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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>assume </b><a NAME="E1:10"/><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> )
 ;<br/>

<div><i><font color="Green">E44</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="10_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:10_1"/><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>
 ;<br/><a NAME="E2:10_1"/>
<font color="Maroon">o</font>,<font color="Maroon">a</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o</font>,<font color="Maroon">b</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/><a NAME="E3:10_1"/><b>then </b>
<font color="Maroon">a</font>,<font color="Maroon">o</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o</font>,<font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/><b>then consider </b><font color="Maroon">y</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font><b> such that </b><br/><a NAME="E5:10_1"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">x</font>,<font color="Maroon">o</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o</font>,<font color="Maroon">y</font> &amp; <font color="Maroon">x</font>,<font color="Maroon">a</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">y</font> )
 <b>by </b><i>, <a class="ref" href="diraf.html#T47">DIRAF:47</a></i>;<br/><a NAME="E6:10_1"/>
<font color="Maroon">o</font>,<font color="Maroon">x</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">o</font>,<font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/><a NAME="E7:10_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">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> )
 <b>by </b><i>, <a class="ref" href="aff_1.html#T13">AFF_1:13</a>, <a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/><b>hence </b><a NAME="E8:10_1"/>
 ex <font color="Olive">y</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="Maroon">x</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">x</font>,<font color="Olive">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="Olive">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="Olive">y</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">a</font>,<font color="Olive">y</font> ) ) )
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:10_2"/><i><font color="Green">E47</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>
 ;<br/><b>consider </b><font color="Maroon">p</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font><b> such that </b><br/><a NAME="E3:10_2"/><i><font color="Green">E48</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>
 <b>by </b><i>, <a class="ref" href="aff_1.html#T22">AFF_1:22</a></i>;<br/><a NAME="E4:10_2"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">p</font> )
 <b>by </b><i>, <a class="ref" href="aff_1.html#T16">AFF_1:16</a></i>;<br/><b>set </b><font color="Maroon">M</font> =  <a href="aff_1.html#K2">Line</a> <font color="Maroon">o</font>,<font color="Maroon">a</font>;<br/><b>set </b><font color="Maroon">K</font> =  <a href="aff_1.html#K2">Line</a> <font color="Maroon">o</font>,<font color="Maroon">p</font>;<br/><b>set </b><font color="Maroon">A</font> =  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">p</font>;<br/><a NAME="E5:10_2"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">o</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">o</font>,<font color="Maroon">a</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">o</font>,<font color="Maroon">a</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">o</font>,<font color="Maroon">p</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">o</font>,<font color="Maroon">p</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">p</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">p</font> )
 <b>by </b><i><a class="ref" href="aff_1.html#T26">AFF_1:26</a></i>;<br/><a NAME="E6:10_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">o</font>,<font color="Maroon">a</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#D2">AFF_1:def 2</a></i>;<br/><a NAME="E7:10_2"/><i><font color="Green">E52</font></i>: 
(  <a href="aff_1.html#K2">Line</a> <font color="Maroon">o</font>,<font color="Maroon">a</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">o</font>,<font color="Maroon">p</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">p</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  )
 <b>by </b><i>, <a class="ref" href="aff_1.html#D3">AFF_1:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">B'</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AFP</font><b> such that </b><br/><a NAME="E9:10_2"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B'</font> &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">p</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">B'</font> )
 <b>by </b><i><a class="ref" href="aff_1.html#T63">AFF_1:63</a></i>;<br/><a NAME="E10:10_2"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">B'</font> <a href="aff_1.html#NR2" target="_self">is_line</a> 
 <b>by </b><i>, <a class="ref" href="aff_1.html#T50">AFF_1:50</a></i>;<br/><a NAME="E11:10_2"/>
not <font color="Maroon">B'</font> <a href="aff_1.html#R4">//</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">o</font>,<font color="Maroon">p</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:10_2_1"/>
<font color="Maroon">B'</font> <a href="aff_1.html#R4">//</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">o</font>,<font color="Maroon">p</font>
 ;<br/>

<a NAME="E2:10_2_1"/><b>then </b>
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">p</font> <a href="aff_1.html#R4">//</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">o</font>,<font color="Maroon">p</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T58">AFF_1:58</a></i>;<br/>
<a NAME="E3:10_2_1"/><b>then </b>
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">o</font>,<font color="Maroon">p</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T59">AFF_1:59</a></i>;<br/>
<b>hence </b><a NAME="E4:10_2_1"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="aff_1.html#T33">AFF_1:33</a></i>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon">p'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font><b> such that </b><br/><a NAME="E13:10_2"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">p'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B'</font> &amp; <font color="Maroon">p'</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">o</font>,<font color="Maroon">p</font> )
 <b>by </b><i>, , <a class="ref" href="aff_1.html#T72">AFF_1:72</a></i>;<br/><b>set </b><font color="Maroon">C</font> =  <a href="aff_1.html#K2">Line</a> <font color="Maroon">p</font>,<font color="Maroon">x</font>;<br/><a NAME="E14:10_2"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">p</font>,<font color="Maroon">x</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">p</font>,<font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="aff_1.html#T26">AFF_1:26</a></i>;<br/><a NAME="E15:10_2"/><i><font color="Green">E58</font></i>: 
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">p</font>,<font color="Maroon">x</font> <a href="aff_1.html#NR2" target="_self">is_line</a> 
 <b>by </b><i>, , <a class="ref" href="aff_1.html#D3">AFF_1:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">D</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AFP</font><b> such that </b><br/><a NAME="E17:10_2"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">p'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">p</font>,<font color="Maroon">x</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">D</font> )
 <b>by </b><i><a class="ref" href="aff_1.html#T63">AFF_1:63</a></i>;<br/><a NAME="E18:10_2"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">D</font> <a href="aff_1.html#NR2" target="_self">is_line</a> 
 <b>by </b><i>, <a class="ref" href="aff_1.html#T50">AFF_1:50</a></i>;<br/><a NAME="E19:10_2"/>
not <font color="Maroon">D</font> <a href="aff_1.html#R4">//</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">o</font>,<font color="Maroon">a</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="10_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:10_2_2"/>
<font color="Maroon">D</font> <a href="aff_1.html#R4">//</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">o</font>,<font color="Maroon">a</font>
 ;<br/>

<a NAME="E2:10_2_2"/><b>then </b>
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">p</font>,<font color="Maroon">x</font> <a href="aff_1.html#R4">//</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">o</font>,<font color="Maroon">a</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T58">AFF_1:58</a></i>;<br/>
<a NAME="E3:10_2_2"/><b>then </b>
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">p</font>,<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">o</font>,<font color="Maroon">a</font>
 
<b>by </b><i>, , <a class="ref" href="aff_1.html#T59">AFF_1:59</a></i>;<br/>
<b>hence </b><a NAME="E4:10_2_2"/>
contradiction
 <b>by </b><i>, , , , <a class="ref" href="aff_1.html#T33">AFF_1:33</a></i>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon">y</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font><b> such that </b><br/><a NAME="E21:10_2"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">o</font>,<font color="Maroon">a</font> )
 <b>by </b><i>, , <a class="ref" href="aff_1.html#T72">AFF_1:72</a></i>;<br/><a NAME="E22:10_2"/>
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">p</font>,<font color="Maroon">p'</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">p'</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">a</font>,<font color="Maroon">y</font> )
 <b>by </b><i>, , , , , , , <a class="ref" href="aff_1.html#T33">AFF_1:33</a>, <a class="ref" href="aff_1.html#T53">AFF_1:53</a></i>;<br/><b>hence </b><a NAME="E23:10_2"/>
 ex <font color="Olive">y</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="Maroon">x</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">x</font>,<font color="Olive">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="Olive">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="Olive">y</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">a</font>,<font color="Olive">y</font> ) ) )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:10"/>
 ex <font color="Olive">y</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="Maroon">x</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">x</font>,<font color="Olive">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="Olive">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="Olive">y</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">a</font>,<font color="Olive">y</font> ) ) )
 <b>by </b><i/>;<br/>


</div>
