</font>,<font color="Maroon" title="c4">c</font>
 ;<br/>

<a NAME="E3:32"/><i><font color="Green" title="E17">A3</font></i>: 
<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font>
 
<b>by </b><i><a class="txt" href="pasch.html#E2:32"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="diraf.html#D3" title="DIRAF:def.3">DIRAF:def 3</a></i>;<br/>
<a NAME="E4:32"/><b>then </b><i><font color="Green" title="E18">A4</font></i>: 
<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">c</font>
 
<b>by </b><i><a class="ref" href="analoaf.html#D5" title="ANALOAF:def.5">ANALOAF:def 5</a></i>;<br/>
<div><i><font color="Green" title="E19">A5</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:32_1"/><i><font color="Green" title="E20">A6</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">a</font>,<font color="Maroon" title="c3">b</font>
 ;<br/><b>consider </b><font color="Maroon" title="c6">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="E3:32_1"/><i><font color="Green" title="E21">A7</font></i>: 
( <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c5">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">x</font> &amp; <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c6">x</font> )
 <b>by </b><i><a class="txt" href="pasch.html#E1:32"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="pasch.html#E4:32"><i><font color="Green" title="E18">A4</font></i></a>, <a class="ref" href="pasch.html#T25" target="_self" title="PASCH:th.25">Th25</a></i>;<br/><a NAME="E4:32_1"/><i><font color="Green" title="E22">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="c4">c</font>
 <b>by </b><i><a class="txt" href="pasch.html#E1:32"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="pasch.html#E2:32"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="diraf.html#T12" title="DIRAF:th.12">DIRAF:12</a></i>;<br/><a NAME="E5:32_1"/><i><font color="Green" title="E23">A9</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="c6">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:32_1_1"/>
<font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font>
 ;<br/>

<a NAME="E2:32_1_1"/><b>then </b>
( <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c2">p</font> &amp; <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">c</font> )
 
<b>by </b><i><a class="txt" href="pasch.html#E2:32"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="pasch.html#E3:32_1"><i><font color="Green" title="E21">A7</font></i></a>, <a class="ref" href="diraf.html#T11" title="DIRAF:th.11">DIRAF:11</a></i>;<br/>
<a NAME="E3:32_1_1"/><b>then </b>
( <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c2">p</font> &amp; <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c2">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">b</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="E4:32_1_1"/><b>then </b>
<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">p</font>
 
<b>by </b><i><a class="txt" href="pasch.html#E4:32_1"><i><font color="Green" title="E22">A8</font></i></a>, <a class="ref" href="diraf.html#T6" title="DIRAF:th.6">DIRAF:6</a></i>;<br/>
<a NAME="E5:32_1_1"/><b>then </b>
(  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">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="c3">b</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c5">a</font> )
 
<b>by </b><i><a class="ref" href="diraf.html#T11" title="DIRAF:th.11">DIRAF:11</a></i>;<br/>
<a NAME="E6:32_1_1"/><b>then </b>
(  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">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="c3">b</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c5">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="E7:32_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="pasch.html#E1:32_1"><i><font color="Green" title="E20">A6</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="E6:32_1"/>
not  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c4">c</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:32_1_2"/><i><font color="Green" title="E24">A10</font></i>: 
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c4">c</font>
 ;<br/>

<a NAME="E2:32_1_2"/>
(  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c6">x</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">x</font>,<font color="Maroon" title="c5">a</font> )
 
<b>by </b><i><a class="txt" href="pasch.html#E3:32_1"><i><font color="Green" title="E21">A7</font></i></a>, <a class="ref" href="diraf.html#T11" title="DIRAF:th.11">DIRAF:11</a></i>;<br/>
<a NAME="E3:32_1_2"/><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">a</font>,<font color="Maroon" title="c6">x</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">x</font>,<font color="Maroon" title="c5">a</font> )
 
<b>by </b><i><a class="ref" href="diraf.html#T34" title="DIRAF:th.34">DIRAF:34</a></i>;<br/>
<a NAME="E4:32_1_2"/><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="c6">x</font>,<font color="Maroon" title="c5">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="c6">x</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="E5:32_1_2"/><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">c</font>,<font color="Maroon" title="c5">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="c3">b</font>,<font color="Maroon" title="c4">c</font> )
 
<b>by </b><i><a class="txt" href="pasch.html#E2:32"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="pasch.html#E5:32_1"><i><font color="Green" title="E23">A9</font></i></a>, <a class="txt" href="pasch.html#E1:32_1_2"><i><font color="Green" title="E24">A10</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/>
<a NAME="E6:32_1_2"/><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">c</font>,<font color="Maroon" title="c5">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="c4">c</font>,<font color="Maroon" title="c3">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="c4">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/>
<b>hence </b><a NAME="E7:32_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="pasch.html#E1:32_1"><i><font color="Green" title="E20">A6</font></i></a>, <a class="txt" href="pasch.html#E4:32_1"><i><font color="Green" title="E22">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><b>hence </b><a NAME="E7:32_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="Maroon" title="c5">a</font>,<font color="Olive" title="b1">x</font> &amp; <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">c</font>,<font color="Olive" title="b1">x</font> )
 <b>by </b><i><a class="txt" href="pasch.html#E1:32"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="pasch.html#E2:32"><i><font color="Green" title="E16">A2</font></i></a>, <a class="txt" href="pasch.html#E3:32_1"><i><font color="Green" title="E21">A7</font></i></a>, <a class="ref" href="pasch.html#T22" target="_self" title="PASCH:th.22">Th22</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green" title="E20">A11</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:32_2"/><i><font color="Green" title="E21">A12</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">a</font>,<font color="Maroon" title="c3">b</font> &amp; <font color="Maroon" title="c4">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font> )
 ;<br/><div><i><font color="Green" title="E22">A13</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:32_2_1"/>
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c3">b</font>
 ;<br/><a NAME="E2:32_2_1"/><b>then </b>
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font>
 <b>by </b><i><a class="txt" href="pasch.html#E2:32"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="diraf.html#T15" title="DIRAF:th.15">DIRAF:15</a></i>;<br/><a NAME="E3:32_2_1"/><b>then </b>
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">a</font>
 <b>by </b><i><a class="ref" href="diraf.html#T13" title="DIRAF:th.13">DIRAF:13</a></i>;<br/><a NAME="E4:32_2_1"/><b>then </b>
<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">a</font>
 <b>by </b><i><a class="ref" href="diraf.html#D3" title="DIRAF:def.3">DIRAF:def 3</a></i>;<br/><a NAME="E5:32_2_1"/><b>then </b>
( <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">a</font> &amp; <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">a</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="E6:32_2_1"/>
(  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c5">a</font> &amp; <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">a</font> )
 <b>by </b><i><a class="txt" href="pasch.html#E1:32_2"><i><font color="Green" title="E21">A12</font></i></a>, <a class="ref" href="analoaf.html#D5" title="ANALOAF:def.5">ANALOAF:def 5</a>, <a class="ref" href="diraf.html#T14" title="DIRAF:th.14">DIRAF:14</a></i>;<br/></div><b>end;</b></div><div><i><font color="Green" title="E23">A14</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:32_2_2"/>
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">b</font>
 ;<br/><a NAME="E2:32_2_2"/><b>then </b>
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font>
 <b>by </b><i><a class="txt" href="pasch.html#E1:32"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="pasch.html#E2:32"><i><font color="Green" title="E16">A2</font></i></a>, <a class="ref" href="diraf.html#T16" title="DIRAF:th.16">DIRAF:16</a></i>;<br/><a NAME="E3:32_2_2"/><b>then </b>
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">a</font>
 <b>by </b><i><a class="ref" href="diraf.html#T13" title="DIRAF:th.13">DIRAF:13</a></i>;<br/><a NAME="E4:32_2_2"/><b>then </b>
<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">a</font>
 <b>by </b><i><a class="ref" href="diraf.html#D3" title="DIRAF:def.3">DIRAF:def 3</a></i>;<br/><a NAME="E5:32_2_2"/><b>then </b>
( <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">a</font> &amp; <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">a</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="E6:32_2_2"/>
(  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c5">a</font> &amp; <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">a</font> )
 <b>by </b><i><a class="txt" href="pasch.html#E1:32_2"><i><font color="Green" title="E21">A12</font></i></a>, <a class="ref" href="analoaf.html#D5" title="ANALOAF:def.5">ANALOAF:def 5</a>, <a class="ref" href="diraf.html#T14" title="DIRAF:th.14">DIRAF:14</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:32_2_3"/>
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">a</font>
 ;<br/><a NAME="E2:32_2_3"/><b>then </b><i><font color="Green" title="E24">A15</font></i>: 
<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">a</font>
 <b>by </b><i><a class="ref" href="diraf.html#D3" title="DIRAF:def.3">DIRAF:def 3</a></i>;<br/><a NAME="E3:32_2_3"/><b>then </b><i><font color="Green" title="E25">A16</font></i>: 
<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font>
 <b>by </b><i><a class="txt" href="pasch.html#E1:32"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="pasch.html#E3:32"><i><font color="Green" title="E17">A3</font></i></a>, <a class="ref" href="analoaf.html#D5" title="ANALOAF:def.5">ANALOAF:def 5</a></i>;<br/><div><i><font color="Green" title="E26">A17</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:32_2_3_1"/><i><font color="Green" title="E27">A18</font></i>: 
<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">c</font>
 ;<br/><a NAME="E2:32_2_3_1"/>
<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c5">a</font>
 <b>by </b><i><a class="txt" href="pasch.html#E2:32_2_3"><i><font color="Green" title="E24">A15</font></i></a>, <a class="ref" href="analoaf.html#D5" title="ANALOAF:def.5">ANALOAF:def 5</a></i>;<br/><a NAME="E3:32_2_3_1"/><b>then </b><i><font color="Green" title="E28">A19</font></i>: 
<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c5">a</font>
 <b>by </b><i><a class="txt" href="pasch.html#E1:32"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="pasch.html#E2:32_2_3"><i><font color="Green" title="E24">A15</font></i></a>, <a class="ref" href="analoaf.html#D5" title="ANALOAF:def.5">ANALOAF:def 5</a></i>;<br/><div><i><font color="Green" title="E29">A20</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_3_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:32_2_3_1_1"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font>
 ;<br/><a NAME="E2:32_2_3_1_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="c5">a</font>,<font color="Maroon" title="c5">a</font> &amp; <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">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="E3:32_2_3_1_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="Maroon" title="c5">a</font>,<font color="Olive" title="b1">x</font> &amp; <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">c</font>,<font color="Olive" title="b1">x</font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_3_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:32_2_3_1_2"/>
<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c5">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c4">c</font>
 ;<br/><a NAME="E2:32_2_3_1_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="c5">a</font>,<font color="Maroon" title="c4">c</font> &amp; <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c4">c</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#D3" title="DIRAF:def.3">DIRAF:def 3</a></i>;<br/><b>hence </b><a NAME="E3:32_2_3_1_2"/>
 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="Maroon" title="c5">a</font>,<font color="Olive" title="b1">x</font> &amp; <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">c</font>,<font color="Olive" title="b1">x</font> )
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:32_2_3_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="Maroon" title="c5">a</font>,<font color="Olive" title="b1">x</font> &amp; <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">c</font>,<font color="Olive" title="b1">x</font> )
 <b>by </b><i><a class="txt" href="pasch.html#E1:32_2_3_1"><i><font color="Green" title="E27">A18</font></i></a>, <a class="txt" href="pasch.html#E3:32_2_3_1"><i><font color="Green" title="E28">A19</font></i></a>, <a class="txt" href="pasch.html#E4:32_2_3_1"><i><font color="Green" title="E29">A20</font></i></a>, <a class="ref" href="analoaf.html#D5" title="ANALOAF:def.5">ANALOAF:def 5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2_3_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:32_2_3_2"/>
<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">a</font>
 ;<br/><a NAME="E2:32_2_3_2"/><b>then </b>
( <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">a</font> &amp; <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">a</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="E3:32_2_3_2"/>
(  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c5">a</font> &amp; <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">a</font> )
 <b>by </b><i><a class="txt" href="pasch.html#E1:32_2"><i><font color="Green" title="E21">A12</font></i></a>, <a class="ref" href="analoaf.html#D5" title="ANALOAF:def.5">ANALOAF:def 5</a>, <a class="ref" href="diraf.html#T14" title="DIRAF:th.14">DIRAF:14</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:32_2_3"/>
 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="Maroon" title="c5">a</font>,<font color="Olive" title="b1">x</font> &amp; <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">c</font>,<font color="Olive" title="b1">x</font> )
 <b>by </b><i><a class="txt" href="pasch.html#E3:32_2_3"><i><font color="Green" title="E25">A16</font></i></a>, <a class="txt" href="pasch.html#E4:32_2_3"><i><font color="Green" title="E26">A17</font></i></a>, <a class="ref" href="analoaf.html#D5" title="ANALOAF:def.5">ANALOAF:def 5</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:32_2"/>
 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="Maroon" title="c5">a</font>,<font color="Olive" title="b1">x</font> &amp; <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">c</font>,<font color="Olive" title="b1">x</font> )
 <b>by </b><i><a class="txt" href="pasch.html#E1:32_2"><i><font color="Green" title="E21">A12</font></i></a>, <a class="txt" href="pasch.html#E2:32_2"><i><font color="Green" title="E22">A13</font></i></a>, <a class="txt" href="pasch.html#E3:32_2"><i><font color="Green" title="E23">A14</font></i></a>, <a class="ref" href="diraf.html#T35" title="DIRAF:th.35">DIRAF:35</a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:32"/>
( <font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</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">a</font>,<font color="Maroon" title="c5">a</font> &amp; <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">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/>
<b>hence </b><a NAME="E8:32"/>
 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="Maroon" title="c5">a</font>,<font color="Olive" title="b1">x</font> &amp; <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">c</font>,<font color="Olive" title="b1">x</font> )
 <b>by </b><i><a class="txt" href="pasch.html#E5:32"><i><font color="Green" title="E19">A5</font></i></a>, <a class="txt" href="pasch.html#E6:32"><i><font color="Green" title="E20">A11</font></i></a></i>;<br/>


</div>
