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<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">a</font>, <font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c4">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/>







<div><i><font color="Green" title="E16">A1</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:34_1"/><i><font color="Green" title="E17">A2</font></i>: 
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font>
 ;<br/><a NAME="E2:34_1"/><i><font color="Green" title="E18">A3</font></i>: 
(  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> implies ( <font color="Maroon" title="c2">a</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> &amp; <font color="Maroon" title="c2">a</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="ref" href="diraf.html#D3" title="DIRAF:def.3">DIRAF:def 3</a></i>;<br/><div><i><font color="Green" title="E19">A4</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:34_1_1"/>
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>
 ;<br/><a NAME="E2:34_1_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="c2">a</font>,<font color="Maroon" title="c3">b</font>
 <b>by </b><i><a class="ref" href="diraf.html#T13" title="DIRAF:th.13">DIRAF:13</a></i>;<br/><a NAME="E3:34_1_1"/><b>then </b>
<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c2">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">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="E4:34_1_1"/><b>then </b>
<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c2">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c3">b</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="E5:34_1_1"/>
( <font color="Maroon" title="c2">a</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="c4">c</font> &amp; <font color="Maroon" title="c2">a</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="c4">c</font> )
 <b>by </b><i><a class="ref" href="diraf.html#T5" title="DIRAF:th.5">DIRAF:5</a>, <a class="ref" href="diraf.html#T7" title="DIRAF:th.7">DIRAF:7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:34_1_2"/>
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c3">b</font>
 ;<br/><a NAME="E2:34_1_2"/><b>then </b>
<font color="Maroon" title="c2">a</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="c3">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="E3:34_1_2"/><b>then </b>
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</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="E4:34_1_2"/>
( <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c2">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> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font> )
 <b>by </b><i><a class="ref" href="diraf.html#T5" title="DIRAF:th.5">DIRAF:5</a>, <a class="ref" href="diraf.html#T7" title="DIRAF:th.7">DIRAF:7</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:34_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/>( <font color="Maroon" title="c2">a</font>,<font color="Olive" title="b1">x</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> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b1">x</font>,<font color="Maroon" title="c4">c</font> )
 <b>by </b><i><a class="txt" href="pasch.html#E1:34_1"><i><font color="Green" title="E17">A2</font></i></a>, <a class="txt" href="pasch.html#E2:34_1"><i><font color="Green" title="E18">A3</font></i></a>, <a class="txt" href="pasch.html#E3:34_1"><i><font color="Green" title="E19">A4</font></i></a>, <a class="ref" href="diraf.html#T35" title="DIRAF:th.35">DIRAF:35</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:34_2"/><i><font color="Green" title="E17">A5</font></i>: 
not  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font>
 ;<br/><a NAME="E2:34_2"/><b>then </b><i><font color="Green" title="E18">A6</font></i>: 
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font> &amp; <font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">c</font> &amp; <font color="Maroon" title="c3">b</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="ref" href="diraf.html#T37" title="DIRAF:th.37">DIRAF:37</a></i>;<br/><b>consider </b><font color="Maroon" title="c5">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="E4:34_2"/><i><font color="Green" title="E19">A7</font></i>: 
(  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">e1</font> &amp; <font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">e1</font> )
 <b>by </b><i><a class="ref" href="diraf.html#T17" title="DIRAF:th.17">DIRAF:17</a></i>;<br/><a NAME="E5:34_2"/><i><font color="Green" title="E20">A8</font></i>: 
<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c2">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">e1</font>
 <b>by </b><i><a class="txt" href="pasch.html#E4:34_2"><i><font color="Green" title="E19">A7</font></i></a>, <a class="ref" href="diraf.html#D3" title="DIRAF:def.3">DIRAF:def 3</a></i>;<br/><b>then consider </b><font color="Maroon" title="c6">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="E7:34_2"/><i><font color="Green" title="E21">A9</font></i>: 
( <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c6">e2</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="c5">e1</font>,<font color="Maroon" title="c6">e2</font> )
 <b>by </b><i><a class="txt" href="pasch.html#E2:34_2"><i><font color="Green" title="E18">A6</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="c7">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="E9:34_2"/><i><font color="Green" title="E22">A10</font></i>: 
(  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c7">e3</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="c7">e3</font> )
 <b>by </b><i><a class="ref" href="diraf.html#T17" title="DIRAF:th.17">DIRAF:17</a></i>;<br/><a NAME="E10:34_2"/><i><font color="Green" title="E23">A11</font></i>: 
<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="c7">e3</font>
 <b>by </b><i><a class="txt" href="pasch.html#E9:34_2"><i><font color="Green" title="E22">A10</font></i></a>, <a class="ref" href="diraf.html#D3" title="DIRAF:def.3">DIRAF:def 3</a></i>;<br/><b>then consider </b><font color="Maroon" title="c8">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="E12:34_2"/><i><font color="Green" title="E24">A12</font></i>: 
( <font color="Maroon" title="c2">a</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="c8">e4</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">e3</font>,<font color="Maroon" title="c8">e4</font> )
 <b>by </b><i><a class="txt" href="pasch.html#E2:34_2"><i><font color="Green" title="E18">A6</font></i></a>, <a class="ref" href="analoaf.html#D5" title="ANALOAF:def.5">ANALOAF:def 5</a></i>;<br/><a NAME="E13:34_2"/>
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c8">e4</font>
 <b>by </b><i><a class="txt" href="pasch.html#E12:34_2"><i><font color="Green" title="E24">A12</font></i></a>, <a class="ref" href="diraf.html#D3" title="DIRAF:def.3">DIRAF:def 3</a></i>;<br/><a NAME="E14:34_2"/><b>then </b><i><font color="Green" title="E25">A13</font></i>: 
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c8">e4</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c2">a</font>
 <b>by </b><i><a class="ref" href="diraf.html#T13" title="DIRAF:th.13">DIRAF:13</a></i>;<br/><a NAME="E15:34_2"/><b>then </b>
( <font color="Maroon" title="c8">e4</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="c2">a</font> &amp; <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c2">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">e1</font> )
 <b>by </b><i><a class="txt" href="pasch.html#E5:34_2"><i><font color="Green" title="E20">A8</font></i></a>, <a class="ref" href="analoaf.html#D5" title="ANALOAF:def.5">ANALOAF:def 5</a>, <a class="ref" href="diraf.html#D3" title="DIRAF:def.3">DIRAF:def 3</a></i>;<br/><a NAME="E16:34_2"/><b>then </b>
<font color="Maroon" title="c8">e4</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">e1</font>
 <b>by </b><i><a class="txt" href="pasch.html#E2:34_2"><i><font color="Green" title="E18">A6</font></i></a>, <a class="ref" href="diraf.html#T6" title="DIRAF:th.6">DIRAF:6</a></i>;<br/><a NAME="E17:34_2"/><b>then </b><i><font color="Green" title="E26">A14</font></i>: 
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c8">e4</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">e1</font>
 <b>by </b><i><a class="ref" href="diraf.html#D3" title="DIRAF:def.3">DIRAF:def 3</a></i>;<br/><a NAME="E18:34_2"/><b>then </b><i><font color="Green" title="E27">A15</font></i>: 
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c5">e1</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c8">e4</font>
 <b>by </b><i><a class="ref" href="diraf.html#T13" title="DIRAF:th.13">DIRAF:13</a></i>;<br/><a NAME="E19:34_2"/><i><font color="Green" title="E28">A16</font></i>: 
<font color="Maroon" title="c8">e4</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">c</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:34_2_1"/>
<font color="Maroon" title="c8">e4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font>
 ;<br/>

<a NAME="E2:34_2_1"/><b>then </b>
( <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">e3</font>,<font color="Maroon" title="c4">c</font> &amp; <font color="Maroon" title="c7">e3</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="c3">b</font> )
 
<b>by </b><i><a class="txt" href="pasch.html#E10:34_2"><i><font color="Green" title="E23">A11</font></i></a>, <a class="txt" href="pasch.html#E12:34_2"><i><font color="Green" title="E24">A12</font></i></a>, <a class="ref" href="diraf.html#T5" title="DIRAF:th.5">DIRAF:5</a></i>;<br/>
<a NAME="E3:34_2_1"/><b>then </b>
<font color="Maroon" title="c2">a</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="c3">b</font>
 
<b>by </b><i><a class="txt" href="pasch.html#E9:34_2"><i><font color="Green" title="E22">A10</font></i></a>, <a class="ref" href="diraf.html#T6" title="DIRAF:th.6">DIRAF:6</a></i>;<br/>
<a NAME="E4:34_2_1"/><b>then </b>
<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">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="ref" href="diraf.html#T5" title="DIRAF:th.5">DIRAF:5</a></i>;<br/>
<a NAME="E5:34_2_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="c2">a</font>,<font color="Maroon" title="c4">c</font> or  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c2">a</font> )
 
<b>by </b><i><a class="ref" href="diraf.html#T11" title="DIRAF:th.11">DIRAF:11</a></i>;<br/>
<a NAME="E6:34_2_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="c2">a</font>,<font color="Maroon" title="c4">c</font> or  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c2">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="E7:34_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="pasch.html#E1:34_2"><i><font color="Green" title="E17">A5</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="c9">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="E21:34_2"/><i><font color="Green" title="E29">A17</font></i>: 
(  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c8">e4</font>,<font color="Maroon" title="c7">e3</font>,<font color="Maroon" title="c9">e5</font> &amp; <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c7">e3</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">e1</font>,<font color="Maroon" title="c9">e5</font> )
 <b>by </b><i><a class="txt" href="pasch.html#E17:34_2"><i><font color="Green" title="E26">A14</font></i></a>, <a class="ref" href="pasch.html#T27" target="_self" title="PASCH:th.27">Th27</a></i>;<br/><a NAME="E22:34_2"/><i><font color="Green" title="E30">A18</font></i>: 
<font color="Maroon" title="c8">e4</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">e3</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:34_2_2"/>
<font color="Maroon" title="c8">e4</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">e3</font>
 ;<br/>

<a NAME="E2:34_2_2"/><b>then </b>
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c7">e3</font>
 
<b>by </b><i><a class="txt" href="pasch.html#E12:34_2"><i><font color="Green" title="E24">A12</font></i></a>, <a class="ref" href="diraf.html#D3" title="DIRAF:def.3">DIRAF:def 3</a></i>;<br/>
<a NAME="E3:34_2_2"/><b>then </b>
(  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c7">e3</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c2">a</font> &amp;  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c7">e3</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c3">b</font> )
 
<b>by </b><i><a class="txt" href="pasch.html#E9:34_2"><i><font color="Green" title="E22">A10</font></i></a>, <a class="ref" href="diraf.html#T13" title="DIRAF:th.13">DIRAF:13</a>, <a class="ref" href="diraf.html#T34" title="DIRAF:th.34">DIRAF:34</a></i>;<br/>
<a NAME="E4:34_2_2"/><b>then </b>
(  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c7">e3</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c2">a</font> &amp;  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c7">e3</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="c7">e3</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c4">c</font> )
 
<b>by </b><i><a class="ref" href="diraf.html#T34" title="DIRAF:th.34">DIRAF:34</a>, <a class="ref" href="diraf.html#T37" title="DIRAF:th.37">DIRAF:37</a></i>;<br/>
<b>hence </b><a NAME="E5:34_2_2"/>
contradiction
 <b>by </b><i><a class="txt" href="pasch.html#E1:34_2"><i><font color="Green" title="E17">A5</font></i></a>, <a class="txt" href="pasch.html#E9:34_2"><i><font color="Green" title="E22">A10</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="E23:34_2"/><b>then </b><i><font color="Green" title="E31">A19</font></i>: 
<font color="Maroon" title="c8">e4</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">e5</font>
 <b>by </b><i><a class="txt" href="pasch.html#E21:34_2"><i><font color="Green" title="E29">A17</font></i></a>, <a class="ref" href="diraf.html#T12" title="DIRAF:th.12">DIRAF:12</a></i>;<br/><a NAME="E24:34_2"/><i><font color="Green" title="E32">A20</font></i>: 
<font color="Maroon" title="c5">e1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">e4</font>
 <b>by </b><i><a class="txt" href="pasch.html#E17:34_2"><i><font color="Green" title="E26">A14</font></i></a>, <a class="txt" href="pasch.html#E19:34_2"><i><font color="Green" title="E28">A16</font></i></a>, <a class="ref" href="diraf.html#T12" title="DIRAF:th.12">DIRAF:12</a></i>;<br/><a NAME="E25:34_2"/><i><font color="Green" title="E33">A21</font></i>: 
not  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c8">e4</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c7">e3</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:34_2_3"/>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c8">e4</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c7">e3</font>
 ;<br/>

<a NAME="E2:34_2_3"/><b>then </b>
(  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c8">e4</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c7">e3</font> &amp;  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c8">e4</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c2">a</font> &amp;  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c8">e4</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c4">c</font> )
 
<b>by </b><i><a class="txt" href="pasch.html#E14:34_2"><i><font color="Green" title="E25">A13</font></i></a>, <a class="ref" href="diraf.html#T34" title="DIRAF:th.34">DIRAF:34</a>, <a class="ref" href="diraf.html#T37" title="DIRAF:th.37">DIRAF:37</a></i>;<br/>
<a NAME="E3:34_2_3"/><b>then </b>
(  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c7">e3</font>,<font color="Maroon" title="c2">a</font> &amp;  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c7">e3</font> )
 
<b>by </b><i><a class="txt" href="pasch.html#E9:34_2"><i><font color="Green" title="E22">A10</font></i></a>, <a class="txt" href="pasch.html#E19:34_2"><i><font color="Green" title="E28">A16</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="E4:34_2_3"/><b>then </b>
(  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c7">e3</font>,<font color="Maroon" title="c2">a</font> &amp;  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c7">e3</font>,<font color="Maroon" title="c3">b</font> &amp;  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c7">e3</font>,<font color="Maroon" title="c4">c</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="E5:34_2_3"/>
contradiction
 <b>by </b><i><a class="txt" href="pasch.html#E1:34_2"><i><font color="Green" title="E17">A5</font></i></a>, <a class="txt" href="pasch.html#E9:34_2"><i><font color="Green" title="E22">A10</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="E26:34_2"/><i><font color="Green" title="E34">A22</font></i>: 
not  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c5">e1</font>,<font color="Maroon" title="c8">e4</font>,<font color="Maroon" title="c7">e3</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:34_2_4"/>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c5">e1</font>,<font color="Maroon" title="c8">e4</font>,<font color="Maroon" title="c7">e3</font>
 ;<br/>

<a NAME="E2:34_2_4"/><b>then </b>
(  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c5">e1</font>,<font color="Maroon" title="c8">e4</font>,<font color="Maroon" title="c7">e3</font> &amp;  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c8">e4</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">e1</font> )
 
<b>by </b><i><a class="txt" href="pasch.html#E17:34_2"><i><font color="Green" title="E26">A14</font></i></a>, <a class="ref" href="diraf.html#T34" title="DIRAF:th.34">DIRAF:34</a></i>;<br/>
<a NAME="E3:34_2_4"/><b>then </b>
(  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c8">e4</font>,<font color="Maroon" title="c5">e1</font>,<font color="Maroon" title="c7">e3</font> &amp;  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c8">e4</font>,<font color="Maroon" title="c5">e1</font>,<font color="Maroon" title="c4">c</font> &amp;  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c8">e4</font>,<font color="Maroon" title="c5">e1</font>,<font color="Maroon" title="c8">e4</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="E4:34_2_4"/>
contradiction
 <b>by </b><i><a class="txt" href="pasch.html#E24:34_2"><i><font color="Green" title="E32">A20</font></i></a>, <a class="txt" href="pasch.html#E25:34_2"><i><font color="Green" title="E33">A21</font></i></a>, <a class="ref" href="diraf.html#T38" title="DIRAF:th.38">DIRAF:38</a></i>;<br/>


</div><b>end;</b></div><a NAME="E27:34_2"/><i><font color="Green" title="E35">A23</font></i>: 
not  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c5">e1</font>,<font color="Maroon" title="c9">e5</font>,<font color="Maroon" title="c8">e4</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2_5"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:34_2_5"/>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c5">e1</font>,<font color="Maroon" title="c9">e5</font>,<font color="Maroon" title="c8">e4</font>
 ;<br/>

<a NAME="E2:34_2_5"/><b>then </b>
(  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c9">e5</font>,<font color="Maroon" title="c8">e4</font>,<font color="Maroon" title="c5">e1</font> &amp;  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c8">e4</font>,<font color="Maroon" title="c7">e3</font>,<font color="Maroon" title="c9">e5</font> )
 
<b>by </b><i><a class="txt" href="pasch.html#E21:34_2"><i><font color="Green" title="E29">A17</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></i>;<br/>
<a NAME="E3:34_2_5"/><b>then </b>
(  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c9">e5</font>,<font color="Maroon" title="c8">e4</font>,<font color="Maroon" title="c5">e1</font> &amp;  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c9">e5</font>,<font color="Maroon" title="c8">e4</font>,<font color="Maroon" title="c7">e3</font> &amp;  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c9">e5</font>,<font color="Maroon" title="c8">e4</font>,<font color="Maroon" title="c8">e4</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="E4:34_2_5"/>
contradiction
 <b>by </b><i><a class="txt" href="pasch.html#E23:34_2"><i><font color="Green" title="E31">A19</font></i></a>, <a class="txt" href="pasch.html#E26:34_2"><i><font color="Green" title="E34">A22</font></i></a>, <a class="ref" href="diraf.html#T38" title="DIRAF:th.38">DIRAF:38</a></i>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon" title="c10">e6</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="E29:34_2"/><i><font color="Green" title="E36">A24</font></i>: 
(  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c5">e1</font>,<font color="Maroon" title="c10">e6</font>,<font color="Maroon" title="c9">e5</font> &amp; <font color="Maroon" title="c9">e5</font>,<font color="Maroon" title="c8">e4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">e6</font>,<font color="Maroon" title="c4">c</font> )
 <b>by </b><i><a class="txt" href="pasch.html#E18:34_2"><i><font color="Green" title="E27">A15</font></i></a>, <a class="ref" href="pasch.html#T28" target="_self" title="PASCH:th.28">Th28</a></i>;<br/><b>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="E31:34_2"/><i><font color="Green" title="E37">A25</font></i>: 
(  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c10">e6</font> &amp; <font color="Maroon" title="c5">e1</font>,<font color="Maroon" title="c10">e6</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c11">x</font> )
 <b>by </b><i><a class="txt" href="pasch.html#E4:34_2"><i><font color="Green" title="E19">A7</font></i></a>, <a class="ref" href="pasch.html#T26" target="_self" title="PASCH:th.26">Th26</a></i>;<br/><a NAME="E32:34_2"/><i><font color="Green" title="E38">A26</font></i>: 
<font color="Maroon" title="c5">e1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">e5</font>
 <b>by </b><i><a class="txt" href="pasch.html#E27:34_2"><i><font color="Green" title="E35">A23</font></i></a>, <a class="ref" href="diraf.html#T37" title="DIRAF:th.37">DIRAF:37</a></i>;<br/><a NAME="E33:34_2"/><i><font color="Green" title="E39">A27</font></i>: 
<font color="Maroon" title="c4">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">e1</font>
 <b>by </b><i><a class="txt" href="pasch.html#E4:34_2"><i><font color="Green" title="E19">A7</font></i></a>, <a class="ref" href="diraf.html#T12" title="DIRAF:th.12">DIRAF:12</a></i>;<br/><a NAME="E34:34_2"/><i><font color="Green" title="E40">A28</font></i>: 
not  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">e1</font>,<font color="Maroon" title="c3">b</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2_6"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:34_2_6"/>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">e1</font>,<font color="Maroon" title="c3">b</font>
 ;<br/>

<a NAME="E2:34_2_6"/><b>then </b>
(  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">e1</font>,<font color="Maroon" title="c3">b</font> &amp;  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">e1</font> )
 
<b>by </b><i><a class="txt" href="pasch.html#E4:34_2"><i><font color="Green" title="E19">A7</font></i></a>, <a class="ref" href="diraf.html#T34" title="DIRAF:th.34">DIRAF:34</a></i>;<br/>
<a NAME="E3:34_2_6"/><b>then </b>
(  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">e1</font>,<font color="Maroon" title="c3">b</font> &amp;  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">e1</font>,<font color="Maroon" title="c2">a</font> &amp;  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">e1</font>,<font color="Maroon" title="c4">c</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="E4:34_2_6"/>
contradiction
 <b>by </b><i><a class="txt" href="pasch.html#E1:34_2"><i><font color="Green" title="E17">A5</font></i></a>, <a class="txt" href="pasch.html#E33:34_2"><i><font color="Green" title="E39">A27</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="E35:34_2"/><i><font color="Green" title="E41">A29</font></i>: 
<font color="Maroon" title="c9">e5</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">e3</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2_7"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:34_2_7"/>
<font color="Maroon" title="c9">e5</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">e3</font>
 ;<br/>

<a NAME="E2:34_2_7"/><b>then </b>
<font color="Maroon" title="c7">e3</font>,<font color="Maroon" title="c4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">e3</font>,<font color="Maroon" title="c5">e1</font>
 
<b>by </b><i><a class="txt" href="pasch.html#E21:34_2"><i><font color="Green" title="E29">A17</font></i></a>, <a class="ref" href="diraf.html#T5" title="DIRAF:th.5">DIRAF:5</a></i>;<br/>
<a NAME="E3:34_2_7"/><b>then </b>
(  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c7">e3</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">e1</font> or  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c7">e3</font>,<font color="Maroon" title="c5">e1</font>,<font color="Maroon" title="c4">c</font> )
 
<b>by </b><i><a class="ref" href="diraf.html#T11" title="DIRAF:th.11">DIRAF:11</a></i>;<br/>
<a NAME="E4:34_2_7"/><b>then </b>
(  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c7">e3</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">e1</font> or  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c7">e3</font>,<font color="Maroon" title="c5">e1</font>,<font color="Maroon" title="c4">c</font> )
 
<b>by </b><i><a class="ref" href="diraf.html#T34" title="DIRAF:th.34">DIRAF:34</a></i>;<br/>
<a NAME="E5:34_2_7"/><b>then </b>
(  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c7">e3</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">e1</font> &amp;  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c7">e3</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c3">b</font> )
 
<b>by </b><i><a class="txt" href="pasch.html#E9:34_2"><i><font color="Green" title="E22">A10</font></i></a>, <a class="ref" href="diraf.html#T13" title="DIRAF:th.13">DIRAF:13</a>, <a class="ref" href="diraf.html#T36" title="DIRAF:th.36">DIRAF:36</a></i>;<br/>
<a NAME="E6:34_2_7"/><b>then </b>
(  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c7">e3</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">e1</font> &amp;  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c7">e3</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="c7">e3</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c4">c</font> )
 
<b>by </b><i><a class="ref" href="diraf.html#T34" title="DIRAF:th.34">DIRAF:34</a>, <a class="ref" href="diraf.html#T37" title="DIRAF:th.37">DIRAF:37</a></i>;<br/>
<b>hence </b><a NAME="E7:34_2_7"/>
contradiction
 <b>by </b><i><a class="txt" href="pasch.html#E9:34_2"><i><font color="Green" title="E22">A10</font></i></a>, <a class="txt" href="pasch.html#E34:34_2"><i><font color="Green" title="E40">A28</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="E36:34_2"/><i><font color="Green" title="E42">A30</font></i>: 
<font color="Maroon" title="c5">e1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">e6</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2_8"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:34_2_8"/><i><font color="Green" title="E43">A31</font></i>: 
<font color="Maroon" title="c5">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">e6</font>
 ;<br/>

<a NAME="E2:34_2_8"/>
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c9">e5</font>,<font color="Maroon" title="c7">e3</font>,<font color="Maroon" title="c8">e4</font>
 
<b>by </b><i><a class="txt" href="pasch.html#E21:34_2"><i><font color="Green" title="E29">A17</font></i></a>, <a class="ref" href="diraf.html#T13" title="DIRAF:th.13">DIRAF:13</a></i>;<br/>
<a NAME="E3:34_2_8"/><b>then </b>
<font color="Maroon" title="c9">e5</font>,<font color="Maroon" title="c7">e3</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">e3</font>,<font color="Maroon" title="c8">e4</font>
 
<b>by </b><i><a class="ref" href="diraf.html#D3" title="DIRAF:def.3">DIRAF:def 3</a></i>;<br/>
<a NAME="E4:34_2_8"/><b>then </b>
( <font color="Maroon" title="c9">e5</font>,<font color="Maroon" title="c7">e3</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">e3</font>,<font color="Maroon" title="c8">e4</font> &amp; <font color="Maroon" title="c9">e5</font>,<font color="Maroon" title="c7">e3</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">e5</font>,<font color="Maroon" title="c8">e4</font> )
 
<b>by </b><i><a class="ref" href="analoaf.html#D5" title="ANALOAF:def.5">ANALOAF:def 5</a></i>;<br/>
<a NAME="E5:34_2_8"/><b>then </b>
<font color="Maroon" title="c9">e5</font>,<font color="Maroon" title="c8">e4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">e3</font>,<font color="Maroon" title="c8">e4</font>
 
<b>by </b><i><a class="txt" href="pasch.html#E35:34_2"><i><font color="Green" title="E41">A29</font></i></a>, <a class="ref" href="analoaf.html#D5" title="ANALOAF:def.5">ANALOAF:def 5</a></i>;<br/>
<a NAME="E6:34_2_8"/><b>then </b>
<font color="Maroon" title="c7">e3</font>,<font color="Maroon" title="c8">e4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">e1</font>,<font color="Maroon" title="c4">c</font>
 
<b>by </b><i><a class="txt" href="pasch.html#E23:34_2"><i><font color="Green" title="E31">A19</font></i></a>, <a class="txt" href="pasch.html#E29:34_2"><i><font color="Green" title="E36">A24</font></i></a>, <a class="txt" href="pasch.html#E1:34_2_8"><i><font color="Green" title="E43">A31</font></i></a>, <a class="ref" href="analoaf.html#D5" title="ANALOAF:def.5">ANALOAF:def 5</a></i>;<br/>
<a NAME="E7:34_2_8"/><b>then </b>
( <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">e1</font>,<font color="Maroon" title="c4">c</font> &amp;  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c5">e1</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font> )
 
<b>by </b><i><a class="txt" href="pasch.html#E4:34_2"><i><font color="Green" title="E19">A7</font></i></a>, <a class="txt" href="pasch.html#E12:34_2"><i><font color="Green" title="E24">A12</font></i></a>, <a class="txt" href="pasch.html#E22:34_2"><i><font color="Green" title="E30">A18</font></i></a>, <a class="ref" href="diraf.html#T6" title="DIRAF:th.6">DIRAF:6</a>, <a class="ref" href="diraf.html#T13" title="DIRAF:th.13">DIRAF:13</a></i>;<br/>
<a NAME="E8:34_2_8"/><b>then </b>
( <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">e1</font>,<font color="Maroon" title="c4">c</font> &amp; <font color="Maroon" title="c5">e1</font>,<font color="Maroon" title="c2">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font> )
 
<b>by </b><i><a class="ref" href="diraf.html#D3" title="DIRAF:def.3">DIRAF:def 3</a></i>;<br/>
<a NAME="E9:34_2_8"/><b>then </b>
( <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">e1</font>,<font color="Maroon" title="c4">c</font> &amp; <font color="Maroon" title="c5">e1</font>,<font color="Maroon" title="c2">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font> &amp; <font color="Maroon" title="c5">e1</font>,<font color="Maroon" title="c2">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">e1</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/>
<a NAME="E10:34_2_8"/><b>then </b>
( <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">e1</font>,<font color="Maroon" title="c4">c</font> &amp; <font color="Maroon" title="c5">e1</font>,<font color="Maroon" title="c4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font> )
 
<b>by </b><i><a class="txt" href="pasch.html#E4:34_2"><i><font color="Green" title="E19">A7</font></i></a>, <a class="ref" href="analoaf.html#D5" title="ANALOAF:def.5">ANALOAF:def 5</a></i>;<br/>
<a NAME="E11:34_2_8"/><b>then </b>
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>
 
<b>by </b><i><a class="txt" href="pasch.html#E33:34_2"><i><font color="Green" title="E39">A27</font></i></a>, <a class="ref" href="diraf.html#T6" title="DIRAF:th.6">DIRAF:6</a></i>;<br/>
<a NAME="E12:34_2_8"/><b>then </b>
( <font color="Maroon" title="c2">a</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> or <font color="Maroon" title="c2">a</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="c3">b</font> )
 
<b>by </b><i><a class="ref" href="diraf.html#T9" title="DIRAF:th.9">DIRAF:9</a></i>;<br/>
<a NAME="E13:34_2_8"/><b>then </b>
(  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> or  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c3">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="E14:34_2_8"/><b>then </b>
(  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">c</font> or  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c3">b</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="E15:34_2_8"/>
contradiction
 <b>by </b><i><a class="txt" href="pasch.html#E1:34_2"><i><font color="Green" title="E17">A5</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="E37:34_2"/>
<font color="Maroon" title="c5">e1</font>,<font color="Maroon" title="c10">e6</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">e6</font>,<font color="Maroon" title="c9">e5</font>
 <b>by </b><i><a class="txt" href="pasch.html#E29:34_2"><i><font color="Green" title="E36">A24</font></i></a>, <a class="ref" href="diraf.html#D3" title="DIRAF:def.3">DIRAF:def 3</a></i>;<br/><a NAME="E38:34_2"/><b>then </b>
<font color="Maroon" title="c5">e1</font>,<font color="Maroon" title="c10">e6</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">e1</font>,<font color="Maroon" title="c9">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="E39:34_2"/><b>then </b>
<font color="Maroon" title="c5">e1</font>,<font color="Maroon" title="c9">e5</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c11">x</font>
 <b>by </b><i><a class="txt" href="pasch.html#E31:34_2"><i><font color="Green" title="E37">A25</font></i></a>, <a class="txt" href="pasch.html#E36:34_2"><i><font color="Green" title="E42">A30</font></i></a>, <a class="ref" href="analoaf.html#D5" title="ANALOAF:def.5">ANALOAF:def 5</a></i>;<br/><a NAME="E40:34_2"/><b>then </b>
( <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c7">e3</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c11">x</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="c4">c</font>,<font color="Maroon" title="c7">e3</font> )
 <b>by </b><i><a class="txt" href="pasch.html#E9:34_2"><i><font color="Green" title="E22">A10</font></i></a>, <a class="txt" href="pasch.html#E21:34_2"><i><font color="Green" title="E29">A17</font></i></a>, <a class="txt" href="pasch.html#E32:34_2"><i><font color="Green" title="E38">A26</font></i></a>, <a class="ref" href="diraf.html#T6" title="DIRAF:th.6">DIRAF:6</a>, <a class="ref" href="diraf.html#D3" title="DIRAF:def.3">DIRAF:def 3</a></i>;<br/><a NAME="E41:34_2"/><b>then </b>
( <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c7">e3</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c11">x</font> &amp; <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c7">e3</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="ref" href="diraf.html#T5" title="DIRAF:th.5">DIRAF:5</a></i>;<br/><a NAME="E42:34_2"/><b>then </b><i><font color="Green" title="E43">A32</font></i>: 
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c11">x</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#E9:34_2"><i><font color="Green" title="E22">A10</font></i></a>, <a class="ref" href="analoaf.html#D5" title="ANALOAF:def.5">ANALOAF:def 5</a></i>;<br/><a NAME="E43:34_2"/>
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c9">e5</font>,<font color="Maroon" title="c7">e3</font>,<font color="Maroon" title="c8">e4</font>
 <b>by </b><i><a class="txt" href="pasch.html#E21:34_2"><i><font color="Green" title="E29">A17</font></i></a>, <a class="ref" href="diraf.html#T13" title="DIRAF:th.13">DIRAF:13</a></i>;<br/><a NAME="E44:34_2"/><b>then </b>
<font color="Maroon" title="c9">e5</font>,<font color="Maroon" title="c7">e3</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">e3</font>,<font color="Maroon" title="c8">e4</font>
 <b>by </b><i><a class="ref" href="diraf.html#D3" title="DIRAF:def.3">DIRAF:def 3</a></i>;<br/><a NAME="E45:34_2"/><b>then </b>
( <font color="Maroon" title="c9">e5</font>,<font color="Maroon" title="c7">e3</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">e3</font>,<font color="Maroon" title="c8">e4</font> &amp; <font color="Maroon" title="c9">e5</font>,<font color="Maroon" title="c7">e3</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">e5</font>,<font color="Maroon" title="c8">e4</font> )
 <b>by </b><i><a class="ref" href="analoaf.html#D5" title="ANALOAF:def.5">ANALOAF:def 5</a></i>;<br/><a NAME="E46:34_2"/><b>then </b>
<font color="Maroon" title="c7">e3</font>,<font color="Maroon" title="c8">e4</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">e5</font>,<font color="Maroon" title="c8">e4</font>
 <b>by </b><i><a class="txt" href="pasch.html#E35:34_2"><i><font color="Green" title="E41">A29</font></i></a>, <a class="ref" href="analoaf.html#D5" title="ANALOAF:def.5">ANALOAF:def 5</a></i>;<br/><a NAME="E47:34_2"/><b>then </b>
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">e5</font>,<font color="Maroon" title="c8">e4</font>
 <b>by </b><i><a class="txt" href="pasch.html#E12:34_2"><i><font color="Green" title="E24">A12</font></i></a>, <a class="txt" href="pasch.html#E22:34_2"><i><font color="Green" title="E30">A18</font></i></a>, <a class="ref" href="diraf.html#T6" title="DIRAF:th.6">DIRAF:6</a></i>;<br/><a NAME="E48:34_2"/><b>then </b><i><font color="Green" title="E44">A33</font></i>: 
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">e6</font>,<font color="Maroon" title="c4">c</font>
 <b>by </b><i><a class="txt" href="pasch.html#E23:34_2"><i><font color="Green" title="E31">A19</font></i></a>, <a class="txt" href="pasch.html#E29:34_2"><i><font color="Green" title="E36">A24</font></i></a>, <a class="ref" href="diraf.html#T6" title="DIRAF:th.6">DIRAF:6</a></i>;<br/><a NAME="E49:34_2"/><i><font color="Green" title="E45">A34</font></i>: 
<font color="Maroon" title="c10">e6</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">c</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2_9"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:34_2_9"/>
<font color="Maroon" title="c10">e6</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font>
 ;<br/>

<a NAME="E2:34_2_9"/><b>then </b>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">c</font>
 
<b>by </b><i><a class="txt" href="pasch.html#E31:34_2"><i><font color="Green" title="E37">A25</font></i></a>, <a class="ref" href="diraf.html#T12" title="DIRAF:th.12">DIRAF:12</a></i>;<br/>
<a NAME="E3:34_2_9"/><b>then </b>
<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c2">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c3">b</font>
 
<b>by </b><i><a class="txt" href="pasch.html#E42:34_2"><i><font color="Green" title="E43">A32</font></i></a>, <a class="ref" href="diraf.html#T5" title="DIRAF:th.5">DIRAF:5</a></i>;<br/>
<a NAME="E4:34_2_9"/><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="c2">a</font>,<font color="Maroon" title="c3">b</font> or  <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="c2">a</font> )
 
<b>by </b><i><a class="ref" href="diraf.html#T11" title="DIRAF:th.11">DIRAF:11</a></i>;<br/>
<a NAME="E5:34_2_9"/><b>then </b>
(  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> or  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">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="E6:34_2_9"/>
contradiction
 <b>by </b><i><a class="txt" href="pasch.html#E1:34_2"><i><font color="Green" title="E17">A5</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="E50:34_2"/><i><font color="Green" title="E46">A35</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">e2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2_10"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:34_2_10"/>
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">e2</font>
 ;<br/>

<a NAME="E2:34_2_10"/><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="c5">e1</font>,<font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c5">e1</font>,<font color="Maroon" title="c2">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font> )
 
<b>by </b><i><a class="txt" href="pasch.html#E5:34_2"><i><font color="Green" title="E20">A8</font></i></a>, <a class="txt" href="pasch.html#E7:34_2"><i><font color="Green" title="E21">A9</font></i></a>, <a class="ref" href="diraf.html#T5" title="DIRAF:th.5">DIRAF:5</a></i>;<br/>
<a NAME="E3:34_2_10"/><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="c2">a</font>,<font color="Maroon" title="c4">c</font>
 
<b>by </b><i><a class="txt" href="pasch.html#E4:34_2"><i><font color="Green" title="E19">A7</font></i></a>, <a class="ref" href="diraf.html#T6" title="DIRAF:th.6">DIRAF:6</a></i>;<br/>
<a NAME="E4:34_2_10"/><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="c4">c</font>,<font color="Maroon" title="c2">a</font>
 
<b>by </b><i><a class="ref" href="diraf.html#T5" title="DIRAF:th.5">DIRAF:5</a></i>;<br/>
<a NAME="E5:34_2_10"/><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="c2">a</font> or  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> )
 
<b>by </b><i><a class="ref" href="diraf.html#T11" title="DIRAF:th.11">DIRAF:11</a></i>;<br/>
<a NAME="E6:34_2_10"/><b>then </b>
(  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font> or  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</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="E7:34_2_10"/>
contradiction
 <b>by </b><i><a class="txt" href="pasch.html#E1:34_2"><i><font color="Green" title="E17">A5</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="E51:34_2"/><i><font color="Green" title="E47">A36</font></i>: 
<font color="Maroon" title="c10">e6</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2_11"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:34_2_11"/>
<font color="Maroon" title="c10">e6</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x</font>
 ;<br/>

<a NAME="E2:34_2_11"/><b>then </b>
<font color="Maroon" title="c10">e6</font>,<font color="Maroon" title="c5">e1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">e6</font>,<font color="Maroon" title="c2">a</font>
 
<b>by </b><i><a class="txt" href="pasch.html#E31:34_2"><i><font color="Green" title="E37">A25</font></i></a>, <a class="ref" href="diraf.html#T5" title="DIRAF:th.5">DIRAF:5</a></i>;<br/>
<a NAME="E3:34_2_11"/><b>then </b>
(  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c10">e6</font>,<font color="Maroon" title="c5">e1</font>,<font color="Maroon" title="c2">a</font> or  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c10">e6</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">e1</font> )
 
<b>by </b><i><a class="ref" href="diraf.html#T11" title="DIRAF:th.11">DIRAF:11</a></i>;<br/>
<a NAME="E4:34_2_11"/><b>then </b>
(  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c10">e6</font>,<font color="Maroon" title="c5">e1</font>,<font color="Maroon" title="c2">a</font> or  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c10">e6</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">e1</font> )
 
<b>by </b><i><a class="ref" href="diraf.html#T34" title="DIRAF:th.34">DIRAF:34</a></i>;<br/>
<a NAME="E5:34_2_11"/><b>then </b>
(  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c10">e6</font>,<font color="Maroon" title="c5">e1</font>,<font color="Maroon" title="c2">a</font> &amp;  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c5">e1</font>,<font color="Maroon" title="c10">e6</font>,<font color="Maroon" title="c9">e5</font> )
 
<b>by </b><i><a class="txt" href="pasch.html#E29:34_2"><i><font color="Green" title="E36">A24</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></i>;<br/>
<a NAME="E6:34_2_11"/><b>then </b>
(  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c10">e6</font>,<font color="Maroon" title="c5">e1</font>,<font color="Maroon" title="c2">a</font> &amp;  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c10">e6</font>,<font color="Maroon" title="c5">e1</font>,<font color="Maroon" title="c9">e5</font> &amp;  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c10">e6</font>,<font color="Maroon" title="c5">e1</font>,<font color="Maroon" title="c5">e1</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="E7:34_2_11"/><b>then </b><i><font color="Green" title="E48">A37</font></i>: 
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c5">e1</font>,<font color="Maroon" title="c9">e5</font>,<font color="Maroon" title="c2">a</font>
 
<b>by </b><i><a class="txt" href="pasch.html#E36:34_2"><i><font color="Green" title="E42">A30</font></i></a>, <a class="ref" href="diraf.html#T38" title="DIRAF:th.38">DIRAF:38</a></i>;<br/>
<a NAME="E8:34_2_11"/>
<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="c5">e1</font>,<font color="Maroon" title="c9">e5</font>
 
<b>by </b><i><a class="txt" href="pasch.html#E9:34_2"><i><font color="Green" title="E22">A10</font></i></a>, <a class="txt" href="pasch.html#E10:34_2"><i><font color="Green" title="E23">A11</font></i></a>, <a class="txt" href="pasch.html#E21:34_2"><i><font color="Green" title="E29">A17</font></i></a>, <a class="ref" href="diraf.html#T6" title="DIRAF:th.6">DIRAF:6</a></i>;<br/>
<a NAME="E9:34_2_11"/><b>then </b>
<font color="Maroon" title="c5">e1</font>,<font color="Maroon" title="c6">e2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">e1</font>,<font color="Maroon" title="c9">e5</font>
 
<b>by </b><i><a class="txt" href="pasch.html#E2:34_2"><i><font color="Green" title="E18">A6</font></i></a>, <a class="txt" href="pasch.html#E7:34_2"><i><font color="Green" title="E21">A9</font></i></a>, <a class="ref" href="analoaf.html#D5" title="ANALOAF:def.5">ANALOAF:def 5</a></i>;<br/>
<a NAME="E10:34_2_11"/><b>then </b>
(  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c5">e1</font>,<font color="Maroon" title="c6">e2</font>,<font color="Maroon" title="c9">e5</font> or  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c5">e1</font>,<font color="Maroon" title="c9">e5</font>,<font color="Maroon" title="c6">e2</font> )
 
<b>by </b><i><a class="ref" href="diraf.html#T11" title="DIRAF:th.11">DIRAF:11</a></i>;<br/>
<a NAME="E11:34_2_11"/><b>then </b>
(  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c5">e1</font>,<font color="Maroon" title="c6">e2</font>,<font color="Maroon" title="c9">e5</font> or  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c5">e1</font>,<font color="Maroon" title="c9">e5</font>,<font color="Maroon" title="c6">e2</font> )
 
<b>by </b><i><a class="ref" href="diraf.html#T34" title="DIRAF:th.34">DIRAF:34</a></i>;<br/>
<a NAME="E12:34_2_11"/><b>then </b>
(  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c5">e1</font>,<font color="Maroon" title="c9">e5</font>,<font color="Maroon" title="c6">e2</font> &amp;  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c5">e1</font>,<font color="Maroon" title="c9">e5</font>,<font color="Maroon" title="c5">e1</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="E13:34_2_11"/><b>then </b>
(  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">e1</font>,<font color="Maroon" title="c6">e2</font> &amp;  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">e1</font> )
 
<b>by </b><i><a class="txt" href="pasch.html#E4:34_2"><i><font color="Green" title="E19">A7</font></i></a>, <a class="txt" href="pasch.html#E32:34_2"><i><font color="Green" title="E38">A26</font></i></a>, <a class="txt" href="pasch.html#E7:34_2_11"><i><font color="Green" title="E48">A37</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="E14:34_2_11"/><b>then </b>
(  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">e1</font>,<font color="Maroon" title="c6">e2</font> &amp;  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">e1</font>,<font color="Maroon" title="c4">c</font> &amp;  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">e1</font>,<font color="Maroon" title="c2">a</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="E15:34_2_11"/><b>then </b>
(  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c6">e2</font>,<font color="Maroon" title="c4">c</font> &amp;  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c6">e2</font> )
 
<b>by </b><i><a class="txt" href="pasch.html#E4:34_2"><i><font color="Green" title="E19">A7</font></i></a>, <a class="txt" href="pasch.html#E7:34_2"><i><font color="Green" title="E21">A9</font></i></a>, <a class="ref" href="diraf.html#T38" title="DIRAF:th.38">DIRAF:38</a>, <a class="ref" href="diraf.html#D3" title="DIRAF:def.3">DIRAF:def 3</a></i>;<br/>
<a NAME="E16:34_2_11"/><b>then </b>
(  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c6">e2</font>,<font color="Maroon" title="c4">c</font> &amp;  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c6">e2</font> )
 
<b>by </b><i><a class="ref" href="diraf.html#T34" title="DIRAF:th.34">DIRAF:34</a></i>;<br/>
<a NAME="E17:34_2_11"/><b>then </b>
(  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c6">e2</font>,<font color="Maroon" title="c4">c</font> &amp;  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c6">e2</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">a</font>,<font color="Maroon" title="c6">e2</font>,<font color="Maroon" title="c2">a</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="E18:34_2_11"/>
contradiction
 <b>by </b><i><a class="txt" href="pasch.html#E1:34_2"><i><font color="Green" title="E17">A5</font></i></a>, <a class="txt" href="pasch.html#E50:34_2"><i><font color="Green" title="E46">A35</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="E52:34_2"/>
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c10">e6</font>,<font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c4">c</font>
 <b>by </b><i><a class="txt" href="pasch.html#E31:34_2"><i><font color="Green" title="E37">A25</font></i></a>, <a class="ref" href="diraf.html#T13" title="DIRAF:th.13">DIRAF:13</a></i>;<br/><a NAME="E53:34_2"/><b>then </b>
<font color="Maroon" title="c10">e6</font>,<font color="Maroon" title="c11">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c4">c</font>
 <b>by </b><i><a class="ref" href="diraf.html#D3" title="DIRAF:def.3">DIRAF:def 3</a></i>;<br/><a NAME="E54:34_2"/><b>then </b>
( <font color="Maroon" title="c10">e6</font>,<font color="Maroon" title="c11">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c4">c</font> &amp; <font color="Maroon" title="c10">e6</font>,<font color="Maroon" title="c11">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c10">e6</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/><a NAME="E55:34_2"/><b>then </b>
<font color="Maroon" title="c10">e6</font>,<font color="Maroon" title="c4">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c4">c</font>
 <b>by </b><i><a class="txt" href="pasch.html#E51:34_2"><i><font color="Green" title="E47">A36</font></i></a>, <a class="ref" href="analoaf.html#D5" title="ANALOAF:def.5">ANALOAF:def 5</a></i>;<br/><a NAME="E56:34_2"/><b>then </b>
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c11">x</font>,<font color="Maroon" title="c4">c</font>
 <b>by </b><i><a class="txt" href="pasch.html#E48:34_2"><i><font color="Green" title="E44">A33</font></i></a>, <a class="txt" href="pasch.html#E49:34_2"><i><font color="Green" title="E45">A34</font></i></a>, <a class="ref" href="diraf.html#T6" title="DIRAF:th.6">DIRAF:6</a></i>;<br/><b>hence </b><a NAME="E57:34_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/>( <font color="Maroon" title="c2">a</font>,<font color="Olive" title="b1">x</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> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b1">x</font>,<font color="Maroon" title="c4">c</font> )
 <b>by </b><i><a class="txt" href="pasch.html#E42:34_2"><i><font color="Green" title="E43">A32</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:34"/>
 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/>( <font color="Maroon" title="c2">a</font>,<font color="Olive" title="b1">x</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> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b1">x</font>,<font color="Maroon" title="c4">c</font> )
 <b>by </b><i><a class="txt" href="pasch.html#E1:34"><i><font color="Green" title="E16">A1</font></i></a></i>;<br/>


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