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<div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">b</font>, <font color="Olive" title="b4">p9</font>, <font color="Olive" title="b5">a9</font>, <font color="Olive" title="b6">b9</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">OAS</font>  st  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font> &amp; <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b4">p9</font>,<font color="Olive" title="b5">a9</font> &amp; not  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">a</font>,<font color="Olive" title="b4">p9</font> &amp;  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Olive" title="b4">p9</font>,<font color="Olive" title="b5">a9</font>,<font color="Olive" title="b6">b9</font> &amp; <font color="Olive" title="b1">p</font>,<font color="Olive" title="b4">p9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b2">a</font>,<font color="Olive" title="b5">a9</font> &amp; <font color="Olive" title="b1">p</font>,<font color="Olive" title="b4">p9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b3">b</font>,<font color="Olive" title="b6">b9</font> holds <br/> <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Olive" title="b4">p9</font>,<font color="Olive" title="b5">a9</font>,<font color="Olive" title="b6">b9</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font>, <font color="Maroon" title="c5">p9</font>, <font color="Maroon" title="c6">a9</font>, <font color="Maroon" title="c7">b9</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font> &amp; <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">p9</font>,<font color="Maroon" title="c6">a9</font> &amp; not  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">p9</font> &amp;  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c5">p9</font>,<font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c7">b9</font> &amp; <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c5">p9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c6">a9</font> &amp; <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c5">p9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c7">b9</font> implies  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c5">p9</font>,<font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c7">b9</font> )</span><br/>













<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:45"/><span class="lab"><font color="Green" title="E19">A1</font></span>: 
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>
 <span class="kw">and </span><br/><a NAME="E2:45"/><span class="lab"><font color="Green" title="E20">A2</font></span>: 
<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">p9</font>,<font color="Maroon" title="c6">a9</font>
 <span class="kw">and </span><br/><a NAME="E3:45"/><span class="lab"><font color="Green" title="E21">A3</font></span>: 
not  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c5">p9</font>
 <span class="kw">and </span><br/><a NAME="E4:45"/><span class="lab"><font color="Green" title="E22">A4</font></span>: 
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c5">p9</font>,<font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c7">b9</font>
 <span class="kw">and </span><br/><a NAME="E5:45"/><span class="lab"><font color="Green" title="E23">A5</font></span>: 
<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c5">p9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c6">a9</font>
 <span class="kw">and </span><br/><a NAME="E6:45"/><span class="lab"><font color="Green" title="E24">A6</font></span>: 
<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c5">p9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c7">b9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c5">p9</font>,<font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c7">b9</font></span><br/>

<a NAME="E7:45"/><span class="lab"><font color="Green" title="E25">A7</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:45"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="ref" href="diraf.html#T37" title="DIRAF:th.37">DIRAF:37</a></span>;<br/>
<a NAME="E8:45"/><span class="lab"><font color="Green" title="E26">A8</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">p9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:45"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="ref" href="diraf.html#T37" title="DIRAF:th.37">DIRAF:37</a></span>;<br/>
<a NAME="E9:45"/><span class="lab"><font color="Green" title="E27">A9</font></span>: 
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="ref" href="diraf.html#T34" title="DIRAF:th.34">DIRAF:34</a></span>;<br/>
<a NAME="E10:45"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A10</font></span>: 
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#T36" title="DIRAF:th.36">DIRAF:36</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1"><span class="kw">now </span></a><div class="add"><a NAME="E1:45_1"/><span class="lab"><font color="Green" title="E29">A11</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="txt" href="#E7:45"><span class="lab"><font color="Green" title="E25">A7</font></span></a>, <a class="ref" href="diraf.html#T12" title="DIRAF:th.12">DIRAF:12</a></span>;<br/><a NAME="E2:45_1"/><span class="lab"><font color="Green" title="E30">A12</font></span>: 
<font color="Maroon" title="c5">p9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">b9</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:45_1_1"/><span class="lab"><font color="Green" title="E31">A13</font></span>: 
<font color="Maroon" title="c5">p9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:45_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">b9</font>,<font color="Maroon" title="c2">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">b9</font>,<font color="Maroon" title="c4">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:45"><span class="lab"><font color="Green" title="E24">A6</font></span></a>, <a class="ref" href="diraf.html#T5" title="DIRAF:th.5">DIRAF:5</a></span>;<br/>
<a NAME="E3:45_1_1"/><span class="kw">then </span>
(  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c7">b9</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">b</font> or  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c7">b9</font>,<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c2">p</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#T11" title="DIRAF:th.11">DIRAF:11</a></span>;<br/>
<a NAME="E4:45_1_1"/><span class="kw">then </span>
(  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c7">b9</font>,<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">b</font> or  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c7">b9</font>,<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c2">p</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#T34" title="DIRAF:th.34">DIRAF:34</a></span>;<br/>
<a NAME="E5:45_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A14</font></span>: 
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c7">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#T36" title="DIRAF:th.36">DIRAF:36</a></span>;<br/>
<a NAME="E6:45_1_1"/>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c2">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#T37" title="DIRAF:th.37">DIRAF:37</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:45_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:45"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="txt" href="#E10:45"><span class="lab"><font color="Green" title="E28">A10</font></span></a>, <a class="txt" href="#E1:45_1"><span class="lab"><font color="Green" title="E29">A11</font></span></a>, <a class="txt" href="#E1:45_1_1"><span class="lab"><font color="Green" title="E31">A13</font></span></a>, <a class="txt" href="#E5:45_1_1"><span class="lab"><font color="Green" title="E32">A14</font></span></a>, <a class="ref" href="diraf.html#T38" title="DIRAF:th.38">DIRAF:38</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E3:45_1"/><span class="lab"><font color="Green" title="E31">A15</font></span>: 
not  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c6">a9</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:45_1_2"/><span class="lab"><font color="Green" title="E32">A16</font></span>: 
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c6">a9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:45_1_2"/>
<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">a</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c5">p9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:45"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="ref" href="diraf.html#D4" title="DIRAF:def.4">DIRAF:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:45_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:45"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="txt" href="#E7:45"><span class="lab"><font color="Green" title="E25">A7</font></span></a>, <a class="txt" href="#E1:45_1_2"><span class="lab"><font color="Green" title="E32">A16</font></span></a>, <a class="ref" href="diraf.html#T39" title="DIRAF:th.39">DIRAF:39</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E4:45_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A17</font></span>: 
<font color="Maroon" title="c3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">a9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#T37" title="DIRAF:th.37">DIRAF:37</a></span>;<br/><div><span class="lab"><font color="Green" title="E33">A18</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_3"><span class="kw">now </span></a><div class="add"><a NAME="E1:45_1_3"/>
<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c6">a9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c5">p9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:45"><span class="lab"><font color="Green" title="E23">A5</font></span></a>, <a class="ref" href="diraf.html#T5" title="DIRAF:th.5">DIRAF:5</a></span>;<br/><a NAME="E2:45_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A19</font></span>: 
<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c6">a9</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c5">p9</font>,<font color="Maroon" title="c2">p</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#D4" title="DIRAF:def.4">DIRAF:def 4</a></span>;<br/><span class="kw">assume </span><a NAME="E3:45_1_3"/>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c5">p9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E4:45_1_3"/><span class="kw">then </span>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c2">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:45_1"><span class="lab"><font color="Green" title="E32">A17</font></span></a>, <a class="txt" href="#E2:45_1_3"><span class="lab"><font color="Green" title="E34">A19</font></span></a>, <a class="ref" href="diraf.html#T39" title="DIRAF:th.39">DIRAF:39</a></span>;<br/><span class="kw">hence </span><a NAME="E5:45_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:45_1"><span class="lab"><font color="Green" title="E31">A15</font></span></a>, <a class="ref" href="diraf.html#T36" title="DIRAF:th.36">DIRAF:36</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E6:45_1"/><span class="lab"><font color="Green" title="E34">A20</font></span>: 
<font color="Maroon" title="c5">p9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">a9</font>
 <span class="kw">and </span><br/><a NAME="E7:45_1"/>
<font color="Maroon" title="c6">a9</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">b9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c5">p9</font>,<font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c7">b9</font></span><br/><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E9:45_1"/><span class="lab"><font color="Green" title="E35">A21</font></span>: 
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c7">b9</font>
 <span class="kw">and </span><br/><a NAME="E10:45_1"/><span class="lab"><font color="Green" title="E36">A22</font></span>: 
<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c7">b9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c8">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="ref" href="pasch.html#T26" target="_self" title="PASCH:th.26">Th26</a></span>;<br/><a NAME="E11:45_1"/>
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c7">b9</font>,<font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c2">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:45_1"><span class="lab"><font color="Green" title="E35">A21</font></span></a>, <a class="ref" href="diraf.html#T13" title="DIRAF:th.13">DIRAF:13</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">OAS</font><span class="kw"> such that </span><br/><a NAME="E13:45_1"/><span class="lab"><font color="Green" title="E37">A23</font></span>: 
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c7">b9</font>,<font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c5">p9</font>
 <span class="kw">and </span><br/><a NAME="E14:45_1"/><span class="lab"><font color="Green" title="E38">A24</font></span>: 
<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c5">p9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pasch.html#T26" target="_self" title="PASCH:th.26">Th26</a></span>;<br/><a NAME="E15:45_1"/>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c7">b9</font>,<font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c5">p9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:45_1"><span class="lab"><font color="Green" title="E37">A23</font></span></a>, <a class="ref" href="diraf.html#T34" title="DIRAF:th.34">DIRAF:34</a></span>;<br/><a NAME="E16:45_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A25</font></span>: 
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c5">p9</font>,<font color="Maroon" title="c7">b9</font>,<font color="Maroon" title="c9">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#T36" title="DIRAF:th.36">DIRAF:36</a></span>;<br/><a NAME="E17:45_1"/><span class="lab"><font color="Green" title="E40">A26</font></span>: 
<font color="Maroon" title="c4">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">b9</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:45_1_4"/>
<font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:45_1_4"/><span class="kw">then </span>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c5">p9</font>,<font color="Maroon" title="c4">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:45"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="ref" href="diraf.html#T36" title="DIRAF:th.36">DIRAF:36</a></span>;<br/>
<a NAME="E3:45_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c5">p9</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c4">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#D5" title="DIRAF:def.5">DIRAF:def 5</a></span>;<br/>
<a NAME="E4:45_1_4"/><span class="kw">then </span>
( <font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c5">p9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c4">b</font> or <font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c5">p9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c6">a9</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#D4" title="DIRAF:def.4">DIRAF:def 4</a></span>;<br/>
<a NAME="E5:45_1_4"/><span class="kw">then </span>
( <font color="Maroon" title="c5">p9</font>,<font color="Maroon" title="c6">a9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c4">b</font> or <font color="Maroon" title="c5">p9</font>,<font color="Maroon" title="c6">a9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c6">a9</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#T5" title="DIRAF:th.5">DIRAF:5</a></span>;<br/>
<a NAME="E6:45_1_4"/><span class="kw">then </span>
( <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c4">b</font> or <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c6">a9</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:45"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="txt" href="#E6:45_1"><span class="lab"><font color="Green" title="E34">A20</font></span></a>, <a class="ref" href="diraf.html#T6" title="DIRAF:th.6">DIRAF:6</a></span>;<br/>
<a NAME="E7:45_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">a</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c6">a9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#D4" title="DIRAF:def.4">DIRAF:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:45_1_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:45"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="txt" href="#E7:45"><span class="lab"><font color="Green" title="E25">A7</font></span></a>, <a class="txt" href="#E3:45_1"><span class="lab"><font color="Green" title="E31">A15</font></span></a>, <a class="ref" href="diraf.html#T34" title="DIRAF:th.34">DIRAF:34</a>, <a class="ref" href="diraf.html#T39" title="DIRAF:th.39">DIRAF:39</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E18:45_1"/><span class="lab"><font color="Green" title="E41">A27</font></span>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">a</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:45_1_5"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:45_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A28</font></span>: 
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c7">b9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:45_1"><span class="lab"><font color="Green" title="E35">A21</font></span></a>, <a class="ref" href="diraf.html#T34" title="DIRAF:th.34">DIRAF:34</a></span>;<br/>
<a NAME="E3:45_1_5"/>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c3">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#T37" title="DIRAF:th.37">DIRAF:37</a></span>;<br/>
<a NAME="E4:45_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A29</font></span>: 
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c7">b9</font>,<font color="Maroon" title="c3">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:45"><span class="lab"><font color="Green" title="E25">A7</font></span></a>, <a class="txt" href="#E9:45"><span class="lab"><font color="Green" title="E27">A9</font></span></a>, <a class="txt" href="#E2:45_1_5"><span class="lab"><font color="Green" title="E42">A28</font></span></a>, <a class="ref" href="diraf.html#T38" title="DIRAF:th.38">DIRAF:38</a></span>;<br/>
<a NAME="E5:45_1_5"/>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c2">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#T37" title="DIRAF:th.37">DIRAF:37</a></span>;<br/>
<a NAME="E6:45_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A30</font></span>: 
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c7">b9</font>,<font color="Maroon" title="c2">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:45"><span class="lab"><font color="Green" title="E25">A7</font></span></a>, <a class="txt" href="#E9:45"><span class="lab"><font color="Green" title="E27">A9</font></span></a>, <a class="txt" href="#E2:45_1_5"><span class="lab"><font color="Green" title="E42">A28</font></span></a>, <a class="ref" href="diraf.html#T38" title="DIRAF:th.38">DIRAF:38</a></span>;<br/>
<a NAME="E7:45_1_5"/>
<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c7">b9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c5">p9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:45"><span class="lab"><font color="Green" title="E24">A6</font></span></a>, <a class="ref" href="diraf.html#T5" title="DIRAF:th.5">DIRAF:5</a></span>;<br/>
<a NAME="E8:45_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c7">b9</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c5">p9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#D4" title="DIRAF:def.4">DIRAF:def 4</a></span>;<br/>
<a NAME="E9:45_1_5"/><span class="kw">then </span>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c7">b9</font>,<font color="Maroon" title="c5">p9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:45_1"><span class="lab"><font color="Green" title="E40">A26</font></span></a>, <a class="txt" href="#E6:45_1_5"><span class="lab"><font color="Green" title="E44">A30</font></span></a>, <a class="ref" href="diraf.html#T39" title="DIRAF:th.39">DIRAF:39</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:45_1_5"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:45"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="txt" href="#E17:45_1"><span class="lab"><font color="Green" title="E40">A26</font></span></a>, <a class="txt" href="#E4:45_1_5"><span class="lab"><font color="Green" title="E43">A29</font></span></a>, <a class="txt" href="#E6:45_1_5"><span class="lab"><font color="Green" title="E44">A30</font></span></a>, <a class="ref" href="diraf.html#T38" title="DIRAF:th.38">DIRAF:38</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E19:45_1"/><span class="lab"><font color="Green" title="E42">A31</font></span>: 
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c5">p9</font>,<font color="Maroon" title="c7">b9</font>,<font color="Maroon" title="c6">a9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:45"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="ref" href="diraf.html#T36" title="DIRAF:th.36">DIRAF:36</a></span>;<br/><a NAME="E20:45_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A32</font></span>: 
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c6">a9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:45_1"><span class="lab"><font color="Green" title="E30">A12</font></span></a>, <a class="txt" href="#E16:45_1"><span class="lab"><font color="Green" title="E39">A25</font></span></a>, <a class="ref" href="diraf.html#T38" title="DIRAF:th.38">DIRAF:38</a></span>;<br/><a NAME="E21:45_1"/><span class="lab"><font color="Green" title="E44">A33</font></span>: 
<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c5">p9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c8">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:45"><span class="lab"><font color="Green" title="E24">A6</font></span></a>, <a class="txt" href="#E10:45_1"><span class="lab"><font color="Green" title="E36">A22</font></span></a>, <a class="txt" href="#E17:45_1"><span class="lab"><font color="Green" title="E40">A26</font></span></a>, <a class="ref" href="diraf.html#T6" title="DIRAF:th.6">DIRAF:6</a></span>;<br/><a NAME="E22:45_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c8">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:45"><span class="lab"><font color="Green" title="E26">A8</font></span></a>, <a class="txt" href="#E14:45_1"><span class="lab"><font color="Green" title="E38">A24</font></span></a>, <a class="ref" href="analoaf.html#D5" title="ANALOAF:def.5">ANALOAF:def 5</a></span>;<br/><a NAME="E23:45_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c8">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c9">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="analoaf.html#D5" title="ANALOAF:def.5">ANALOAF:def 5</a></span>;<br/><a NAME="E24:45_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c5">p9</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c9">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:45_1"><span class="lab"><font color="Green" title="E41">A27</font></span></a>, <a class="txt" href="#E21:45_1"><span class="lab"><font color="Green" title="E44">A33</font></span></a>, <a class="ref" href="diraf.html#T6" title="DIRAF:th.6">DIRAF:6</a></span>;<br/><a NAME="E25:45_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c9">y</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c6">a9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:45"><span class="lab"><font color="Green" title="E23">A5</font></span></a>, <a class="txt" href="#E8:45"><span class="lab"><font color="Green" title="E26">A8</font></span></a>, <a class="ref" href="analoaf.html#D5" title="ANALOAF:def.5">ANALOAF:def 5</a></span>;<br/><a NAME="E26:45_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c9">y</font> <a href="diraf.html#R2" title="DIRAF:pred.2">'||'</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c6">a9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#D4" title="DIRAF:def.4">DIRAF:def 4</a></span>;<br/><a NAME="E27:45_1"/><span class="kw">then </span>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c6">a9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#D5" title="DIRAF:def.5">DIRAF:def 5</a></span>;<br/><a NAME="E28:45_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A34</font></span>: 
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c3">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#T36" title="DIRAF:th.36">DIRAF:36</a></span>;<br/><a NAME="E29:45_1"/>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c5">p9</font>,<font color="Maroon" title="c7">b9</font>,<font color="Maroon" title="c5">p9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#T37" title="DIRAF:th.37">DIRAF:37</a></span>;<br/><a NAME="E30:45_1"/><span class="kw">then </span>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c5">p9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:45_1"><span class="lab"><font color="Green" title="E30">A12</font></span></a>, <a class="txt" href="#E16:45_1"><span class="lab"><font color="Green" title="E39">A25</font></span></a>, <a class="txt" href="#E19:45_1"><span class="lab"><font color="Green" title="E42">A31</font></span></a>, <a class="ref" href="diraf.html#T38" title="DIRAF:th.38">DIRAF:38</a></span>;<br/><a NAME="E31:45_1"/><span class="kw">then </span>
( <font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a9</font> or  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c5">p9</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:45_1"><span class="lab"><font color="Green" title="E45">A34</font></span></a>, <a class="txt" href="#E20:45_1"><span class="lab"><font color="Green" title="E43">A32</font></span></a>, <a class="ref" href="diraf.html#T38" title="DIRAF:th.38">DIRAF:38</a></span>;<br/><span class="kw">hence </span><a NAME="E32:45_1"/>
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c5">p9</font>,<font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c7">b9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:45_1"><span class="lab"><font color="Green" title="E37">A23</font></span></a>, <a class="txt" href="#E5:45_1"><span class="lab"><font color="Green" title="E33">A18</font></span></a>, <a class="ref" href="diraf.html#T13" title="DIRAF:th.13">DIRAF:13</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E12:45"/>
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c5">p9</font>,<font color="Maroon" title="c6">a9</font>,<font color="Maroon" title="c7">b9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#T14" title="DIRAF:th.14">DIRAF:14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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