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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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">a'</font>, <font color="Olive" title="b4">b'</font>, <font color="Olive" title="b5">c</font>, <font color="Olive" title="b6">c'</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">OAS</font>  st not  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b3">a'</font> &amp; <font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">a'</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Olive" title="b2">b</font>,<font color="Olive" title="b4">b'</font> &amp; <font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">a'</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Olive" title="b5">c</font>,<font color="Olive" title="b6">c'</font> &amp;  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font>,<font color="Olive" title="b5">c</font> &amp;  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Olive" title="b3">a'</font>,<font color="Olive" title="b4">b'</font>,<font color="Olive" title="b6">c'</font> holds <br/> <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Olive" title="b3">a'</font>,<font color="Olive" title="b4">b'</font>,<font color="Olive" title="b6">c'</font></span><br/><b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c4">a'</font>, <font color="Maroon" title="c5">b'</font>, <font color="Maroon" title="c6">c</font>, <font color="Maroon" title="c7">c'</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">OAS</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( 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">a'</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">a'</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">b'</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">a'</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">c'</font> &amp;  <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="c6">c</font> &amp;  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c4">a'</font>,<font color="Maroon" title="c5">b'</font>,<font color="Maroon" title="c7">c'</font> implies  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c4">a'</font>,<font color="Maroon" title="c5">b'</font>,<font color="Maroon" title="c7">c'</font> )</span><br/>













<b>assume </b><b>that </b><br/><a NAME="E1:16"/><i><font color="Green" title="E8">A1</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">a'</font>
 <b>and </b><br/><a NAME="E2:16"/><i><font color="Green" title="E9">A2</font></i>: 
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">a'</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">b'</font>
 <b>and </b><br/><a NAME="E3:16"/><i><font color="Green" title="E10">A3</font></i>: 
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">a'</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">c'</font>
 <b>and </b><br/><a NAME="E4:16"/><i><font color="Green" title="E11">A4</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="c6">c</font>
 <b>and </b><br/><a NAME="E5:16"/><i><font color="Green" title="E12">A5</font></i>: 
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c4">a'</font>,<font color="Maroon" title="c5">b'</font>,<font color="Maroon" title="c7">c'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c4">a'</font>,<font color="Maroon" title="c5">b'</font>,<font color="Maroon" title="c7">c'</font></span><br/>

<a NAME="E6:16"/><i><font color="Green" title="E13">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="c4">a'</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="c3">b</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">a'</font> )
 
<b>by </b><i><a class="txt" href="#E1:16"><i><font color="Green" title="E8">A1</font></i></a>, <a class="ref" href="diraf.html#T37" title="DIRAF:th.37">DIRAF:37</a></i>;<br/>
<a NAME="E7:16"/><i><font color="Green" title="E14">A7</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="c6">c</font>
 
<b>by </b><i><a class="txt" href="#E4:16"><i><font color="Green" title="E11">A4</font></i></a>, <a class="ref" href="diraf.html#T34" title="DIRAF:th.34">DIRAF:34</a></i>;<br/>
<a NAME="E8:16"/><i><font color="Green" title="E15">A8</font></i>: 
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c5">b'</font>,<font color="Maroon" title="c7">c'</font>,<font color="Maroon" title="c4">a'</font>
 
<b>by </b><i><a class="txt" href="#E5:16"><i><font color="Green" title="E12">A5</font></i></a>, <a class="ref" href="diraf.html#T36" title="DIRAF:th.36">DIRAF:36</a></i>;<br/>
<a NAME="E9:16"/><i><font color="Green" title="E16">A9</font></i>: 
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c7">c'</font>,<font color="Maroon" title="c5">b'</font>,<font color="Maroon" title="c4">a'</font>
 
<b>by </b><i><a class="txt" href="#E5:16"><i><font color="Green" title="E12">A5</font></i></a>, <a class="ref" href="diraf.html#T36" title="DIRAF:th.36">DIRAF:36</a></i>;<br/>
<a NAME="E10:16"/><i><font color="Green" title="E17">A10</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">c</font>
 
<b>by </b><i><a class="txt" href="#E4:16"><i><font color="Green" title="E11">A4</font></i></a>, <a class="txt" href="#E6:16"><i><font color="Green" title="E13">A6</font></i></a>, <a class="ref" href="diraf.html#T12" title="DIRAF:th.12">DIRAF:12</a></i>;<br/>
<a NAME="E11:16"/><i><font color="Green" title="E18">A11</font></i>: 
<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">b'</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">c'</font>
 
<b>by </b><i><a class="txt" href="#E2:16"><i><font color="Green" title="E9">A2</font></i></a>, <a class="txt" href="#E3:16"><i><font color="Green" title="E10">A3</font></i></a>, <a class="txt" href="#E6:16"><i><font color="Green" title="E13">A6</font></i></a>, <a class="ref" href="diraf.html#T28" title="DIRAF:th.28">DIRAF:28</a></i>;<br/>
<a NAME="E12:16"/><i><font color="Green" title="E19">A12</font></i>: 
( <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b'</font> implies  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c4">a'</font>,<font color="Maroon" title="c5">b'</font>,<font color="Maroon" title="c7">c'</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:16_1"/><i><font color="Green" title="E20">A13</font></i>: 
<font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c4">a'</font>,<font color="Maroon" title="c5">b'</font>,<font color="Maroon" title="c7">c'</font></span><br/>

<a NAME="E2:16_1"/><b>then </b><i><font color="Green" title="E21">A14</font></i>: 
(  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">b'</font>,<font color="Maroon" title="c6">c</font> &amp;  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c5">b'</font>,<font color="Maroon" title="c4">a'</font>,<font color="Maroon" title="c7">c'</font> )
 
<b>by </b><i><a class="txt" href="#E4:16"><i><font color="Green" title="E11">A4</font></i></a>, <a class="txt" href="#E5:16"><i><font color="Green" title="E12">A5</font></i></a>, <a class="ref" href="diraf.html#T36" title="DIRAF:th.36">DIRAF:36</a></i>;<br/>
<a NAME="E3:16_1"/><i><font color="Green" title="E22">A15</font></i>: 
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">a'</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c7">c'</font>,<font color="Maroon" title="c6">c</font>
 
<b>by </b><i><a class="txt" href="#E3:16"><i><font color="Green" title="E10">A3</font></i></a>, <a class="ref" href="diraf.html#T27" title="DIRAF:th.27">DIRAF:27</a></i>;<br/>
<a NAME="E4:16_1"/>
not  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c5">b'</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">a'</font>
 
<b>by </b><i><a class="txt" href="#E1:16"><i><font color="Green" title="E8">A1</font></i></a>, <a class="txt" href="#E1:16_1"><i><font color="Green" title="E20">A13</font></i></a>, <a class="ref" href="diraf.html#T36" title="DIRAF:th.36">DIRAF:36</a></i>;<br/>
<b>hence </b><a NAME="E5:16_1"/>
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c4">a'</font>,<font color="Maroon" title="c5">b'</font>,<font color="Maroon" title="c7">c'</font>
 <b>by </b><i><a class="txt" href="#E2:16_1"><i><font color="Green" title="E21">A14</font></i></a>, <a class="txt" href="#E3:16_1"><i><font color="Green" title="E22">A15</font></i></a>, <a class="ref" href="pasch.html#T13" target="_self" title="PASCH:th.13">Th13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:16_2"/><i><font color="Green" title="E20">A16</font></i>: 
( <font color="Maroon" title="c5">b'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">c'</font> &amp; <font color="Maroon" title="c4">a'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b'</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="c5">b'</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c4">a'</font>,<font color="Maroon" title="c5">b'</font>,<font color="Maroon" title="c7">c'</font></span><br/><a NAME="E2:16_2"/><i><font color="Green" title="E21">A17</font></i>: 
not  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">b'</font>,<font color="Maroon" title="c4">a'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:16_2_1"/><i><font color="Green" title="E22">A18</font></i>: 
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">b'</font>,<font color="Maroon" title="c4">a'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:16_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="c5">b'</font>,<font color="Maroon" title="c4">a'</font> &amp; <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">b'</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c4">a'</font>,<font color="Maroon" title="c2">a</font> )
 
<b>by </b><i><a class="txt" href="#E2:16"><i><font color="Green" title="E9">A2</font></i></a>, <a class="ref" href="diraf.html#T27" title="DIRAF:th.27">DIRAF:27</a></i>;<br/>
<a NAME="E3:16_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="c5">b'</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="c5">b'</font>,<font color="Maroon" title="c3">b</font> )
 
<b>by </b><i><a class="txt" href="#E1:16_2"><i><font color="Green" title="E20">A16</font></i></a>, <a class="ref" href="diraf.html#T37" title="DIRAF:th.37">DIRAF:37</a>, <a class="ref" href="diraf.html#T39" title="DIRAF:th.39">DIRAF:39</a></i>;<br/>
<b>hence </b><a NAME="E4:16_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:16"><i><font color="Green" title="E8">A1</font></i></a>, <a class="txt" href="#E1:16_2"><i><font color="Green" title="E20">A16</font></i></a>, <a class="txt" href="#E1:16_2_1"><i><font color="Green" title="E22">A18</font></i></a>, <a class="ref" href="diraf.html#T38" title="DIRAF:th.38">DIRAF:38</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E3:16_2"/><i><font color="Green" title="E22">A19</font></i>: 
( <font color="Maroon" title="c6">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">c'</font> implies  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c4">a'</font>,<font color="Maroon" title="c5">b'</font>,<font color="Maroon" title="c7">c'</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:16_2_2"/><i><font color="Green" title="E23">A20</font></i>: 
<font color="Maroon" title="c6">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">c'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c4">a'</font>,<font color="Maroon" title="c5">b'</font>,<font color="Maroon" title="c7">c'</font></span><br/>

<a NAME="E2:16_2_2"/><i><font color="Green" title="E24">A21</font></i>: 
<font color="Maroon" title="c5">b'</font>,<font color="Maroon" title="c3">b</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">a'</font>
 
<b>by </b><i><a class="txt" href="#E2:16"><i><font color="Green" title="E9">A2</font></i></a>, <a class="ref" href="diraf.html#T27" title="DIRAF:th.27">DIRAF:27</a></i>;<br/>
<a NAME="E3:16_2_2"/><i><font color="Green" title="E25">A22</font></i>: 
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c7">c'</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font>
 
<b>by </b><i><a class="txt" href="#E4:16"><i><font color="Green" title="E11">A4</font></i></a>, <a class="txt" href="#E1:16_2_2"><i><font color="Green" title="E23">A20</font></i></a>, <a class="ref" href="diraf.html#T13" title="DIRAF:th.13">DIRAF:13</a></i>;<br/>
<a NAME="E4:16_2_2"/>
not  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c7">c'</font>,<font color="Maroon" title="c5">b'</font>,<font color="Maroon" title="c3">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:16_2_2_1"/>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c7">c'</font>,<font color="Maroon" title="c5">b'</font>,<font color="Maroon" title="c3">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:16_2_2_1"/><b>then </b>
(  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c7">c'</font>,<font color="Maroon" title="c5">b'</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">c'</font>,<font color="Maroon" title="c5">b'</font>,<font color="Maroon" title="c5">b'</font> )
 
<b>by </b><i><a class="ref" href="diraf.html#T37" title="DIRAF:th.37">DIRAF:37</a></i>;<br/>
<b>hence </b><a NAME="E3:16_2_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E9:16"><i><font color="Green" title="E16">A9</font></i></a>, <a class="txt" href="#E1:16_2"><i><font color="Green" title="E20">A16</font></i></a>, <a class="txt" href="#E2:16_2"><i><font color="Green" title="E21">A17</font></i></a>, <a class="ref" href="diraf.html#T38" title="DIRAF:th.38">DIRAF:38</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E5:16_2_2"/><b>then </b>
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c7">c'</font>,<font color="Maroon" title="c5">b'</font>,<font color="Maroon" title="c4">a'</font>
 
<b>by </b><i><a class="txt" href="#E9:16"><i><font color="Green" title="E16">A9</font></i></a>, <a class="txt" href="#E2:16_2_2"><i><font color="Green" title="E24">A21</font></i></a>, <a class="txt" href="#E3:16_2_2"><i><font color="Green" title="E25">A22</font></i></a>, <a class="ref" href="pasch.html#T15" target="_self" title="PASCH:th.15">Th15</a></i>;<br/>
<b>hence </b><a NAME="E6:16_2_2"/>
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c4">a'</font>,<font color="Maroon" title="c5">b'</font>,<font color="Maroon" title="c7">c'</font>
 <b>by </b><i><a class="ref" href="diraf.html#T13" title="DIRAF:th.13">DIRAF:13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:16_2_3"/><i><font color="Green" title="E23">A23</font></i>: 
<font color="Maroon" title="c6">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">c'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c4">a'</font>,<font color="Maroon" title="c5">b'</font>,<font color="Maroon" title="c7">c'</font></span><br/><a NAME="E2:16_2_3"/>
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c6">c</font>
 <b>by </b><i><a class="txt" href="#E7:16"><i><font color="Green" title="E14">A7</font></i></a>, <a class="ref" href="diraf.html#D5" title="DIRAF:def.5">DIRAF:def 5</a></i>;<br/><a NAME="E3:16_2_3"/><b>then </b>
<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c2">a</font> <a href="diraf.html#R2" title="DIRAF: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#T27" title="DIRAF:th.27">DIRAF:27</a></i>;<br/><b>then consider </b><font color="Maroon" title="c8">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="E5:16_2_3"/><i><font color="Green" title="E24">A24</font></i>: 
( <font color="Maroon" title="c7">c'</font>,<font color="Maroon" title="c2">a</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c8">x</font> &amp; <font color="Maroon" title="c7">c'</font>,<font color="Maroon" title="c6">c</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c8">x</font> )
 <b>by </b><i><a class="txt" href="#E10:16"><i><font color="Green" title="E17">A10</font></i></a>, <a class="ref" href="diraf.html#T32" title="DIRAF:th.32">DIRAF:32</a></i>;<br/><a NAME="E6:16_2_3"/>
<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">c'</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c8">x</font>
 <b>by </b><i><a class="txt" href="#E5:16_2_3"><i><font color="Green" title="E24">A24</font></i></a>, <a class="ref" href="diraf.html#T27" title="DIRAF:th.27">DIRAF:27</a></i>;<br/><a NAME="E7:16_2_3"/><b>then </b>
( <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c7">c'</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c8">x</font> &amp; <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">b'</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c8">x</font> )
 <b>by </b><i><a class="txt" href="#E11:16"><i><font color="Green" title="E18">A11</font></i></a>, <a class="txt" href="#E1:16_2_3"><i><font color="Green" title="E23">A23</font></i></a>, <a class="txt" href="#E5:16_2_3"><i><font color="Green" title="E24">A24</font></i></a>, <a class="ref" href="diraf.html#T27" title="DIRAF:th.27">DIRAF:27</a>, <a class="ref" href="diraf.html#T28" title="DIRAF:th.28">DIRAF:28</a></i>;<br/><a NAME="E8:16_2_3"/><b>then </b><i><font color="Green" title="E25">A25</font></i>: 
(  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c7">c'</font>,<font color="Maroon" title="c8">x</font> &amp;  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">b'</font>,<font color="Maroon" title="c8">x</font> )
 <b>by </b><i><a class="ref" href="diraf.html#D5" title="DIRAF:def.5">DIRAF:def 5</a></i>;<br/><a NAME="E9:16_2_3"/><b>then </b><i><font color="Green" title="E26">A26</font></i>: 
(  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">b'</font> &amp;  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c7">c'</font> )
 <b>by </b><i><a class="ref" href="diraf.html#T36" title="DIRAF:th.36">DIRAF:36</a></i>;<br/><a NAME="E10:16_2_3"/><i><font color="Green" title="E27">A27</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:16_2_3_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:16_2_3_1"/><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="c7">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="c3">b</font>,<font color="Maroon" title="c2">a</font> &amp;  <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="c3">b</font> )
 
<b>by </b><i><a class="txt" href="#E8:16_2_3"><i><font color="Green" title="E25">A25</font></i></a>, <a class="ref" href="diraf.html#T36" title="DIRAF:th.36">DIRAF:36</a>, <a class="ref" href="diraf.html#T37" title="DIRAF:th.37">DIRAF:37</a></i>;<br/>
<a NAME="E3:16_2_3_1"/><b>then </b>
(  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">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="c6">c</font>,<font color="Maroon" title="c7">c'</font>,<font color="Maroon" title="c3">b</font> )
 
<b>by </b><i><a class="txt" href="#E6:16"><i><font color="Green" title="E13">A6</font></i></a>, <a class="txt" href="#E7:16"><i><font color="Green" title="E14">A7</font></i></a>, <a class="ref" href="diraf.html#T38" title="DIRAF:th.38">DIRAF:38</a></i>;<br/>
<a NAME="E4:16_2_3_1"/><b>then </b>
<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">c'</font> <a href="diraf.html#R2" title="DIRAF: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#T40" title="DIRAF:th.40">DIRAF:40</a></i>;<br/>
<a NAME="E5:16_2_3_1"/><b>then </b>
<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">b'</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>
 
<b>by </b><i><a class="txt" href="#E11:16"><i><font color="Green" title="E18">A11</font></i></a>, <a class="txt" href="#E1:16_2_3"><i><font color="Green" title="E23">A23</font></i></a>, <a class="ref" href="diraf.html#T28" title="DIRAF:th.28">DIRAF:28</a></i>;<br/>
<a NAME="E6:16_2_3_1"/><b>then </b>
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">a'</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>
 
<b>by </b><i><a class="txt" href="#E2:16"><i><font color="Green" title="E9">A2</font></i></a>, <a class="txt" href="#E1:16_2"><i><font color="Green" title="E20">A16</font></i></a>, <a class="ref" href="diraf.html#T28" title="DIRAF:th.28">DIRAF:28</a></i>;<br/>
<a NAME="E7:16_2_3_1"/><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="c4">a'</font>,<font color="Maroon" title="c3">b</font>
 
<b>by </b><i><a class="ref" href="diraf.html#D5" title="DIRAF:def.5">DIRAF:def 5</a></i>;<br/>
<b>hence </b><a NAME="E8:16_2_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:16"><i><font color="Green" title="E8">A1</font></i></a>, <a class="ref" href="diraf.html#T36" title="DIRAF:th.36">DIRAF:36</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E11:16_2_3"/><i><font color="Green" title="E28">A28</font></i>: 
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c7">c'</font>
 <b>by </b><i><a class="txt" href="#E8:16_2_3"><i><font color="Green" title="E25">A25</font></i></a>, <a class="ref" href="diraf.html#T36" title="DIRAF:th.36">DIRAF:36</a></i>;<br/><a NAME="E12:16_2_3"/><i><font color="Green" title="E29">A29</font></i>: 
<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c3">b</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">c'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_3_2"><b>proof </b></a><div class="add">

<a NAME="E1:16_2_3_2"/>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c5">b'</font>
 
<b>by </b><i><a class="txt" href="#E8:16_2_3"><i><font color="Green" title="E25">A25</font></i></a>, <a class="ref" href="diraf.html#T36" title="DIRAF:th.36">DIRAF:36</a></i>;<br/>
<a NAME="E2:16_2_3_2"/><b>then </b>
<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c8">x</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">b'</font>
 
<b>by </b><i><a class="ref" href="diraf.html#D5" title="DIRAF:def.5">DIRAF:def 5</a></i>;<br/>
<a NAME="E3:16_2_3_2"/><b>then </b>
<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c8">x</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">c'</font>
 
<b>by </b><i><a class="txt" href="#E11:16"><i><font color="Green" title="E18">A11</font></i></a>, <a class="txt" href="#E1:16_2"><i><font color="Green" title="E20">A16</font></i></a>, <a class="ref" href="diraf.html#T28" title="DIRAF:th.28">DIRAF:28</a></i>;<br/>
<b>hence </b><a NAME="E4:16_2_3_2"/>
<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c3">b</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">c'</font>
 <b>by </b><i><a class="ref" href="diraf.html#T27" title="DIRAF:th.27">DIRAF:27</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E13:16_2_3"/>
not  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c3">b</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_3_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:16_2_3_3"/>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c3">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:16_2_3_3"/><b>then </b>
(  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c3">b</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">x</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c3">b</font> )
 
<b>by </b><i><a class="ref" href="diraf.html#T36" title="DIRAF:th.36">DIRAF:36</a>, <a class="ref" href="diraf.html#T37" title="DIRAF:th.37">DIRAF:37</a></i>;<br/>
<a NAME="E3:16_2_3_3"/><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">b'</font>,<font color="Maroon" title="c2">a</font> &amp; <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">b'</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">a'</font> )
 
<b>by </b><i><a class="txt" href="#E2:16"><i><font color="Green" title="E9">A2</font></i></a>, <a class="txt" href="#E9:16_2_3"><i><font color="Green" title="E26">A26</font></i></a>, <a class="txt" href="#E10:16_2_3"><i><font color="Green" title="E27">A27</font></i></a>, <a class="ref" href="diraf.html#T27" title="DIRAF:th.27">DIRAF:27</a>, <a class="ref" href="diraf.html#T38" title="DIRAF:th.38">DIRAF:38</a></i>;<br/>
<b>hence </b><a NAME="E4:16_2_3_3"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:16_2"><i><font color="Green" title="E20">A16</font></i></a>, <a class="txt" href="#E2:16_2"><i><font color="Green" title="E21">A17</font></i></a>, <a class="ref" href="diraf.html#T39" title="DIRAF:th.39">DIRAF:39</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E14:16_2_3"/><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="c8">x</font>,<font color="Maroon" title="c7">c'</font>
 <b>by </b><i><a class="txt" href="#E4:16"><i><font color="Green" title="E11">A4</font></i></a>, <a class="txt" href="#E11:16_2_3"><i><font color="Green" title="E28">A28</font></i></a>, <a class="txt" href="#E12:16_2_3"><i><font color="Green" title="E29">A29</font></i></a>, <a class="ref" href="pasch.html#T15" target="_self" title="PASCH:th.15">Th15</a></i>;<br/><a NAME="E15:16_2_3"/><b>then </b><i><font color="Green" title="E30">A30</font></i>: 
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c7">c'</font>,<font color="Maroon" title="c8">x</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="E16:16_2_3"/><i><font color="Green" title="E31">A31</font></i>: 
<font color="Maroon" title="c5">b'</font>,<font color="Maroon" title="c8">x</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">a'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_3_4"><b>proof </b></a><div class="add">

<a NAME="E1:16_2_3_4"/>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c5">b'</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c8">x</font>
 
<b>by </b><i><a class="txt" href="#E8:16_2_3"><i><font color="Green" title="E25">A25</font></i></a>, <a class="ref" href="diraf.html#T36" title="DIRAF:th.36">DIRAF:36</a></i>;<br/>
<a NAME="E2:16_2_3_4"/><b>then </b>
<font color="Maroon" title="c5">b'</font>,<font color="Maroon" title="c3">b</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c5">b'</font>,<font color="Maroon" title="c8">x</font>
 
<b>by </b><i><a class="ref" href="diraf.html#D5" title="DIRAF:def.5">DIRAF:def 5</a></i>;<br/>
<a NAME="E3:16_2_3_4"/><b>then </b>
<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">b'</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c5">b'</font>,<font color="Maroon" title="c8">x</font>
 
<b>by </b><i><a class="ref" href="diraf.html#T27" title="DIRAF:th.27">DIRAF:27</a></i>;<br/>
<b>hence </b><a NAME="E4:16_2_3_4"/>
<font color="Maroon" title="c5">b'</font>,<font color="Maroon" title="c8">x</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">a'</font>
 <b>by </b><i><a class="txt" href="#E2:16"><i><font color="Green" title="E9">A2</font></i></a>, <a class="txt" href="#E1:16_2"><i><font color="Green" title="E20">A16</font></i></a>, <a class="ref" href="diraf.html#T28" title="DIRAF:th.28">DIRAF:28</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E17:16_2_3"/><i><font color="Green" title="E32">A32</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">b'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_3_5"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:16_2_3_5"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:16_2_3_5"/><b>then </b>
(  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c5">b'</font>,<font color="Maroon" title="c7">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="c5">b'</font>,<font color="Maroon" title="c7">c'</font>,<font color="Maroon" title="c5">b'</font> )
 
<b>by </b><i><a class="txt" href="#E8:16_2_3"><i><font color="Green" title="E25">A25</font></i></a>, <a class="ref" href="diraf.html#T36" title="DIRAF:th.36">DIRAF:36</a>, <a class="ref" href="diraf.html#T37" title="DIRAF:th.37">DIRAF:37</a></i>;<br/>
<a NAME="E3:16_2_3_5"/><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="c4">a'</font>,<font color="Maroon" title="c5">b'</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">a'</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c5">b'</font>,<font color="Maroon" title="c3">b</font> )
 
<b>by </b><i><a class="txt" href="#E2:16"><i><font color="Green" title="E9">A2</font></i></a>, <a class="txt" href="#E8:16"><i><font color="Green" title="E15">A8</font></i></a>, <a class="txt" href="#E1:16_2"><i><font color="Green" title="E20">A16</font></i></a>, <a class="ref" href="diraf.html#T27" title="DIRAF:th.27">DIRAF:27</a>, <a class="ref" href="diraf.html#T38" title="DIRAF:th.38">DIRAF:38</a></i>;<br/>
<a NAME="E4:16_2_3_5"/><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="c4">a'</font>,<font color="Maroon" title="c3">b</font>
 
<b>by </b><i><a class="txt" href="#E6:16"><i><font color="Green" title="E13">A6</font></i></a>, <a class="ref" href="diraf.html#T39" title="DIRAF:th.39">DIRAF:39</a></i>;<br/>
<b>hence </b><a NAME="E5:16_2_3_5"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:16"><i><font color="Green" title="E8">A1</font></i></a>, <a class="ref" href="diraf.html#T36" title="DIRAF:th.36">DIRAF:36</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E18:16_2_3"/>
not  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c7">c'</font>,<font color="Maroon" title="c5">b'</font>,<font color="Maroon" title="c8">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_3_6"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:16_2_3_6"/>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c7">c'</font>,<font color="Maroon" title="c5">b'</font>,<font color="Maroon" title="c8">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:16_2_3_6"/><b>then </b>
(  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c5">b'</font>,<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c7">c'</font> &amp;  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c5">b'</font>,<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c5">b'</font> &amp;  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c5">b'</font>,<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c3">b</font> )
 
<b>by </b><i><a class="txt" href="#E8:16_2_3"><i><font color="Green" title="E25">A25</font></i></a>, <a class="ref" href="diraf.html#T36" title="DIRAF:th.36">DIRAF:36</a>, <a class="ref" href="diraf.html#T37" title="DIRAF:th.37">DIRAF:37</a></i>;<br/>
<a NAME="E3:16_2_3_6"/><b>then </b>
(  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c7">c'</font>,<font color="Maroon" title="c5">b'</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">c'</font>,<font color="Maroon" title="c5">b'</font>,<font color="Maroon" title="c5">b'</font> )
 
<b>by </b><i><a class="txt" href="#E17:16_2_3"><i><font color="Green" title="E32">A32</font></i></a>, <a class="ref" href="diraf.html#T38" title="DIRAF:th.38">DIRAF:38</a></i>;<br/>
<b>hence </b><a NAME="E4:16_2_3_6"/>
contradiction
 <b>by </b><i><a class="txt" href="#E9:16"><i><font color="Green" title="E16">A9</font></i></a>, <a class="txt" href="#E1:16_2"><i><font color="Green" title="E20">A16</font></i></a>, <a class="txt" href="#E2:16_2"><i><font color="Green" title="E21">A17</font></i></a>, <a class="ref" href="diraf.html#T38" title="DIRAF:th.38">DIRAF:38</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E19:16_2_3"/><b>then </b>
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c7">c'</font>,<font color="Maroon" title="c5">b'</font>,<font color="Maroon" title="c4">a'</font>
 <b>by </b><i><a class="txt" href="#E9:16"><i><font color="Green" title="E16">A9</font></i></a>, <a class="txt" href="#E15:16_2_3"><i><font color="Green" title="E30">A30</font></i></a>, <a class="txt" href="#E16:16_2_3"><i><font color="Green" title="E31">A31</font></i></a>, <a class="ref" href="pasch.html#T15" target="_self" title="PASCH:th.15">Th15</a></i>;<br/><b>hence </b><a NAME="E20:16_2_3"/>
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c4">a'</font>,<font color="Maroon" title="c5">b'</font>,<font color="Maroon" title="c7">c'</font>
 <b>by </b><i><a class="ref" href="diraf.html#T13" title="DIRAF:th.13">DIRAF:13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>hence </b><a NAME="E5:16_2"/>
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c4">a'</font>,<font color="Maroon" title="c5">b'</font>,<font color="Maroon" title="c7">c'</font>
 <b>by </b><i><a class="txt" href="#E3:16_2"><i><font color="Green" title="E22">A19</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E14:16"/>
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c4">a'</font>,<font color="Maroon" title="c5">b'</font>,<font color="Maroon" title="c7">c'</font>
 <b>by </b><i><a class="txt" href="#E12:16"><i><font color="Green" title="E19">A12</font></i></a>, <a class="ref" href="diraf.html#T14" title="DIRAF:th.14">DIRAF:14</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
