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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">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="b2">b</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="b1">a</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="b6">y</font>,<font color="Olive" title="b4">x</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="c3">b</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="c2">a</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="Olive" title="b1">y</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c4">d</font> ) )</span><br/>











<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:41"/><span class="lab"><font color="Green" title="E18">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="E2:41"/><span class="lab"><font color="Green" title="E19">A2</font></span>: 
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c6">c</font>
 <span class="kw">and </span><br/><a NAME="E3:41"/><span class="lab"><font color="Green" title="E20">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="c2">a</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="Olive" title="b1">y</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c4">d</font> )</span><br/>

<div><span class="lab"><font color="Green" title="E21">A4</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:41_1"/><span class="lab"><font color="Green" title="E22">A5</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>
 ; <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="c2">a</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="c4">d</font>,<font color="Maroon" title="c5">x</font>,<font color="Olive" title="b1">y</font> &amp;  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c2">a</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="Olive" title="b1">y</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c4">d</font> )</span><br/><span class="kw">take </span><font color="Maroon" title="c7">y</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"> (  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c7">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="c4">d</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c7">y</font> &amp;  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c7">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="c7">y</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c4">d</font> )</span><br/><span class="kw">thus </span><a NAME="E2:41_1"/>
(  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c7">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="c4">d</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c7">y</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:41"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="txt" href="#E1:41_1"><span class="lab"><font color="Green" title="E22">A5</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"> (  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c7">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="c7">y</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c4">d</font> )</span><br/><span class="kw">thus </span><a NAME="E3:41_1"/>
(  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c7">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="c7">y</font>,<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="#E2:41"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="txt" href="#E1:41_1"><span class="lab"><font color="Green" title="E22">A5</font></span></a>, <a class="ref" href="diraf.html#T13" title="DIRAF:th.13">DIRAF:13</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>
<a NAME="E5:41"/><span class="lab"><font color="Green" title="E22">A6</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="#E1:41"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="ref" href="diraf.html#T13" title="DIRAF:th.13">DIRAF:13</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E23">A7</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:41_2"/><span class="lab"><font color="Green" title="E24">A8</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="E2:41_2"/><span class="lab"><font color="Green" title="E25">A9</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="c3">b</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="c2">a</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="Olive" title="b1">y</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c4">d</font> )</span><br/><a NAME="E3:41_2"/>
<font color="Maroon" title="c4">d</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="txt" href="#E5:41"><span class="lab"><font color="Green" title="E22">A6</font></span></a>, <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="E5:41_2"/><span class="lab"><font color="Green" title="E26">A10</font></span>: 
<font color="Maroon" title="c5">x</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="c7">e1</font>
 <span class="kw">and </span><br/><a NAME="E6:41_2"/><span class="lab"><font color="Green" title="E27">A11</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="c2">a</font>,<font color="Maroon" title="c7">e1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41_2"><span class="lab"><font color="Green" title="E24">A8</font></span></a>, <a class="ref" href="analoaf.html#D5" title="ANALOAF:def.5">ANALOAF:def 5</a></span>;<br/><a NAME="E7:41_2"/><span class="lab"><font color="Green" title="E28">A12</font></span>: 
 <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="c7">e1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:41_2"><span class="lab"><font color="Green" title="E26">A10</font></span></a>, <a class="ref" href="diraf.html#D3" title="DIRAF:def.3">DIRAF:def 3</a></span>;<br/><a NAME="E8:41_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">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="c3">b</font>,<font color="Maroon" title="c5">x</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="E9:41_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A14</font></span>: 
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c7">e1</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="#E2:41"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="txt" href="#E2:41_2"><span class="lab"><font color="Green" title="E25">A9</font></span></a>, <a class="ref" href="diraf.html#T16" title="DIRAF:th.16">DIRAF:16</a></span>;<br/><a NAME="E10:41_2"/><span class="lab"><font color="Green" title="E31">A15</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="41_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:41_2_1"/>
<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="E2:41_2_1"/><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="c3">b</font>,<font color="Maroon" title="c5">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:41"><span class="lab"><font color="Green" title="E19">A2</font></span></a>, <a class="txt" href="#E2:41_2"><span class="lab"><font color="Green" title="E25">A9</font></span></a>, <a class="txt" href="#E8:41_2"><span class="lab"><font color="Green" title="E29">A13</font></span></a>, <a class="ref" href="diraf.html#T12" title="DIRAF:th.12">DIRAF:12</a>, <a class="ref" href="diraf.html#T16" title="DIRAF:th.16">DIRAF:16</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:41_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:41_2"><span class="lab"><font color="Green" title="E25">A9</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="E11:41_2"/><span class="lab"><font color="Green" title="E32">A16</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="c7">e1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:41_2"><span class="lab"><font color="Green" title="E25">A9</font></span></a>, <a class="txt" href="#E7:41_2"><span class="lab"><font color="Green" title="E28">A12</font></span></a>, <a class="ref" href="diraf.html#T12" title="DIRAF:th.12">DIRAF:12</a></span>;<br/><a NAME="E12:41_2"/><span class="lab"><font color="Green" title="E33">A17</font></span>: 
not  <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c7">e1</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2_2"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:41_2_2"/>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c3">b</font>,<font color="Maroon" title="c7">e1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:41_2"><span class="lab"><font color="Green" title="E28">A12</font></span></a>, <a class="ref" href="diraf.html#T34" title="DIRAF:th.34">DIRAF:34</a></span>;<br/>
<a NAME="E2:41_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A18</font></span>: 
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c7">e1</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/>
<span class="kw">assume </span><a NAME="E3:41_2_2"/>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c2">a</font>,<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="E4:41_2_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="c6">c</font>,<font color="Maroon" title="c7">e1</font>,<font color="Maroon" title="c2">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="E5:41_2_2"/>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c7">e1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:41_2"><span class="lab"><font color="Green" title="E30">A14</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="E6:41_2_2"/><span class="kw">then </span><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="c6">c</font>,<font color="Maroon" title="c7">e1</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="E7:41_2_2"/><span class="lab"><font color="Green" title="E37">A21</font></span>: 
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">e1</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="E8:41_2_2"/>
 <a href="diraf.html#R3" title="DIRAF:pred.3">LIN</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c7">e1</font>,<font color="Maroon" title="c7">e1</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="E9:41_2_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="c7">e1</font>,<font color="Maroon" title="c3">b</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:41_2"><span class="lab"><font color="Green" title="E32">A16</font></span></a>, <a class="txt" href="#E6:41_2_2"><span class="lab"><font color="Green" title="E36">A20</font></span></a>, <a class="txt" href="#E2:41_2_2"><span class="lab"><font color="Green" title="E34">A18</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="E10:41_2_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:41"><span class="lab"><font color="Green" title="E20">A3</font></span></a>, <a class="txt" href="#E10:41_2"><span class="lab"><font color="Green" title="E31">A15</font></span></a>, <a class="txt" href="#E4:41_2_2"><span class="lab"><font color="Green" title="E35">A19</font></span></a>, <a class="txt" href="#E7:41_2_2"><span class="lab"><font color="Green" title="E37">A21</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="E13:41_2"/>
 <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="c7">e1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:41_2"><span class="lab"><font color="Green" title="E30">A14</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="c8">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="E15:41_2"/><span class="lab"><font color="Green" title="E34">A22</font></span>: 
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c6">c</font>,<font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c2">a</font>
 <span class="kw">and </span><br/><a NAME="E16:41_2"/><span class="lab"><font color="Green" title="E35">A23</font></span>: 
<font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c7">e1</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c5">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:41_2"><span class="lab"><font color="Green" title="E33">A17</font></span></a>, <a class="ref" href="pasch.html#T28" target="_self" title="PASCH:th.28">Th28</a></span>;<br/><a NAME="E17:41_2"/>
<font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">e1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:41_2"><span class="lab"><font color="Green" title="E33">A17</font></span></a>, <a class="ref" href="diraf.html#T37" title="DIRAF:th.37">DIRAF:37</a></span>;<br/><a NAME="E18:41_2"/><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="c8">y</font>,<font color="Maroon" title="c5">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:41_2"><span class="lab"><font color="Green" title="E27">A11</font></span></a>, <a class="txt" href="#E16:41_2"><span class="lab"><font color="Green" title="E35">A23</font></span></a>, <a class="ref" href="diraf.html#T6" title="DIRAF:th.6">DIRAF:6</a></span>;<br/><a NAME="E19:41_2"/><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="c5">x</font>,<font color="Maroon" title="c8">y</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="E20:41_2"/><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="c8">y</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="E21:41_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A24</font></span>: 
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c8">y</font>,<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#T13" title="DIRAF:th.13">DIRAF:13</a></span>;<br/><a NAME="E22:41_2"/>
 <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c8">y</font>,<font color="Maroon" title="c6">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:41_2"><span class="lab"><font color="Green" title="E34">A22</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="E23:41_2"/>
 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="c2">a</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="Olive" title="b1">y</font>,<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="#E21:41_2"><span class="lab"><font color="Green" title="E36">A24</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>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_3"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:41_3"/>
<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="E2:41_3"/><span class="lab"><font color="Green" title="E24">A25</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><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="c2">a</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="Olive" title="b1">y</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c4">d</font> )</span><br/><span class="kw">take </span><font color="Maroon" title="c7">y</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"> (  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c7">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="c7">y</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c4">d</font> )</span><br/><span class="kw">thus </span><a NAME="E3:41_3"/>
(  <a href="diraf.html#R1" title="DIRAF:pred.1">Mid</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c7">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="c7">y</font>,<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="#E1:41"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="txt" href="#E2:41_3"><span class="lab"><font color="Green" title="E24">A25</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="E8:41"/>
 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="c2">a</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="Olive" title="b1">y</font>,<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="#E6:41"><span class="lab"><font color="Green" title="E23">A7</font></span></a>, <a class="txt" href="#E4:41"><span class="lab"><font color="Green" title="E21">A4</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>
