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<b>let </b><font color="Maroon">AFP</font> be   <a href="diraf.html#NM2">AffinPlane</a>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font>;<br/><b>let </b><font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font>;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:8"/><i><font color="Green">E23</font></i>: 
not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">x</font>
 <b>and </b><br/><a NAME="E2:8"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
 <b>and </b><br/><a NAME="E3:8"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font>
 ;<br/>

<b>thus </b><a NAME="E4:8"/>
not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">a</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:8_1"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">a</font>
 ;<br/>

<a NAME="E2:8_1"/><b>then </b>
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">a</font> &amp; <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> )
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/>
<a NAME="E3:8_1"/><b>then </b>
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">a</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">b</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">x</font> )
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T16">AFF_1:16</a>, <a class="ref" href="aff_1.html#T18">AFF_1:18</a></i>;<br/>
<b>hence </b><a NAME="E4:8_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="aff_1.html#T17">AFF_1:17</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E5:8"/>
not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">b</font>,<font color="Maroon">a</font>,<font color="Maroon">y</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:8_2"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">b</font>,<font color="Maroon">a</font>,<font color="Maroon">y</font>
 ;<br/>

<a NAME="E2:8_2"/><b>then </b>
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">b</font>,<font color="Maroon">a</font>,<font color="Maroon">y</font> &amp; <font color="Maroon">b</font>,<font color="Maroon">a</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">y</font>,<font color="Maroon">x</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> )
 
<b>by </b><i>, , <a class="ref" href="aff_1.html#T13">AFF_1:13</a>, <a class="ref" href="aff_1.html#T16">AFF_1:16</a></i>;<br/>
<a NAME="E3:8_2"/><b>then </b>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">b</font>,<font color="Maroon">a</font>,<font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T18">AFF_1:18</a></i>;<br/>
<b>hence </b><a NAME="E4:8_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E6:8"/>
not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">y</font>,<font color="Maroon">x</font>,<font color="Maroon">b</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:8_3"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">y</font>,<font color="Maroon">x</font>,<font color="Maroon">b</font>
 ;<br/>

<a NAME="E2:8_3"/><b>then </b>
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">y</font>,<font color="Maroon">x</font>,<font color="Maroon">b</font> &amp; <font color="Maroon">y</font>,<font color="Maroon">x</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">a</font> )
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/>
<a NAME="E3:8_3"/><b>then </b>
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">y</font>,<font color="Maroon">x</font>,<font color="Maroon">b</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">y</font>,<font color="Maroon">x</font>,<font color="Maroon">a</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">y</font>,<font color="Maroon">x</font>,<font color="Maroon">x</font> )
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T16">AFF_1:16</a>, <a class="ref" href="aff_1.html#T18">AFF_1:18</a></i>;<br/>
<b>hence </b><a NAME="E4:8_3"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="aff_1.html#T17">AFF_1:17</a></i>;<br/>


</div><b>end;</b></div>


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