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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">c</font>, <font color="Olive" title="b3">b</font>, <font color="Olive" title="b4">d</font>, <font color="Olive" title="b5">a</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">c</font>,<font color="Olive" title="b3">b</font> &amp; <font color="Olive" title="b2">c</font>,<font color="Olive" title="b4">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b3">b</font>,<font color="Olive" title="b5">a</font> &amp; <font color="Olive" title="b1">p</font>,<font color="Olive" title="b4">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b5">a</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="b5">a</font>,<font color="Olive" title="b3">b</font> &amp; <font color="Olive" title="b1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">c</font> holds <br/> <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Olive" title="b1">p</font>,<font color="Olive" title="b4">d</font>,<font color="Olive" title="b5">a</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c3">c</font>, <font color="Maroon" title="c4">b</font>, <font color="Maroon" title="c5">d</font>, <font color="Maroon" title="c6">a</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">c</font>,<font color="Maroon" title="c4">b</font> &amp; <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c6">a</font> &amp; <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c5">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">a</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="c6">a</font>,<font color="Maroon" title="c4">b</font> &amp; <font color="Maroon" title="c2">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">c</font> implies  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c6">a</font> )</span><br/>











<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:28"/><span class="lab"><font color="Green" title="E10">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">c</font>,<font color="Maroon" title="c4">b</font>
 <span class="kw">and </span><br/><a NAME="E2:28"/><span class="lab"><font color="Green" title="E11">A2</font></span>: 
<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c6">a</font>
 <span class="kw">and </span><br/><a NAME="E3:28"/><span class="lab"><font color="Green" title="E12">A3</font></span>: 
<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c5">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">a</font>
 <span class="kw">and </span><br/><a NAME="E4:28"/><span class="lab"><font color="Green" title="E13">A4</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="c6">a</font>,<font color="Maroon" title="c4">b</font>
 <span class="kw">and </span><br/><a NAME="E5:28"/><span class="lab"><font color="Green" title="E14">A5</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">c</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="c5">d</font>,<font color="Maroon" title="c6">a</font></span><br/>

<a NAME="E6:28"/><span class="lab"><font color="Green" title="E15">A6</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">c</font>,<font color="Maroon" title="c4">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="ref" href="diraf.html#T34" title="DIRAF:th.34">DIRAF:34</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:28_1"/><span class="lab"><font color="Green" title="E16">A7</font></span>: 
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c5">d</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="c5">d</font>,<font color="Maroon" title="c6">a</font></span><br/><div><span class="lab"><font color="Green" title="E17">A8</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1"><span class="kw">now </span></a><div class="add"><a NAME="E1:28_1_1"/><span class="lab"><font color="Green" title="E18">A9</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="c6">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:28"><span class="lab"><font color="Green" title="E13">A4</font></span></a>, <a class="ref" href="diraf.html#T37" title="DIRAF:th.37">DIRAF:37</a></span>;<br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:28_1_1"/><span class="lab"><font color="Green" title="E19">A10</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="c3">c</font>
 <span class="kw">and </span><br/><a NAME="E3:28_1_1"/><span class="lab"><font color="Green" title="E20">A11</font></span>: 
<font color="Maroon" title="c5">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">a</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="c5">d</font>,<font color="Maroon" title="c6">a</font></span><br/><span class="kw">assume </span><a NAME="E4:28_1_1"/>
not  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c6">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="E5:28_1_1"/><span class="kw">then </span>
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c5">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:28"><span class="lab"><font color="Green" title="E12">A3</font></span></a>, <a class="ref" href="diraf.html#T11" title="DIRAF:th.11">DIRAF:11</a></span>;<br/><a NAME="E6:28_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c5">d</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#D3" title="DIRAF:def.3">DIRAF:def 3</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">e1</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="E8:28_1_1"/><span class="lab"><font color="Green" title="E21">A12</font></span>: 
<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c6">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">e1</font>
 <span class="kw">and </span><br/><a NAME="E9:28_1_1"/><span class="lab"><font color="Green" title="E22">A13</font></span>: 
<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c2">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c7">e1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28_1_1"><span class="lab"><font color="Green" title="E18">A9</font></span></a>, <a class="ref" href="analoaf.html#D5" title="ANALOAF:def.5">ANALOAF:def 5</a></span>;<br/><a NAME="E10:28_1_1"/><span class="lab"><font color="Green" title="E23">A14</font></span>: 
<font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c7">e1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c2">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:28_1_1"><span class="lab"><font color="Green" title="E22">A13</font></span></a>, <a class="ref" href="diraf.html#T5" title="DIRAF:th.5">DIRAF:5</a></span>;<br/><a NAME="E11:28_1_1"/><span class="lab"><font color="Green" title="E24">A15</font></span>: 
<font color="Maroon" title="c3">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">e1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:28_1_1_1"/>
<font color="Maroon" title="c3">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">e1</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="E2:28_1_1_1"/><span class="kw">then </span>
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c3">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:28_1_1"><span class="lab"><font color="Green" title="E21">A12</font></span></a>, <a class="ref" href="diraf.html#D3" title="DIRAF:def.3">DIRAF:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:28_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:28"><span class="lab"><font color="Green" title="E13">A4</font></span></a>, <a class="txt" href="#E6:28"><span class="lab"><font color="Green" title="E15">A6</font></span></a>, <a class="ref" href="diraf.html#T12" title="DIRAF:th.12">DIRAF:12</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="E12:28_1_1"/>
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c2">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="ref" href="diraf.html#T13" title="DIRAF:th.13">DIRAF:13</a></span>;<br/><a NAME="E13:28_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A16</font></span>: 
<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c2">p</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#D3" title="DIRAF:def.3">DIRAF:def 3</a></span>;<br/><a NAME="E14:28_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A17</font></span>: 
<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c2">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">c</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/><span class="kw">consider </span><font color="Maroon" title="c8">e2</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="E16:28_1_1"/><span class="lab"><font color="Green" title="E27">A18</font></span>: 
<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c6">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">e2</font>
 <span class="kw">and </span><br/><a NAME="E17:28_1_1"/><span class="lab"><font color="Green" title="E28">A19</font></span>: 
<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">e1</font>,<font color="Maroon" title="c8">e2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:28"><span class="lab"><font color="Green" title="E13">A4</font></span></a>, <a class="txt" href="#E6:28"><span class="lab"><font color="Green" title="E15">A6</font></span></a>, <a class="txt" href="#E8:28_1_1"><span class="lab"><font color="Green" title="E21">A12</font></span></a>, <a class="ref" href="analoaf.html#D5" title="ANALOAF:def.5">ANALOAF:def 5</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c9">e3</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="E19:28_1_1"/><span class="lab"><font color="Green" title="E29">A20</font></span>: 
<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">e2</font>,<font color="Maroon" title="c9">e3</font>
 <span class="kw">and </span><br/><a NAME="E20:28_1_1"/><span class="lab"><font color="Green" title="E30">A21</font></span>: 
<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c8">e2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c9">e3</font>
 <span class="kw">and </span><br/><a NAME="E21:28_1_1"/>
<font color="Maroon" title="c3">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">e3</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="E22:28_1_1"/><span class="lab"><font color="Green" title="E31">A22</font></span>: 
<font color="Maroon" title="c6">a</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="#E4:28"><span class="lab"><font color="Green" title="E13">A4</font></span></a>, <a class="ref" href="diraf.html#T37" title="DIRAF:th.37">DIRAF:37</a></span>;<br/><a NAME="E23:28_1_1"/><span class="lab"><font color="Green" title="E32">A23</font></span>: 
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">e1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:28_1_1"><span class="lab"><font color="Green" title="E21">A12</font></span></a>, <a class="ref" href="diraf.html#D3" title="DIRAF:def.3">DIRAF:def 3</a></span>;<br/><a NAME="E24:28_1_1"/><span class="lab"><font color="Green" title="E33">A24</font></span>: 
<font color="Maroon" title="c5">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">e1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_2"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:28_1_1_2"/>
(  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c6">a</font> or  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c5">d</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:28"><span class="lab"><font color="Green" title="E12">A3</font></span></a>, <a class="ref" href="diraf.html#T11" title="DIRAF:th.11">DIRAF:11</a></span>;<br/>
<a NAME="E2:28_1_1_2"/><span class="kw">then </span>
(  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c6">a</font> or  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c5">d</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="E3:28_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A25</font></span>: 
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c2">p</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="E4:28_1_1_2"/><span class="lab"><font color="Green" title="E35">A26</font></span>: 
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c6">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/>
<span class="kw">assume </span><a NAME="E5:28_1_1_2"/>
<font color="Maroon" title="c5">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">e1</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="E6:28_1_1_2"/><span class="kw">then </span>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c5">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:28_1_1"><span class="lab"><font color="Green" title="E32">A23</font></span></a>, <a class="ref" href="diraf.html#T34" title="DIRAF:th.34">DIRAF:34</a></span>;<br/>
<a NAME="E7:28_1_1_2"/><span class="kw">then </span>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c3">c</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="E8:28_1_1_2"/><span class="kw">then </span>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c4">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:28"><span class="lab"><font color="Green" title="E14">A5</font></span></a>, <a class="txt" href="#E6:28"><span class="lab"><font color="Green" title="E15">A6</font></span></a>, <a class="txt" href="#E3:28_1_1_2"><span class="lab"><font color="Green" title="E34">A25</font></span></a>, <a class="ref" href="diraf.html#T41" title="DIRAF:th.41">DIRAF:41</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:28_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:28"><span class="lab"><font color="Green" title="E13">A4</font></span></a>, <a class="txt" href="#E3:28_1_1"><span class="lab"><font color="Green" title="E20">A11</font></span></a>, <a class="txt" href="#E3:28_1_1_2"><span class="lab"><font color="Green" title="E34">A25</font></span></a>, <a class="txt" href="#E4:28_1_1_2"><span class="lab"><font color="Green" title="E35">A26</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="E25:28_1_1"/>
<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c6">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c8">e2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:28_1_1"><span class="lab"><font color="Green" title="E27">A18</font></span></a>, <a class="ref" href="analoaf.html#D5" title="ANALOAF:def.5">ANALOAF:def 5</a></span>;<br/><a NAME="E26:28_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A27</font></span>: 
<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c8">e2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:28"><span class="lab"><font color="Green" title="E11">A2</font></span></a>, <a class="txt" href="#E22:28_1_1"><span class="lab"><font color="Green" title="E31">A22</font></span></a>, <a class="ref" href="diraf.html#T6" title="DIRAF:th.6">DIRAF:6</a></span>;<br/><a NAME="E27:28_1_1"/>
<font color="Maroon" title="c4">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">e2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:28_1_1"><span class="lab"><font color="Green" title="E31">A22</font></span></a>, <a class="txt" href="#E16:28_1_1"><span class="lab"><font color="Green" title="E27">A18</font></span></a>, <a class="ref" href="analoaf.html#D5" title="ANALOAF:def.5">ANALOAF:def 5</a></span>;<br/><a NAME="E28:28_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c9">e3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:28_1_1"><span class="lab"><font color="Green" title="E30">A21</font></span></a>, <a class="txt" href="#E26:28_1_1"><span class="lab"><font color="Green" title="E34">A27</font></span></a>, <a class="ref" href="diraf.html#T6" title="DIRAF:th.6">DIRAF:6</a></span>;<br/><a NAME="E29:28_1_1"/><span class="kw">then </span>
(  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c9">e3</font> or  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c9">e3</font>,<font color="Maroon" title="c5">d</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="E30:28_1_1"/><span class="kw">then </span>
(  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c9">e3</font> or  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c9">e3</font>,<font color="Maroon" title="c5">d</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="E31:28_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A28</font></span>: 
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c9">e3</font>,<font color="Maroon" title="c3">c</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="E32:28_1_1"/>
<font color="Maroon" title="c7">e1</font>,<font color="Maroon" title="c8">e2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c2">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:28_1_1"><span class="lab"><font color="Green" title="E19">A10</font></span></a>, <a class="txt" href="#E17:28_1_1"><span class="lab"><font color="Green" title="E28">A19</font></span></a>, <a class="txt" href="#E13:28_1_1"><span class="lab"><font color="Green" title="E25">A16</font></span></a>, <a class="ref" href="analoaf.html#D5" title="ANALOAF:def.5">ANALOAF:def 5</a></span>;<br/><a NAME="E33:28_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A29</font></span>: 
<font color="Maroon" title="c7">e1</font>,<font color="Maroon" title="c8">e2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c7">e1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:28"><span class="lab"><font color="Green" title="E14">A5</font></span></a>, <a class="txt" href="#E9:28_1_1"><span class="lab"><font color="Green" title="E22">A13</font></span></a>, <a class="ref" href="diraf.html#T6" title="DIRAF:th.6">DIRAF:6</a></span>;<br/><a NAME="E34:28_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c7">e1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">e1</font>,<font color="Maroon" title="c8">e2</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="E35:28_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A30</font></span>: 
<font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c7">e1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c8">e2</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="E36:28_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c8">e2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c7">e1</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="E37:28_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c8">e2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c2">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:28_1_1"><span class="lab"><font color="Green" title="E33">A24</font></span></a>, <a class="txt" href="#E10:28_1_1"><span class="lab"><font color="Green" title="E23">A14</font></span></a>, <a class="ref" href="diraf.html#T6" title="DIRAF:th.6">DIRAF:6</a></span>;<br/><a NAME="E38:28_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c8">e2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:28"><span class="lab"><font color="Green" title="E14">A5</font></span></a>, <a class="txt" href="#E14:28_1_1"><span class="lab"><font color="Green" title="E26">A17</font></span></a>, <a class="ref" href="diraf.html#T6" title="DIRAF:th.6">DIRAF:6</a></span>;<br/><a NAME="E39:28_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A31</font></span>: 
<font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c8">e2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">e2</font>,<font color="Maroon" title="c9">e3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:28_1_1"><span class="lab"><font color="Green" title="E19">A10</font></span></a>, <a class="txt" href="#E19:28_1_1"><span class="lab"><font color="Green" title="E29">A20</font></span></a>, <a class="ref" href="diraf.html#T6" title="DIRAF:th.6">DIRAF:6</a></span>;<br/><a NAME="E40:28_1_1"/><span class="kw">then </span>
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c8">e2</font>,<font color="Maroon" title="c9">e3</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#D3" title="DIRAF:def.3">DIRAF:def 3</a></span>;<br/><a NAME="E41:28_1_1"/><span class="kw">then </span>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c8">e2</font>,<font color="Maroon" title="c9">e3</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="E42:28_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A32</font></span>: 
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c9">e3</font>,<font color="Maroon" title="c8">e2</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="E43:28_1_1"/><span class="lab"><font color="Green" title="E40">A33</font></span>: 
<font color="Maroon" title="c5">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">e2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:28_1_1"><span class="lab"><font color="Green" title="E33">A24</font></span></a>, <a class="txt" href="#E33:28_1_1"><span class="lab"><font color="Green" title="E36">A29</font></span></a>, <a class="ref" href="analoaf.html#D5" title="ANALOAF:def.5">ANALOAF:def 5</a></span>;<br/><a NAME="E44:28_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A34</font></span>: 
<font color="Maroon" title="c5">d</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">e3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:28_1_1"><span class="lab"><font color="Green" title="E38">A31</font></span></a>, <a class="ref" href="analoaf.html#D5" title="ANALOAF:def.5">ANALOAF:def 5</a></span>;<br/><a NAME="E45:28_1_1"/>
<font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c8">e2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c9">e3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:28_1_1"><span class="lab"><font color="Green" title="E38">A31</font></span></a>, <a class="ref" href="analoaf.html#D5" title="ANALOAF:def.5">ANALOAF:def 5</a></span>;<br/><a NAME="E46:28_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c7">e1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c9">e3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:28_1_1"><span class="lab"><font color="Green" title="E37">A30</font></span></a>, <a class="txt" href="#E43:28_1_1"><span class="lab"><font color="Green" title="E40">A33</font></span></a>, <a class="ref" href="diraf.html#T6" title="DIRAF:th.6">DIRAF:6</a></span>;<br/><a NAME="E47:28_1_1"/><span class="kw">then </span>
(  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c7">e1</font>,<font color="Maroon" title="c9">e3</font> or  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c9">e3</font>,<font color="Maroon" title="c7">e1</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="E48:28_1_1"/><span class="kw">then </span>
(  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c7">e1</font>,<font color="Maroon" title="c9">e3</font> or  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c9">e3</font>,<font color="Maroon" title="c7">e1</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="E49:28_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A35</font></span>: 
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c9">e3</font>,<font color="Maroon" title="c7">e1</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="E50:28_1_1"/>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">e1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:28_1_1"><span class="lab"><font color="Green" title="E32">A23</font></span></a>, <a class="ref" href="diraf.html#T34" title="DIRAF:th.34">DIRAF:34</a></span>;<br/><a NAME="E51:28_1_1"/><span class="kw">then </span>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c7">e1</font>,<font color="Maroon" title="c6">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="E52:28_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A36</font></span>: 
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c9">e3</font>,<font color="Maroon" title="c6">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:28_1_1"><span class="lab"><font color="Green" title="E24">A15</font></span></a>, <a class="txt" href="#E31:28_1_1"><span class="lab"><font color="Green" title="E35">A28</font></span></a>, <a class="txt" href="#E49:28_1_1"><span class="lab"><font color="Green" title="E42">A35</font></span></a>, <a class="ref" href="diraf.html#T41" title="DIRAF:th.41">DIRAF:41</a></span>;<br/><a NAME="E53:28_1_1"/><span class="lab"><font color="Green" title="E44">A37</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">e1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_3"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:28_1_1_3"/>
<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c5">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28_1"><span class="lab"><font color="Green" title="E16">A7</font></span></a>, <a class="ref" href="diraf.html#D3" title="DIRAF:def.3">DIRAF:def 3</a></span>;<br/>
<a NAME="E2:28_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A38</font></span>: 
<font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c6">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c2">p</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/>
<span class="kw">assume </span><a NAME="E3:28_1_1_3"/>
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">e1</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="E4:28_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c2">p</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c2">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:28_1_1"><span class="lab"><font color="Green" title="E20">A11</font></span></a>, <a class="txt" href="#E9:28_1_1"><span class="lab"><font color="Green" title="E22">A13</font></span></a>, <a class="txt" href="#E2:28_1_1_3"><span class="lab"><font color="Green" title="E45">A38</font></span></a>, <a class="ref" href="diraf.html#T6" title="DIRAF:th.6">DIRAF:6</a></span>;<br/>
<a NAME="E5:28_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">a</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="E6:28_1_1_3"/><span class="kw">then </span>
(  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c6">a</font> or  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c3">c</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="E7:28_1_1_3"/><span class="kw">then </span>
(  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c6">a</font> or  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c3">c</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="E8:28_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A39</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">c</font>,<font color="Maroon" title="c6">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="E9:28_1_1_3"/>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c3">c</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="E10:28_1_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:28"><span class="lab"><font color="Green" title="E13">A4</font></span></a>, <a class="txt" href="#E5:28"><span class="lab"><font color="Green" title="E14">A5</font></span></a>, <a class="txt" href="#E6:28"><span class="lab"><font color="Green" title="E15">A6</font></span></a>, <a class="txt" href="#E8:28_1_1_3"><span class="lab"><font color="Green" title="E46">A39</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="E54:28_1_1"/><span class="lab"><font color="Green" title="E45">A40</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">e2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:28_1_1_4"/><span class="lab"><font color="Green" title="E46">A41</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">e2</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="E2:28_1_1_4"/>
<font color="Maroon" title="c7">e1</font>,<font color="Maroon" title="c6">a</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c3">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:28_1_1"><span class="lab"><font color="Green" title="E21">A12</font></span></a>, <a class="ref" href="diraf.html#T5" title="DIRAF:th.5">DIRAF:5</a></span>;<br/>
<a NAME="E3:28_1_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c3">c</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c3">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:28_1_1"><span class="lab"><font color="Green" title="E28">A19</font></span></a>, <a class="txt" href="#E53:28_1_1"><span class="lab"><font color="Green" title="E44">A37</font></span></a>, <a class="txt" href="#E1:28_1_1_4"><span class="lab"><font color="Green" title="E46">A41</font></span></a>, <a class="ref" href="diraf.html#T6" title="DIRAF:th.6">DIRAF:6</a></span>;<br/>
<a NAME="E4:28_1_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c6">a</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="E5:28_1_1_4"/><span class="kw">then </span>
(  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c6">a</font> or  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c4">b</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="E6:28_1_1_4"/><span class="kw">then </span>
(  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c6">a</font> or  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c4">b</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="E7:28_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A42</font></span>: 
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c6">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="E8:28_1_1_4"/><span class="lab"><font color="Green" title="E48">A43</font></span>: 
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c3">c</font>,<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c4">b</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="E9:28_1_1_4"/>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c3">c</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="txt" href="#E6:28"><span class="lab"><font color="Green" title="E15">A6</font></span></a>, <a class="ref" href="diraf.html#T36" title="DIRAF:th.36">DIRAF:36</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:28_1_1_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:28"><span class="lab"><font color="Green" title="E13">A4</font></span></a>, <a class="txt" href="#E2:28_1_1"><span class="lab"><font color="Green" title="E19">A10</font></span></a>, <a class="txt" href="#E7:28_1_1_4"><span class="lab"><font color="Green" title="E47">A42</font></span></a>, <a class="txt" href="#E8:28_1_1_4"><span class="lab"><font color="Green" title="E48">A43</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="E55:28_1_1"/>
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">e2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:28_1_1"><span class="lab"><font color="Green" title="E27">A18</font></span></a>, <a class="ref" href="diraf.html#D3" title="DIRAF:def.3">DIRAF:def 3</a></span>;<br/><a NAME="E56:28_1_1"/><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="c6">a</font>,<font color="Maroon" title="c8">e2</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="E57:28_1_1"/><span class="kw">then </span>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">e2</font>,<font color="Maroon" title="c4">b</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="E58:28_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A44</font></span>: 
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c9">e3</font>,<font color="Maroon" title="c4">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E54:28_1_1"><span class="lab"><font color="Green" title="E45">A40</font></span></a>, <a class="txt" href="#E52:28_1_1"><span class="lab"><font color="Green" title="E43">A36</font></span></a>, <a class="txt" href="#E42:28_1_1"><span class="lab"><font color="Green" title="E39">A32</font></span></a>, <a class="ref" href="diraf.html#T41" title="DIRAF:th.41">DIRAF:41</a></span>;<br/><a NAME="E59:28_1_1"/>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c3">c</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="txt" href="#E6:28"><span class="lab"><font color="Green" title="E15">A6</font></span></a>, <a class="ref" href="diraf.html#T36" title="DIRAF:th.36">DIRAF:36</a></span>;<br/><a NAME="E60:28_1_1"/><span class="kw">then </span>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c9">e3</font>,<font color="Maroon" title="c2">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:28_1_1"><span class="lab"><font color="Green" title="E19">A10</font></span></a>, <a class="txt" href="#E31:28_1_1"><span class="lab"><font color="Green" title="E35">A28</font></span></a>, <a class="txt" href="#E58:28_1_1"><span class="lab"><font color="Green" title="E46">A44</font></span></a>, <a class="ref" href="diraf.html#T41" title="DIRAF:th.41">DIRAF:41</a></span>;<br/><span class="kw">hence </span><a NAME="E61:28_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:28"><span class="lab"><font color="Green" title="E13">A4</font></span></a>, <a class="txt" href="#E44:28_1_1"><span class="lab"><font color="Green" title="E41">A34</font></span></a>, <a class="txt" href="#E52:28_1_1"><span class="lab"><font color="Green" title="E43">A36</font></span></a>, <a class="txt" href="#E58:28_1_1"><span class="lab"><font color="Green" title="E46">A44</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="E3:28_1"/><span class="lab"><font color="Green" title="E18">A45</font></span>: 
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c5">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28_1"><span class="lab"><font color="Green" title="E16">A7</font></span></a>, <a class="ref" href="diraf.html#T34" title="DIRAF:th.34">DIRAF:34</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="28_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:28_1_2"/>
<font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">d</font></span><br/><a NAME="E2:28_1_2"/><span class="kw">then </span>
(  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c6">a</font> or  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c5">d</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:28"><span class="lab"><font color="Green" title="E11">A2</font></span></a>, <a class="ref" href="diraf.html#T11" title="DIRAF:th.11">DIRAF:11</a></span>;<br/><a NAME="E3:28_1_2"/><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="c5">d</font>,<font color="Maroon" title="c6">a</font> or  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c5">d</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="E4:28_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A46</font></span>: 
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c4">b</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="E5:28_1_2"/><span class="lab"><font color="Green" title="E20">A47</font></span>: 
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c6">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="E6:28_1_2"/>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c2">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:28_1"><span class="lab"><font color="Green" title="E18">A45</font></span></a>, <a class="ref" href="diraf.html#T36" title="DIRAF:th.36">DIRAF:36</a></span>;<br/><span class="kw">hence </span><a NAME="E7:28_1_2"/>
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:28"><span class="lab"><font color="Green" title="E13">A4</font></span></a>, <a class="txt" href="#E4:28_1_2"><span class="lab"><font color="Green" title="E19">A46</font></span></a>, <a class="txt" href="#E5:28_1_2"><span class="lab"><font color="Green" title="E20">A47</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><span class="kw">hence </span><a NAME="E5:28_1"/>
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c6">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:28_1"><span class="lab"><font color="Green" title="E17">A8</font></span></a>, <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/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E8:28"/>
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c6">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:28"><span class="lab"><font color="Green" title="E12">A3</font></span></a>, <a class="ref" href="diraf.html#T11" title="DIRAF:th.11">DIRAF:11</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>
