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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">c</font>, <font color="Olive" title="b4">d1</font>, <font color="Olive" title="b5">d2</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">c</font> &amp; <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Olive" title="b3">c</font>,<font color="Olive" title="b4">d1</font> &amp; <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Olive" title="b3">c</font>,<font color="Olive" title="b5">d2</font> &amp; <font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">c</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Olive" title="b2">b</font>,<font color="Olive" title="b4">d1</font> &amp; <font color="Olive" title="b1">a</font>,<font color="Olive" title="b3">c</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Olive" title="b2">b</font>,<font color="Olive" title="b5">d2</font> holds <br/><font color="Olive" title="b4">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">d2</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">c</font>, <font color="Maroon" title="c5">d1</font>, <font color="Maroon" title="c6">d2</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">c</font> &amp; <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="c4">c</font>,<font color="Maroon" title="c5">d1</font> &amp; <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="c4">c</font>,<font color="Maroon" title="c6">d2</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">d1</font> &amp; <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c6">d2</font> implies <font color="Maroon" title="c5">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">d2</font> )</span><br/>











<b>assume </b><b>that </b><br/><a NAME="E1:17"/><i><font color="Green" title="E5">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">c</font>
 <b>and </b><br/><a NAME="E2:17"/><i><font color="Green" title="E6">A2</font></i>: 
<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="c4">c</font>,<font color="Maroon" title="c5">d1</font>
 <b>and </b><br/><a NAME="E3:17"/><i><font color="Green" title="E7">A3</font></i>: 
<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="c4">c</font>,<font color="Maroon" title="c6">d2</font>
 <b>and </b><br/><a NAME="E4:17"/><i><font color="Green" title="E8">A4</font></i>: 
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">d1</font>
 <b>and </b><br/><a NAME="E5:17"/><i><font color="Green" title="E9">A5</font></i>: 
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">c</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c6">d2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">d2</font></span><br/>

<b>assume </b><a NAME="E6:17"/><i><font color="Green" title="E10">A6</font></i>: 
<font color="Maroon" title="c5">d1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">d2</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="E7:17"/><i><font color="Green" title="E11">A7</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> )
 
<b>by </b><i><a class="txt" href="pasch.html#E1:17"><i><font color="Green" title="E5">A1</font></i></a>, <a class="ref" href="diraf.html#T37" title="DIRAF:th.37">DIRAF:37</a></i>;<br/>
<a NAME="E8:17"/><b>then </b><i><font color="Green" title="E12">A8</font></i>: 
<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d1</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c6">d2</font>
 
<b>by </b><i><a class="txt" href="pasch.html#E2:17"><i><font color="Green" title="E6">A2</font></i></a>, <a class="txt" href="pasch.html#E3:17"><i><font color="Green" title="E7">A3</font></i></a>, <a class="ref" href="diraf.html#T28" title="DIRAF:th.28">DIRAF:28</a></i>;<br/>
<a NAME="E9:17"/>
<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">d1</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c6">d2</font>
 
<b>by </b><i><a class="txt" href="pasch.html#E4:17"><i><font color="Green" title="E8">A4</font></i></a>, <a class="txt" href="pasch.html#E5:17"><i><font color="Green" title="E9">A5</font></i></a>, <a class="txt" href="pasch.html#E7:17"><i><font color="Green" title="E11">A7</font></i></a>, <a class="ref" href="diraf.html#T28" title="DIRAF:th.28">DIRAF:28</a></i>;<br/>
<a NAME="E10:17"/><b>then </b><i><font color="Green" title="E13">A9</font></i>: 
(  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c6">d2</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">d1</font>,<font color="Maroon" title="c6">d2</font> )
 
<b>by </b><i><a class="txt" href="pasch.html#E8:17"><i><font color="Green" title="E12">A8</font></i></a>, <a class="ref" href="diraf.html#D5" title="DIRAF:def.5">DIRAF:def 5</a></i>;<br/>
<a NAME="E11:17"/><b>then </b><i><font color="Green" title="E14">A10</font></i>: 
(  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c6">d2</font>,<font color="Maroon" title="c4">c</font> &amp;  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c6">d2</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></i>;<br/>
<a NAME="E12:17"/>
(  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c6">d2</font>,<font color="Maroon" title="c4">c</font> &amp;  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c6">d2</font>,<font color="Maroon" title="c5">d1</font> )
 
<b>by </b><i><a class="txt" href="pasch.html#E10:17"><i><font color="Green" title="E13">A9</font></i></a>, <a class="ref" href="diraf.html#T36" title="DIRAF:th.36">DIRAF:36</a>, <a class="ref" href="diraf.html#T37" title="DIRAF:th.37">DIRAF:37</a></i>;<br/>
<a NAME="E13:17"/><b>then </b>
<font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c6">d2</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c5">d1</font>
 
<b>by </b><i><a class="ref" href="diraf.html#T40" title="DIRAF:th.40">DIRAF:40</a></i>;<br/>
<a NAME="E14:17"/><b>then </b><i><font color="Green" title="E15">A11</font></i>: 
( <font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c6">d2</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> or <font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">d1</font> )
 
<b>by </b><i><a class="txt" href="pasch.html#E2:17"><i><font color="Green" title="E6">A2</font></i></a>, <a class="ref" href="diraf.html#T28" title="DIRAF:th.28">DIRAF:28</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:17_1"/>
<font color="Maroon" title="c4">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">d1</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="E2:17_1"/><b>then </b>
<font color="Maroon" title="c4">c</font>,<font color="Maroon" title="c2">a</font> <a href="diraf.html#R2" title="DIRAF: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#E4:17"><i><font color="Green" title="E8">A4</font></i></a>, <a class="ref" href="diraf.html#T27" title="DIRAF:th.27">DIRAF:27</a></i>;<br/><a NAME="E3:17_1"/><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>
 <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="E4:17_1"/>
contradiction
 <b>by </b><i><a class="txt" href="pasch.html#E1:17"><i><font color="Green" title="E5">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="E16:17"/><b>then </b>
<font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c6">d2</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font>
 
<b>by </b><i><a class="txt" href="pasch.html#E14:17"><i><font color="Green" title="E15">A11</font></i></a>, <a class="ref" href="diraf.html#T27" title="DIRAF:th.27">DIRAF:27</a></i>;<br/>
<a NAME="E17:17"/><b>then </b>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c5">d1</font>,<font color="Maroon" title="c6">d2</font>,<font color="Maroon" title="c2">a</font>
 
<b>by </b><i><a class="txt" href="pasch.html#E6:17"><i><font color="Green" title="E10">A6</font></i></a>, <a class="txt" href="pasch.html#E11:17"><i><font color="Green" title="E14">A10</font></i></a>, <a class="ref" href="diraf.html#T39" title="DIRAF:th.39">DIRAF:39</a></i>;<br/>
<b>hence </b><a NAME="E18:17"/>
contradiction
 <b>by </b><i><a class="txt" href="pasch.html#E1:17"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="pasch.html#E6:17"><i><font color="Green" title="E10">A6</font></i></a>, <a class="txt" href="pasch.html#E11:17"><i><font color="Green" title="E14">A10</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/>


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