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<div class="add">

<b>let </b><font color="Maroon" title="c1">OAS</font> be   <a href="analoaf.html#NM1" title="ANALOAF:NM.1">OAffinSpace</a>;<br/><b>let </b><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>;<br/>











<b>assume </b><a NAME="E1:39"/><i><font color="Green" title="E18">A1</font></i>: 
(  <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> )
 ;<br/>

<a NAME="E2:39"/><b>then </b><i><font color="Green" title="E19">A2</font></i>: 
(  <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> &amp;  <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="c3">b</font> )
 
<b>by </b><i><a class="ref" href="diraf.html#T13" title="DIRAF:th.13">DIRAF:13</a></i>;<br/>
<div><i><font color="Green" title="E20">A3</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:39_1"/><i><font color="Green" title="E21">A4</font></i>: 
( <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> &amp; <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> )
 ;<br/><a NAME="E2:39_1"/>
<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>
 <b>by </b><i><a class="txt" href="pasch.html#E2:39"><i><font color="Green" title="E19">A2</font></i></a>, <a class="ref" href="diraf.html#D3" title="DIRAF:def.3">DIRAF:def 3</a></i>;<br/><b>then consider </b><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><b> such that </b><br/><a NAME="E4:39_1"/><i><font color="Green" title="E22">A5</font></i>: 
( <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> &amp; <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> )
 <b>by </b><i><a class="txt" href="pasch.html#E1:39_1"><i><font color="Green" title="E21">A4</font></i></a>, <a class="ref" href="analoaf.html#D5" title="ANALOAF:def.5">ANALOAF:def 5</a></i>;<br/><a NAME="E5:39_1"/><i><font color="Green" title="E23">A6</font></i>: 
 <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>
 <b>by </b><i><a class="txt" href="pasch.html#E4:39_1"><i><font color="Green" title="E22">A5</font></i></a>, <a class="ref" href="diraf.html#D3" title="DIRAF:def.3">DIRAF:def 3</a></i>;<br/><a NAME="E6:39_1"/><b>then </b>
 <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>
 <b>by </b><i><a class="ref" href="diraf.html#T13" title="DIRAF:th.13">DIRAF:13</a></i>;<br/><a NAME="E7:39_1"/><b>then </b><i><font color="Green" title="E24">A7</font></i>: 
 <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>
 <b>by </b><i><a class="txt" href="pasch.html#E1:39"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="pasch.html#E1:39_1"><i><font color="Green" title="E21">A4</font></i></a>, <a class="ref" href="diraf.html#T16" title="DIRAF:th.16">DIRAF:16</a></i>;<br/><a NAME="E8:39_1"/><b>then </b><i><font color="Green" title="E25">A8</font></i>: 
 <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>
 <b>by </b><i><a class="ref" href="diraf.html#T13" title="DIRAF:th.13">DIRAF:13</a></i>;<br/><a NAME="E9:39_1"/><i><font color="Green" title="E26">A9</font></i>: 
<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="39_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:39_1_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>
 ;<br/>

<a NAME="E2:39_1_1"/><b>then </b>
 <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>
 
<b>by </b><i><a class="txt" href="pasch.html#E5:39_1"><i><font color="Green" title="E23">A6</font></i></a>, <a class="txt" href="pasch.html#E7:39_1"><i><font color="Green" title="E24">A7</font></i></a>, <a class="ref" href="diraf.html#T12" title="DIRAF:th.12">DIRAF:12</a></i>;<br/>
<b>hence </b><a NAME="E3:39_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="pasch.html#E1:39_1"><i><font color="Green" title="E21">A4</font></i></a>, <a class="ref" href="diraf.html#T12" title="DIRAF:th.12">DIRAF:12</a></i>;<br/>


</div><b>end;</b></div><a NAME="E10:39_1"/><i><font color="Green" title="E27">A10</font></i>: 
<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>
 <b>by </b><i><a class="txt" href="pasch.html#E1:39_1"><i><font color="Green" title="E21">A4</font></i></a>, <a class="txt" href="pasch.html#E5:39_1"><i><font color="Green" title="E23">A6</font></i></a>, <a class="ref" href="diraf.html#T12" title="DIRAF:th.12">DIRAF:12</a></i>;<br/><a NAME="E11:39_1"/><i><font color="Green" title="E28">A11</font></i>: 
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="39_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:39_1_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>
 ;<br/>

<a NAME="E2:39_1_2"/><b>then </b><i><font color="Green" title="E29">A12</font></i>: 
 <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>
 
<b>by </b><i><a class="ref" href="diraf.html#T36" title="DIRAF:th.36">DIRAF:36</a></i>;<br/>
<a NAME="E3:39_1_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>
 
<b>by </b><i><a class="txt" href="pasch.html#E8:39_1"><i><font color="Green" title="E25">A8</font></i></a>, <a class="ref" href="diraf.html#T34" title="DIRAF:th.34">DIRAF:34</a></i>;<br/>
<a NAME="E4:39_1_2"/><b>then </b>
(  <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> &amp;  <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> )
 
<b>by </b><i><a class="txt" href="pasch.html#E5:39_1"><i><font color="Green" title="E23">A6</font></i></a>, <a class="ref" href="diraf.html#T34" title="DIRAF:th.34">DIRAF:34</a>, <a class="ref" href="diraf.html#T36" title="DIRAF:th.36">DIRAF:36</a></i>;<br/>
<a NAME="E5:39_1_2"/><b>then </b>
(  <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> &amp;  <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> &amp;  <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> )
 
<b>by </b><i><a class="ref" href="diraf.html#T36" title="DIRAF:th.36">DIRAF:36</a>, <a class="ref" href="diraf.html#T37" title="DIRAF:th.37">DIRAF:37</a></i>;<br/>
<a NAME="E6:39_1_2"/><b>then </b>
(  <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> &amp;  <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> )
 
<b>by </b><i><a class="txt" href="pasch.html#E10:39_1"><i><font color="Green" title="E27">A10</font></i></a>, <a class="ref" href="diraf.html#T37" title="DIRAF:th.37">DIRAF:37</a>, <a class="ref" href="diraf.html#T41" title="DIRAF:th.41">DIRAF:41</a></i>;<br/>
<b>hence </b><a NAME="E7:39_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="pasch.html#E1:39"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="pasch.html#E9:39_1"><i><font color="Green" title="E26">A9</font></i></a>, <a class="txt" href="pasch.html#E2:39_1_2"><i><font color="Green" title="E29">A12</font></i></a>, <a class="ref" href="diraf.html#T38" title="DIRAF:th.38">DIRAF:38</a></i>;<br/>


</div><b>end;</b></div><b>then consider </b><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><b> such that </b><br/><a NAME="E13:39_1"/><i><font color="Green" title="E29">A13</font></i>: 
(  <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> &amp; <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> )
 <b>by </b><i><a class="txt" href="pasch.html#E8:39_1"><i><font color="Green" title="E25">A8</font></i></a>, <a class="ref" href="pasch.html#T28" target="_self" title="PASCH:th.28">Th28</a></i>;<br/><a NAME="E14:39_1"/>
<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>
 <b>by </b><i><a class="txt" href="pasch.html#E11:39_1"><i><font color="Green" title="E28">A11</font></i></a>, <a class="ref" href="diraf.html#T37" title="DIRAF:th.37">DIRAF:37</a></i>;<br/><a NAME="E15:39_1"/><b>then </b>
<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>
 <b>by </b><i><a class="txt" href="pasch.html#E4:39_1"><i><font color="Green" title="E22">A5</font></i></a>, <a class="txt" href="pasch.html#E13:39_1"><i><font color="Green" title="E29">A13</font></i></a>, <a class="ref" href="diraf.html#T6" title="DIRAF:th.6">DIRAF:6</a></i>;<br/><a NAME="E16:39_1"/><b>then </b>
<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>
 <b>by </b><i><a class="ref" href="diraf.html#T5" title="DIRAF:th.5">DIRAF:5</a></i>;<br/><a NAME="E17:39_1"/><b>then </b>
 <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>
 <b>by </b><i><a class="ref" href="diraf.html#D3" title="DIRAF:def.3">DIRAF:def 3</a></i>;<br/><a NAME="E18:39_1"/><b>then </b>
(  <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> &amp;  <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> )
 <b>by </b><i><a class="txt" href="pasch.html#E13:39_1"><i><font color="Green" title="E29">A13</font></i></a>, <a class="ref" href="diraf.html#T13" title="DIRAF:th.13">DIRAF:13</a></i>;<br/><b>hence </b><a NAME="E19:39_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="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> )
 ;<br/></div><b>end;</b></div>
<div><i><font color="Green" title="E21">A14</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="39_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:39_2"/><i><font color="Green" title="E22">A15</font></i>: 
( <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> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> )
 ;<br/><b>take </b><font color="Maroon" title="c7">y</font> = <font color="Maroon" title="c2">a</font>;<br/><b>thus </b><a NAME="E2:39_2"/>
(  <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> )
 <b>by </b><i><a class="txt" href="pasch.html#E1:39"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="pasch.html#E1:39_2"><i><font color="Green" title="E22">A15</font></i></a>, <a class="ref" href="diraf.html#T14" title="DIRAF:th.14">DIRAF:14</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:39_3"/><i><font color="Green" title="E22">A16</font></i>: 
<font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">d</font>
 ;<br/><b>take </b><font color="Maroon" title="c7">y</font> = <font color="Maroon" title="c6">c</font>;<br/><b>thus </b><a NAME="E2:39_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="c4">d</font>,<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c7">y</font> )
 <b>by </b><i><a class="txt" href="pasch.html#E1:39"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="pasch.html#E1:39_3"><i><font color="Green" title="E22">A16</font></i></a>, <a class="ref" href="diraf.html#T14" title="DIRAF:th.14">DIRAF:14</a></i>;<br/><b>thus </b><a NAME="E3:39_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> )
 <b>by </b><i><a class="txt" href="pasch.html#E1:39"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="pasch.html#E1:39_3"><i><font color="Green" title="E22">A16</font></i></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></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:39"/>
 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> )
 <b>by </b><i><a class="txt" href="pasch.html#E3:39"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="pasch.html#E4:39"><i><font color="Green" title="E21">A14</font></i></a></i>;<br/>


</div>
