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











<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:43"/><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">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">d</font>
 <span class="kw">and </span><br/><a NAME="E2:43"/><span class="lab"><font color="Green" title="E20">A2</font></span>: 
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">c</font>
 <span class="kw">and </span><br/><a NAME="E3:43"/><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">a</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c6">c</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">OAS</font> st <br/>(  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c3">b</font>,<font color="Olive" title="b1">y</font>,<font color="Maroon" title="c6">c</font> &amp;  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c5">x</font>,<font color="Olive" title="b1">y</font>,<font color="Maroon" title="c4">d</font> )</span><br/>

<a NAME="E4:43"/><span class="lab"><font color="Green" title="E22">A4</font></span>: 
<font color="Maroon" title="c3">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43"><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="E5:43"/><span class="lab"><font color="Green" title="E23">A5</font></span>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43"><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/>
<div><span class="lab"><font color="Green" title="E24">A6</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:43_1"/><span class="lab"><font color="Green" title="E25">A7</font></span>: 
<font color="Maroon" title="c3">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">d</font>
 <span class="kw">and </span><br/><a NAME="E2:43_1"/><span class="lab"><font color="Green" title="E26">A8</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">c</font>
 <span class="kw">and </span><br/><a NAME="E3:43_1"/><span class="lab"><font color="Green" title="E27">A9</font></span>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">OAS</font> st <br/>(  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c3">b</font>,<font color="Olive" title="b1">y</font>,<font color="Maroon" title="c6">c</font> &amp;  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c5">x</font>,<font color="Olive" title="b1">y</font>,<font color="Maroon" title="c4">d</font> )</span><br/><a NAME="E4:43_1"/><span class="lab"><font color="Green" title="E28">A10</font></span>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="txt" href="#E1:43_1"><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/><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="E6:43_1"/><span class="lab"><font color="Green" title="E29">A11</font></span>: 
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c7">e1</font>
 <span class="kw">and </span><br/><a NAME="E7:43_1"/><span class="lab"><font color="Green" title="E30">A12</font></span>: 
<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c4">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">e1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="txt" href="#E3:43_1"><span class="lab"><font color="Green" title="E27">A9</font></span></a>, <a class="ref" href="pasch.html#T27" target="_self" title="PASCH:th.27">Th27</a></span>;<br/><a NAME="E8:43_1"/><span class="lab"><font color="Green" title="E31">A13</font></span>: 
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c7">e1</font>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c2">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:43_1"><span class="lab"><font color="Green" title="E29">A11</font></span></a>, <a class="ref" href="diraf.html#T13" title="DIRAF:th.13">DIRAF:13</a></span>;<br/><a NAME="E9:43_1"/><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="c4">d</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="ref" href="diraf.html#T13" title="DIRAF:th.13">DIRAF:13</a>, <a class="ref" href="diraf.html#T34" title="DIRAF:th.34">DIRAF:34</a></span>;<br/><a NAME="E10:43_1"/><span class="lab"><font color="Green" title="E33">A15</font></span>: 
not  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c3">b</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:43_1_1"/><span class="lab"><font color="Green" title="E34">A16</font></span>: 
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c2">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="E2:43_1_1"/><span class="lab"><font color="Green" title="E35">A17</font></span>: 
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c3">b</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="E3:43_1_1"/>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="ref" href="diraf.html#T34" title="DIRAF:th.34">DIRAF:34</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:43_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="txt" href="#E3:43_1"><span class="lab"><font color="Green" title="E27">A9</font></span></a>, <a class="txt" href="#E2:43_1_1"><span class="lab"><font color="Green" title="E35">A17</font></span></a>, <a class="txt" href="#E1:43_1_1"><span class="lab"><font color="Green" title="E34">A16</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="E11:43_1"/><span class="lab"><font color="Green" title="E34">A18</font></span>: 
not  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c2">a</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:43_1_2"/>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c2">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="E2:43_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A19</font></span>: 
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">x</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="E3:43_1_2"/><span class="lab"><font color="Green" title="E36">A20</font></span>: 
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c2">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="E4:43_1_2"/>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:43_1"><span class="lab"><font color="Green" title="E32">A14</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="E5:43_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:43_1"><span class="lab"><font color="Green" title="E28">A10</font></span></a>, <a class="txt" href="#E10:43_1"><span class="lab"><font color="Green" title="E33">A15</font></span></a>, <a class="txt" href="#E2:43_1_2"><span class="lab"><font color="Green" title="E35">A19</font></span></a>, <a class="txt" href="#E3:43_1_2"><span class="lab"><font color="Green" title="E36">A20</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="E12:43_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A21</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">d</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="E13:43_1"/><span class="lab"><font color="Green" title="E36">A22</font></span>: 
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="ref" href="diraf.html#T13" title="DIRAF:th.13">DIRAF:13</a></span>;<br/><a NAME="E14:43_1"/>
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c7">e1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="txt" href="#E6:43_1"><span class="lab"><font color="Green" title="E29">A11</font></span></a>, <a class="ref" href="diraf.html#T15" title="DIRAF:th.15">DIRAF:15</a></span>;<br/><a NAME="E15:43_1"/><span class="kw">then </span>
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c7">e1</font>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c3">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#T13" title="DIRAF:th.13">DIRAF:13</a></span>;<br/><a NAME="E16:43_1"/><span class="kw">then </span>
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c7">e1</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:43_1"><span class="lab"><font color="Green" title="E36">A22</font></span></a>, <a class="txt" href="#E8:43_1"><span class="lab"><font color="Green" title="E31">A13</font></span></a>, <a class="ref" href="diraf.html#T16" title="DIRAF:th.16">DIRAF:16</a></span>;<br/><a NAME="E17:43_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A23</font></span>: 
<font color="Maroon" title="c7">e1</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</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="E18:43_1"/>
<font color="Maroon" title="c7">e1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_1"><span class="lab"><font color="Green" title="E25">A7</font></span></a>, <a class="txt" href="#E13:43_1"><span class="lab"><font color="Green" title="E36">A22</font></span></a>, <a class="txt" href="#E8:43_1"><span class="lab"><font color="Green" title="E31">A13</font></span></a>, <a class="ref" href="diraf.html#T18" title="DIRAF:th.18">DIRAF:18</a></span>;<br/><span class="kw">then </span><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="E20:43_1"/><span class="lab"><font color="Green" title="E38">A24</font></span>: 
<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c8">e2</font>
 <span class="kw">and </span><br/><a NAME="E21:43_1"/><span class="lab"><font color="Green" title="E39">A25</font></span>: 
<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">e1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c8">e2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:43_1"><span class="lab"><font color="Green" title="E37">A23</font></span></a>, <a class="ref" href="analoaf.html#D5" title="ANALOAF:def.5">ANALOAF:def 5</a></span>;<br/><a NAME="E22:43_1"/><span class="lab"><font color="Green" title="E40">A26</font></span>: 
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c8">e2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:43_1"><span class="lab"><font color="Green" title="E38">A24</font></span></a>, <a class="ref" href="diraf.html#D3" title="DIRAF:def.3">DIRAF:def 3</a></span>;<br/><a NAME="E23:43_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A27</font></span>: 
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c3">b</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="E24:43_1"/>
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c2">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43"><span class="lab"><font color="Green" title="E20">A2</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="E26:43_1"/><span class="lab"><font color="Green" title="E42">A28</font></span>: 
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c8">e2</font>
 <span class="kw">and </span><br/><a NAME="E27:43_1"/><span class="lab"><font color="Green" title="E43">A29</font></span>: 
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c8">e2</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">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="E28:43_1"/><span class="lab"><font color="Green" title="E44">A30</font></span>: 
(  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c9">y</font> or  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c3">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:43_1"><span class="lab"><font color="Green" title="E40">A26</font></span></a>, <a class="txt" href="#E26:43_1"><span class="lab"><font color="Green" title="E42">A28</font></span></a>, <a class="ref" href="diraf.html#T21" title="DIRAF:th.21">DIRAF:21</a></span>;<br/><a NAME="E29:43_1"/><span class="lab"><font color="Green" title="E45">A31</font></span>: 
<font color="Maroon" title="c6">c</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="#E4:43"><span class="lab"><font color="Green" title="E22">A4</font></span></a>, <a class="txt" href="#E20:43_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/><div><span class="lab"><font color="Green" title="E46">A32</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_3"><span class="kw">now </span></a><div class="add"><a NAME="E1:43_1_3"/>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c8">e2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:43_1"><span class="lab"><font color="Green" title="E40">A26</font></span></a>, <a class="ref" href="diraf.html#T34" title="DIRAF:th.34">DIRAF:34</a></span>;<br/><a NAME="E2:43_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A33</font></span>: 
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c8">e2</font>,<font color="Maroon" title="c3">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="E3:43_1_3"/>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c8">e2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:43_1"><span class="lab"><font color="Green" title="E42">A28</font></span></a>, <a class="ref" href="diraf.html#T34" title="DIRAF:th.34">DIRAF:34</a></span>;<br/><a NAME="E4:43_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A34</font></span>: 
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c8">e2</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="E5:43_1_3"/>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c8">e2</font>,<font color="Maroon" title="c6">c</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="E6:43_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A35</font></span>: 
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c6">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:43_1"><span class="lab"><font color="Green" title="E45">A31</font></span></a>, <a class="txt" href="#E4:43_1_3"><span class="lab"><font color="Green" title="E48">A34</font></span></a>, <a class="txt" href="#E2:43_1_3"><span class="lab"><font color="Green" title="E47">A33</font></span></a>, <a class="ref" href="diraf.html#T38" title="DIRAF:th.38">DIRAF:38</a></span>;<br/><a NAME="E7:43_1_3"/>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="ref" href="diraf.html#T34" title="DIRAF:th.34">DIRAF:34</a></span>;<br/><a NAME="E8:43_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A36</font></span>: 
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c6">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="E9:43_1_3"/><span class="lab"><font color="Green" title="E51">A37</font></span>: 
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c2">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="ref" href="diraf.html#T13" title="DIRAF:th.13">DIRAF:13</a></span>;<br/><span class="kw">assume </span><a NAME="E10:43_1_3"/><span class="lab"><font color="Green" title="E52">A38</font></span>: 
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c9">y</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/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">c9</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="E12:43_1_3"/><span class="lab"><font color="Green" title="E53">A39</font></span>: 
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c10">c9</font>,<font color="Maroon" title="c2">a</font>
 <span class="kw">and </span><br/><a NAME="E13:43_1_3"/><span class="lab"><font color="Green" title="E54">A40</font></span>: 
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c10">c9</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c9">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:43_1"><span class="lab"><font color="Green" title="E36">A22</font></span></a>, <a class="txt" href="#E11:43_1"><span class="lab"><font color="Green" title="E34">A18</font></span></a>, <a class="ref" href="pasch.html#T35" target="_self" title="PASCH:th.35">Th35</a></span>;<br/><a NAME="E14:43_1_3"/>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c10">c9</font>,<font color="Maroon" title="c2">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:43_1_3"><span class="lab"><font color="Green" title="E53">A39</font></span></a>, <a class="ref" href="diraf.html#T34" title="DIRAF:th.34">DIRAF:34</a></span>;<br/><a NAME="E15:43_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A41</font></span>: 
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c10">c9</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="E16:43_1_3"/><span class="lab"><font color="Green" title="E56">A42</font></span>: 
<font color="Maroon" title="c3">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">y</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:43_1_3_1"/>
<font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y</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="E2:43_1_3_1"/><span class="kw">then </span>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c3">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:43_1_3"><span class="lab"><font color="Green" title="E52">A38</font></span></a>, <a class="ref" href="diraf.html#T34" title="DIRAF:th.34">DIRAF:34</a></span>;<br/>
<a NAME="E3:43_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A43</font></span>: 
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">x</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="E4:43_1_3_1"/><span class="lab"><font color="Green" title="E58">A44</font></span>: 
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c3">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="E5:43_1_3_1"/>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="ref" href="diraf.html#T34" title="DIRAF:th.34">DIRAF:34</a></span>;<br/>
<a NAME="E6:43_1_3_1"/><span class="kw">then </span>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c6">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_1"><span class="lab"><font color="Green" title="E27">A9</font></span></a>, <a class="txt" href="#E9:43_1"><span class="lab"><font color="Green" title="E32">A14</font></span></a>, <a class="txt" href="#E3:43_1_3_1"><span class="lab"><font color="Green" title="E57">A43</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="E7:43_1_3_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="txt" href="#E1:43_1"><span class="lab"><font color="Green" title="E25">A7</font></span></a>, <a class="txt" href="#E9:43_1"><span class="lab"><font color="Green" title="E32">A14</font></span></a>, <a class="txt" href="#E4:43_1_3_1"><span class="lab"><font color="Green" title="E58">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="E17:43_1_3"/>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c10">c9</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c9">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:43_1_3"><span class="lab"><font color="Green" title="E54">A40</font></span></a>, <a class="ref" href="diraf.html#T34" title="DIRAF:th.34">DIRAF:34</a></span>;<br/><a NAME="E18:43_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A45</font></span>: 
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c10">c9</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="E19:43_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A46</font></span>: 
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c10">c9</font>,<font color="Maroon" title="c6">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:43_1_3"><span class="lab"><font color="Green" title="E56">A42</font></span></a>, <a class="txt" href="#E6:43_1_3"><span class="lab"><font color="Green" title="E49">A35</font></span></a>, <a class="ref" href="diraf.html#T38" title="DIRAF:th.38">DIRAF:38</a></span>;<br/><a NAME="E20:43_1_3"/>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c3">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="E21:43_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A47</font></span>: 
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c10">c9</font>,<font color="Maroon" title="c3">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:43_1_3"><span class="lab"><font color="Green" title="E56">A42</font></span></a>, <a class="txt" href="#E6:43_1_3"><span class="lab"><font color="Green" title="E49">A35</font></span></a>, <a class="txt" href="#E18:43_1_3"><span class="lab"><font color="Green" title="E57">A45</font></span></a>, <a class="ref" href="diraf.html#T38" title="DIRAF:th.38">DIRAF:38</a></span>;<br/><a NAME="E22:43_1_3"/>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c2">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="E23:43_1_3"/><span class="kw">then </span>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c10">c9</font>,<font color="Maroon" title="c2">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43_1"><span class="lab"><font color="Green" title="E27">A9</font></span></a>, <a class="txt" href="#E8:43_1_3"><span class="lab"><font color="Green" title="E50">A36</font></span></a>, <a class="txt" href="#E15:43_1_3"><span class="lab"><font color="Green" title="E55">A41</font></span></a>, <a class="ref" href="diraf.html#T38" title="DIRAF:th.38">DIRAF:38</a></span>;<br/><a NAME="E24:43_1_3"/><span class="kw">then </span>
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c2">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="txt" href="#E12:43_1_3"><span class="lab"><font color="Green" title="E53">A39</font></span></a>, <a class="txt" href="#E21:43_1_3"><span class="lab"><font color="Green" title="E59">A47</font></span></a>, <a class="txt" href="#E19:43_1_3"><span class="lab"><font color="Green" title="E58">A46</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="E25:43_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43_1"><span class="lab"><font color="Green" title="E26">A8</font></span></a>, <a class="txt" href="#E9:43_1_3"><span class="lab"><font color="Green" title="E51">A37</font></span></a>, <a class="ref" href="diraf.html#T18" title="DIRAF:th.18">DIRAF:18</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="E31:43_1"/><span class="lab"><font color="Green" title="E47">A48</font></span>: 
<font color="Maroon" title="c2">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="43_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:43_1_4"/>
<font color="Maroon" title="c2">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="E2:43_1_4"/><span class="kw">then </span>
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:43_1"><span class="lab"><font color="Green" title="E38">A24</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="E3:43_1_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="ref" href="diraf.html#T13" title="DIRAF:th.13">DIRAF:13</a>, <a class="ref" href="diraf.html#T34" title="DIRAF:th.34">DIRAF:34</a></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="E32:43_1"/><span class="lab"><font color="Green" title="E48">A49</font></span>: 
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c7">e1</font>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c2">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:43_1"><span class="lab"><font color="Green" title="E29">A11</font></span></a>, <a class="ref" href="diraf.html#T13" title="DIRAF:th.13">DIRAF:13</a>, <a class="ref" href="diraf.html#T34" title="DIRAF:th.34">DIRAF:34</a></span>;<br/><a NAME="E33:43_1"/><span class="lab"><font color="Green" title="E49">A50</font></span>: 
<font color="Maroon" title="c6">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="43_1_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:43_1_5"/>
<font color="Maroon" title="c6">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="E2:43_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A51</font></span>: 
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c6">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:43_1"><span class="lab"><font color="Green" title="E48">A49</font></span></a>, <a class="ref" href="diraf.html#T36" title="DIRAF:th.36">DIRAF:36</a></span>;<br/>
<a NAME="E3:43_1_5"/><span class="lab"><font color="Green" title="E51">A52</font></span>: 
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c2">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="E4:43_1_5"/>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:43_1"><span class="lab"><font color="Green" title="E32">A14</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="E5:43_1_5"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="txt" href="#E4:43_1"><span class="lab"><font color="Green" title="E28">A10</font></span></a>, <a class="txt" href="#E2:43_1_5"><span class="lab"><font color="Green" title="E50">A51</font></span></a>, <a class="txt" href="#E3:43_1_5"><span class="lab"><font color="Green" title="E51">A52</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="E34:43_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c4">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c8">e2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:43_1"><span class="lab"><font color="Green" title="E30">A12</font></span></a>, <a class="txt" href="#E21:43_1"><span class="lab"><font color="Green" title="E39">A25</font></span></a>, <a class="ref" href="diraf.html#T6" title="DIRAF:th.6">DIRAF:6</a></span>;<br/><a NAME="E35:43_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c4">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c9">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:43_1"><span class="lab"><font color="Green" title="E47">A48</font></span></a>, <a class="txt" href="#E27:43_1"><span class="lab"><font color="Green" title="E43">A29</font></span></a>, <a class="ref" href="diraf.html#T6" title="DIRAF:th.6">DIRAF:6</a></span>;<br/><a NAME="E36:43_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A53</font></span>: 
(  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c9">y</font> or  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c4">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="E37:43_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A54</font></span>: 
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c5">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:43_1"><span class="lab"><font color="Green" title="E46">A32</font></span></a>, <a class="ref" href="diraf.html#T13" title="DIRAF:th.13">DIRAF:13</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_6"><span class="kw">now </span></a><div class="add"><a NAME="E1:43_1_6"/><span class="lab"><font color="Green" title="E52">A55</font></span>: 
<font color="Maroon" title="c3">b</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="43_1_6_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:43_1_6_1"/><span class="lab"><font color="Green" title="E53">A56</font></span>: 
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="ref" href="diraf.html#D3" title="DIRAF:def.3">DIRAF:def 3</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:43_1_6_1"/><span class="lab"><font color="Green" title="E54">A57</font></span>: 
<font color="Maroon" title="c3">b</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="E3:43_1_6_1"/>
<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">e1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c4">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:43_1"><span class="lab"><font color="Green" title="E30">A12</font></span></a>, <a class="ref" href="diraf.html#T5" title="DIRAF:th.5">DIRAF:5</a></span>;<br/>
<a NAME="E4:43_1_6_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c4">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:43_1"><span class="lab"><font color="Green" title="E39">A25</font></span></a>, <a class="txt" href="#E33:43_1"><span class="lab"><font color="Green" title="E49">A50</font></span></a>, <a class="txt" href="#E2:43_1_6_1"><span class="lab"><font color="Green" title="E54">A57</font></span></a>, <a class="ref" href="analoaf.html#D5" title="ANALOAF:def.5">ANALOAF:def 5</a></span>;<br/>
<a NAME="E5:43_1_6_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c4">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:43"><span class="lab"><font color="Green" title="E23">A5</font></span></a>, <a class="txt" href="#E1:43_1_6_1"><span class="lab"><font color="Green" title="E53">A56</font></span></a>, <a class="ref" href="analoaf.html#D5" title="ANALOAF:def.5">ANALOAF:def 5</a></span>;<br/>
<a NAME="E6:43_1_6_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">x</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c3">b</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="E7:43_1_6_1"/><span class="kw">then </span>
(  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c3">b</font> or  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">x</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="E8:43_1_6_1"/><span class="kw">then </span>
(  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c3">b</font> or  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">x</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="E9:43_1_6_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A58</font></span>: 
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">x</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="E10:43_1_6_1"/>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c3">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="E11:43_1_6_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A59</font></span>: 
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c3">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_1"><span class="lab"><font color="Green" title="E25">A7</font></span></a>, <a class="txt" href="#E9:43_1"><span class="lab"><font color="Green" title="E32">A14</font></span></a>, <a class="txt" href="#E9:43_1_6_1"><span class="lab"><font color="Green" title="E55">A58</font></span></a>, <a class="ref" href="diraf.html#T38" title="DIRAF:th.38">DIRAF:38</a></span>;<br/>
<a NAME="E12:43_1_6_1"/><span class="lab"><font color="Green" title="E57">A60</font></span>: 
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">c</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="ref" href="diraf.html#T34" title="DIRAF:th.34">DIRAF:34</a></span>;<br/>
<a NAME="E13:43_1_6_1"/>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c2">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_1"><span class="lab"><font color="Green" title="E25">A7</font></span></a>, <a class="txt" href="#E9:43_1"><span class="lab"><font color="Green" title="E32">A14</font></span></a>, <a class="txt" href="#E9:43_1_6_1"><span class="lab"><font color="Green" title="E55">A58</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="E14:43_1_6_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="txt" href="#E3:43_1"><span class="lab"><font color="Green" title="E27">A9</font></span></a>, <a class="txt" href="#E11:43_1_6_1"><span class="lab"><font color="Green" title="E56">A59</font></span></a>, <a class="txt" href="#E12:43_1_6_1"><span class="lab"><font color="Green" title="E57">A60</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="E2:43_1_6"/><span class="lab"><font color="Green" title="E53">A61</font></span>: 
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c2">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:43_1"><span class="lab"><font color="Green" title="E32">A14</font></span></a>, <a class="ref" href="diraf.html#T36" title="DIRAF:th.36">DIRAF:36</a></span>;<br/><a NAME="E3:43_1_6"/><span class="lab"><font color="Green" title="E54">A62</font></span>: 
<font color="Maroon" title="c9">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">d</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_6_2"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:43_1_6_2"/><span class="lab"><font color="Green" title="E55">A63</font></span>: 
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c8">e2</font>,<font color="Maroon" title="c6">c</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="E2:43_1_6_2"/><span class="lab"><font color="Green" title="E56">A64</font></span>: 
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c8">e2</font>,<font color="Maroon" title="c3">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:43_1"><span class="lab"><font color="Green" title="E41">A27</font></span></a>, <a class="ref" href="diraf.html#T36" title="DIRAF:th.36">DIRAF:36</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:43_1_6_2"/>
<font color="Maroon" title="c9">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">d</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="E4:43_1_6_2"/><span class="kw">then </span>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c8">e2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:43_1"><span class="lab"><font color="Green" title="E42">A28</font></span></a>, <a class="ref" href="diraf.html#T34" title="DIRAF:th.34">DIRAF:34</a></span>;<br/>
<a NAME="E5:43_1_6_2"/><span class="kw">then </span>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c8">e2</font>,<font color="Maroon" title="c4">d</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="E6:43_1_6_2"/><span class="kw">then </span>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c8">e2</font>,<font color="Maroon" title="c2">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_1"><span class="lab"><font color="Green" title="E25">A7</font></span></a>, <a class="txt" href="#E9:43_1"><span class="lab"><font color="Green" title="E32">A14</font></span></a>, <a class="txt" href="#E2:43_1_6_2"><span class="lab"><font color="Green" title="E56">A64</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="E7:43_1_6_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="txt" href="#E29:43_1"><span class="lab"><font color="Green" title="E45">A31</font></span></a>, <a class="txt" href="#E2:43_1_6_2"><span class="lab"><font color="Green" title="E56">A64</font></span></a>, <a class="txt" href="#E1:43_1_6_2"><span class="lab"><font color="Green" title="E55">A63</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="E4:43_1_6"/><span class="lab"><font color="Green" title="E55">A65</font></span>: 
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c8">e2</font>,<font color="Maroon" title="c6">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:43_1"><span class="lab"><font color="Green" title="E41">A27</font></span></a>, <a class="ref" href="diraf.html#T36" title="DIRAF:th.36">DIRAF:36</a></span>;<br/><span class="kw">assume </span><a NAME="E5:43_1_6"/><span class="lab"><font color="Green" title="E56">A66</font></span>: 
not  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c3">b</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="E6:43_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A67</font></span>: 
<font color="Maroon" title="c9">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#T14" title="DIRAF:th.14">DIRAF:14</a></span>;<br/><a NAME="E7:43_1_6"/>
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c8">e2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:43_1"><span class="lab"><font color="Green" title="E42">A28</font></span></a>, <a class="txt" href="#E28:43_1"><span class="lab"><font color="Green" title="E44">A30</font></span></a>, <a class="txt" href="#E5:43_1_6"><span class="lab"><font color="Green" title="E56">A66</font></span></a>, <a class="ref" href="diraf.html#T15" title="DIRAF:th.15">DIRAF:15</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">z</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="E9:43_1_6"/><span class="lab"><font color="Green" title="E58">A68</font></span>: 
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c10">z</font>
 <span class="kw">and </span><br/><a NAME="E10:43_1_6"/><span class="lab"><font color="Green" title="E59">A69</font></span>: 
<font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c4">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">e2</font>,<font color="Maroon" title="c10">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:43_1_6"><span class="lab"><font color="Green" title="E57">A67</font></span></a>, <a class="ref" href="pasch.html#T27" target="_self" title="PASCH:th.27">Th27</a></span>;<br/><a NAME="E11:43_1_6"/><span class="lab"><font color="Green" title="E60">A70</font></span>: 
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c8">e2</font>,<font color="Maroon" title="c2">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/><div><span class="lab"><font color="Green" title="E61">A71</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_6_3"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:43_1_6_3"/><span class="lab"><font color="Green" title="E62">A72</font></span>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">z</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="E2:43_1_6_3"/>
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c2">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="ref" href="diraf.html#T13" title="DIRAF:th.13">DIRAF:13</a></span>;<br/><span class="kw">hence </span><a NAME="E3:43_1_6_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_1"><span class="lab"><font color="Green" title="E25">A7</font></span></a>, <a class="txt" href="#E9:43_1_6"><span class="lab"><font color="Green" title="E58">A68</font></span></a>, <a class="txt" href="#E1:43_1_6_3"><span class="lab"><font color="Green" title="E62">A72</font></span></a>, <a class="ref" href="diraf.html#T18" title="DIRAF:th.18">DIRAF:18</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="E13:43_1_6"/>
<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c9">y</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c5">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:43_1"><span class="lab"><font color="Green" title="E51">A54</font></span></a>, <a class="ref" href="diraf.html#D3" title="DIRAF:def.3">DIRAF:def 3</a></span>;<br/><a NAME="E14:43_1_6"/><span class="kw">then </span>
<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c9">y</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c5">x</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="E15:43_1_6"/><span class="kw">then </span>
<font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c4">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c4">d</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="E16:43_1_6"/><span class="kw">then </span>
<font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c4">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">e1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:43_1"><span class="lab"><font color="Green" title="E30">A12</font></span></a>, <a class="txt" href="#E12:43_1"><span class="lab"><font color="Green" title="E35">A21</font></span></a>, <a class="ref" href="diraf.html#T6" title="DIRAF:th.6">DIRAF:6</a></span>;<br/><a NAME="E17:43_1_6"/><span class="kw">then </span>
<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">e1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">e2</font>,<font color="Maroon" title="c10">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:43_1_6"><span class="lab"><font color="Green" title="E59">A69</font></span></a>, <a class="txt" href="#E3:43_1_6"><span class="lab"><font color="Green" title="E54">A62</font></span></a>, <a class="ref" href="analoaf.html#D5" title="ANALOAF:def.5">ANALOAF:def 5</a></span>;<br/><a NAME="E18:43_1_6"/><span class="kw">then </span>
<font color="Maroon" title="c2">a</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="c10">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:43_1"><span class="lab"><font color="Green" title="E39">A25</font></span></a>, <a class="txt" href="#E33:43_1"><span class="lab"><font color="Green" title="E49">A50</font></span></a>, <a class="ref" href="analoaf.html#D5" title="ANALOAF:def.5">ANALOAF:def 5</a></span>;<br/><a NAME="E19:43_1_6"/><span class="kw">then </span>
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c8">e2</font>,<font color="Maroon" title="c10">z</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="E20:43_1_6"/><span class="kw">then </span>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c8">e2</font>,<font color="Maroon" title="c10">z</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="E21:43_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A73</font></span>: 
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c10">z</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="E22:43_1_6"/><span class="lab"><font color="Green" title="E63">A74</font></span>: 
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c10">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:43_1_6"><span class="lab"><font color="Green" title="E58">A68</font></span></a>, <a class="ref" href="diraf.html#T34" title="DIRAF:th.34">DIRAF:34</a></span>;<br/><a NAME="E23:43_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A75</font></span>: 
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c10">z</font>,<font color="Maroon" title="c2">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_1"><span class="lab"><font color="Green" title="E25">A7</font></span></a>, <a class="txt" href="#E2:43_1_6"><span class="lab"><font color="Green" title="E53">A61</font></span></a>, <a class="ref" href="diraf.html#T38" title="DIRAF:th.38">DIRAF:38</a></span>;<br/><a NAME="E24:43_1_6"/>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c3">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="E25:43_1_6"/><span class="kw">then </span>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c10">z</font>,<font color="Maroon" title="c3">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_1"><span class="lab"><font color="Green" title="E25">A7</font></span></a>, <a class="txt" href="#E22:43_1_6"><span class="lab"><font color="Green" title="E63">A74</font></span></a>, <a class="txt" href="#E2:43_1_6"><span class="lab"><font color="Green" title="E53">A61</font></span></a>, <a class="ref" href="diraf.html#T38" title="DIRAF:th.38">DIRAF:38</a></span>;<br/><a NAME="E26:43_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A76</font></span>: 
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c8">e2</font>,<font color="Maroon" title="c3">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:43_1_6"><span class="lab"><font color="Green" title="E62">A73</font></span></a>, <a class="txt" href="#E23:43_1_6"><span class="lab"><font color="Green" title="E64">A75</font></span></a>, <a class="txt" href="#E12:43_1_6"><span class="lab"><font color="Green" title="E61">A71</font></span></a>, <a class="ref" href="diraf.html#T38" title="DIRAF:th.38">DIRAF:38</a></span>;<br/><a NAME="E27:43_1_6"/>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c8">e2</font>,<font color="Maroon" title="c8">e2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:43_1_6"><span class="lab"><font color="Green" title="E62">A73</font></span></a>, <a class="txt" href="#E23:43_1_6"><span class="lab"><font color="Green" title="E64">A75</font></span></a>, <a class="txt" href="#E12:43_1_6"><span class="lab"><font color="Green" title="E61">A71</font></span></a>, <a class="ref" href="diraf.html#T38" title="DIRAF:th.38">DIRAF:38</a></span>;<br/><a NAME="E28:43_1_6"/><span class="kw">then </span>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c8">e2</font>,<font color="Maroon" title="c6">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43_1_6"><span class="lab"><font color="Green" title="E52">A55</font></span></a>, <a class="txt" href="#E26:43_1_6"><span class="lab"><font color="Green" title="E65">A76</font></span></a>, <a class="txt" href="#E4:43_1_6"><span class="lab"><font color="Green" title="E55">A65</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="E29:43_1_6"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:43"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="txt" href="#E31:43_1"><span class="lab"><font color="Green" title="E47">A48</font></span></a>, <a class="txt" href="#E26:43_1_6"><span class="lab"><font color="Green" title="E65">A76</font></span></a>, <a class="txt" href="#E11:43_1_6"><span class="lab"><font color="Green" title="E60">A70</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="E39:43_1"/><span class="kw">then </span>
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c6">c</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#T13" title="DIRAF:th.13">DIRAF:13</a></span>;<br/><span class="kw">hence </span><a NAME="E40:43_1"/>
 ex <font color="Olive" title="b1">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">OAS</font> st <br/>(  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c3">b</font>,<font color="Olive" title="b1">y</font>,<font color="Maroon" title="c6">c</font> &amp;  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c5">x</font>,<font color="Olive" title="b1">y</font>,<font color="Maroon" title="c4">d</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:43_1"><span class="lab"><font color="Green" title="E50">A53</font></span></a>, <a class="txt" href="#E30:43_1"><span class="lab"><font color="Green" title="E46">A32</font></span></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="E7:43"/><span class="lab"><font color="Green" title="E25">A77</font></span>: 
( <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">d</font> implies (  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c6">c</font> &amp;  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c4">d</font>,<font color="Maroon" title="c4">d</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#T14" title="DIRAF:th.14">DIRAF:14</a></span>;<br/>
<a NAME="E8:43"/><span class="lab"><font color="Green" title="E26">A78</font></span>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">c</font> implies (  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c6">c</font> &amp;  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c4">d</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="diraf.html#T14" title="DIRAF:th.14">DIRAF:14</a></span>;<br/>
<a NAME="E9:43"/>
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">x</font> implies (  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c6">c</font> &amp;  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c4">d</font> ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:43"><span class="lab"><font color="Green" title="E19">A1</font></span></a>, <a class="ref" href="diraf.html#T14" title="DIRAF:th.14">DIRAF:14</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:43"/>
 ex <font color="Olive" title="b1">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">OAS</font> st <br/>(  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c3">b</font>,<font color="Olive" title="b1">y</font>,<font color="Maroon" title="c6">c</font> &amp;  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c5">x</font>,<font color="Olive" title="b1">y</font>,<font color="Maroon" title="c4">d</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:43"><span class="lab"><font color="Green" title="E24">A6</font></span></a>, <a class="txt" href="#E7:43"><span class="lab"><font color="Green" title="E25">A77</font></span></a>, <a class="txt" href="#E8:43"><span class="lab"><font color="Green" title="E26">A78</font></span></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>
