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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">o</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font>;<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="E1:12"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">a</font>,<font color="Maroon">b</font> )
 <b>and </b><br/><a NAME="E2:12"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>and </b><br/><a NAME="E3:12"/><i><font color="Green">E45</font></i>: 
( ( not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">a</font>,<font color="Maroon">x</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">x</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">y</font> ) or (  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">a</font>,<font color="Maroon">x</font> &amp;  ex <font color="Olive">p</font>, <font color="Olive">p'</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font> st <br/>( not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">a</font>,<font color="Olive">p</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Olive">p</font>,<font color="Olive">p'</font> &amp; <font color="Maroon">a</font>,<font color="Olive">p</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Olive">p'</font> &amp; <font color="Olive">p</font>,<font color="Maroon">x</font> <a href="analoaf.html#R2">//</a> <font color="Olive">p'</font>,<font color="Maroon">y</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">a</font>,<font color="Maroon">y</font> ) ) )
 ;<br/>

<div><i><font color="Green">E46</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="12_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:12_1"/><i><font color="Green">E47</font></i>: 
not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">a</font>,<font color="Maroon">x</font>
 ;<br/><a NAME="E2:12_1"/>
( <font color="Maroon">a</font>,<font color="Maroon">x</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a</font>,<font color="Maroon">x</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">a</font>,<font color="Maroon">a</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">x</font>,<font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="aff_1.html#T11">AFF_1:11</a>, <a class="ref" href="aff_1.html#T16">AFF_1:16</a></i>;<br/><b>hence </b><a NAME="E3:12_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i>, , , <a class="ref" href="aff_1.html#T70">AFF_1:70</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:12_2"/><i><font color="Green">E48</font></i>: 
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">a</font>,<font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">o</font> )
 ;<br/><b>then consider </b><font color="Maroon">p</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font>, <font color="Maroon">q</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font><b> such that </b><br/><a NAME="E3:12_2"/><i><font color="Green">E49</font></i>: 
( not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">a</font>,<font color="Maroon">p</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">p</font>,<font color="Maroon">q</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">p</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">q</font> &amp; <font color="Maroon">p</font>,<font color="Maroon">x</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">q</font>,<font color="Maroon">y</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">a</font>,<font color="Maroon">y</font> )
 <b>by </b><i/>;<br/><a NAME="E4:12_2"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">a</font>,<font color="Maroon">p</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a</font>,<font color="Maroon">q</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">p</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">a</font>,<font color="Maroon">p</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">a</font>,<font color="Maroon">a</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">p</font>,<font color="Maroon">p</font> )
 <b>by </b><i>, , <a class="ref" href="aff_1.html#T11">AFF_1:11</a>, <a class="ref" href="aff_1.html#T16">AFF_1:16</a></i>;<br/><a NAME="E5:12_2"/><i><font color="Green">E51</font></i>: 
not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">p</font>,<font color="Maroon">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="12_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:12_2_1"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">p</font>,<font color="Maroon">x</font>
 ;<br/>

<a NAME="E2:12_2_1"/><b>then </b>
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">x</font>,<font color="Maroon">o</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">x</font>,<font color="Maroon">p</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">x</font>,<font color="Maroon">a</font> )
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T15">AFF_1:15</a>, <a class="ref" href="aff_1.html#T16">AFF_1:16</a></i>;<br/>
<b>hence </b><a NAME="E3:12_2_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><a NAME="E6:12_2"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">a</font>,<font color="Maroon">o</font>
 <b>by </b><i><a class="ref" href="aff_1.html#T16">AFF_1:16</a></i>;<br/><a NAME="E7:12_2"/><b>then </b><i><font color="Green">E52</font></i>: 
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font>
 <b>by </b><i>, , , <a class="ref" href="aff_1.html#T17">AFF_1:17</a></i>;<br/><a NAME="E8:12_2"/>
( <font color="Maroon">p</font>,<font color="Maroon">x</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p</font>,<font color="Maroon">x</font> &amp; <font color="Maroon">p</font>,<font color="Maroon">x</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p</font>,<font color="Maroon">y</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">p</font>,<font color="Maroon">p</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">x</font>,<font color="Maroon">x</font> )
 <b>by </b><i>, , <a class="ref" href="aff_1.html#T11">AFF_1:11</a>, <a class="ref" href="aff_1.html#T16">AFF_1:16</a>, <a class="ref" href="aff_1.html#T70">AFF_1:70</a></i>;<br/><b>hence </b><a NAME="E9:12_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i>, , <a class="ref" href="aff_1.html#T70">AFF_1:70</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:12"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i>, , , </i>;<br/>


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