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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>let </b><font color="Maroon">r</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="E1:13"/><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="E2:13"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">AFP</font> <a href="aff_2.html#NR4" target="_self">satisfies_DES</a> 
 <b>and </b><br/><a NAME="E3:13"/><i><font color="Green">E45</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>
 <b>and </b><br/><a NAME="E4:13"/><i><font color="Green">E46</font></i>: 
 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> )
 <b>and </b><br/><a NAME="E5:13"/><i><font color="Green">E47</font></i>: 
 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">r</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">a</font>,<font color="Maroon">r</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>, <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="E7:13"/><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> &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; <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/>;<br/>
<b>consider </b><font color="Maroon">q</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="E9:13"/><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">q</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">q</font>,<font color="Maroon">q'</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">q</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b</font>,<font color="Maroon">q'</font> &amp; <font color="Maroon">q</font>,<font color="Maroon">x</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">q'</font>,<font color="Maroon">r</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">a</font>,<font color="Maroon">r</font> )
 <b>by </b><i/>;<br/>
<a NAME="E10:13"/><i><font color="Green">E50</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">p</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q</font> )
 
<b>by </b><i>, , <a class="ref" href="aff_1.html#T16">AFF_1:16</a></i>;<br/>
<a NAME="E11:13"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> implies <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:13_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 ;<br/>

<a NAME="E2:13_1"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> )
 
<b>by </b><i>, , , , </i>;<br/>
<b>hence </b><a NAME="E3:13_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E12:13"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> implies <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:13_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">o</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 ;<br/>

<a NAME="E2:13_2"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o</font>
 
<b>by </b><i>, , , </i>;<br/>
<b>hence </b><a NAME="E3:13_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 <b>by </b><i>, , , , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:13"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> implies <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:13_3"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> )
 ;<br/>

<a NAME="E2:13_3"/><b>then </b><i><font color="Green">E57</font></i>: 
<font color="Maroon">p'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q'</font>
 
<b>by </b><i>, , , <a class="ref" href="aff_1.html#T70">AFF_1:70</a></i>;<br/>
<a NAME="E3:13_3"/>
 <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="E4:13_3"/><b>then </b><i><font color="Green">E58</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> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">x</font>,<font color="Maroon">r</font> )
 
<b>by </b><i>, , , , <a class="ref" href="aff_1.html#T17">AFF_1:17</a></i>;<br/>
<a NAME="E5:13_3"/>
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="13_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:13_3_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="E2:13_3_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="E3:13_3_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>hence </b><a NAME="E6:13_3"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 <b>by </b><i>, , , , , <a class="ref" href="aff_1.html#T70">AFF_1:70</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E14:13"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> implies <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_4"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:13_4"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 ;<br/>

<a NAME="E2:13_4"/><i><font color="Green">E62</font></i>: 
not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">p</font>,<font color="Maroon">a</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/>
<a NAME="E3:13_4"/>
( <font color="Maroon">p</font>,<font color="Maroon">a</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p'</font>,<font color="Maroon">b</font> &amp; <font color="Maroon">p</font>,<font color="Maroon">a</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p'</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="E4:13_4"/><b>then </b><i><font color="Green">E63</font></i>: 
<font color="Maroon">b</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/>
<a NAME="E5:13_4"/><i><font color="Green">E64</font></i>: 
not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">q</font>,<font color="Maroon">a</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/>
<a NAME="E6:13_4"/>
( <font color="Maroon">q</font>,<font color="Maroon">a</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">q'</font>,<font color="Maroon">b</font> &amp; <font color="Maroon">q</font>,<font color="Maroon">a</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">q'</font>,<font color="Maroon">r</font> )
 
<b>by </b><i>, , <a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/>
<b>hence </b><a NAME="E7:13_4"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 <b>by </b><i>, , , , <a class="ref" href="aff_1.html#T70">AFF_1:70</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E15:13"/>
( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> implies <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_5"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:13_5"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q</font> &amp; <font color="Maroon">o</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font> )
 ;<br/>

<div><i><font color="Green">E66</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="13_5_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:13_5_1"/><i><font color="Green">E67</font></i>: 
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">p</font>,<font color="Maroon">q</font>
 ;<br/><a NAME="E2:13_5_1"/><b>then </b>
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">q</font>,<font color="Maroon">o</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">q</font>,<font color="Maroon">p</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">q</font>,<font color="Maroon">q'</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/><a NAME="E3:13_5_1"/><b>then </b><i><font color="Green">E68</font></i>: 
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">p</font>,<font color="Maroon">q'</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#T17">AFF_1:17</a></i>;<br/><a NAME="E4:13_5_1"/>
<font color="Maroon">x</font>,<font color="Maroon">p</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">y</font>,<font color="Maroon">p'</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/><a NAME="E5:13_5_1"/><b>then </b>
<font color="Maroon">x</font>,<font color="Maroon">q</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">y</font>,<font color="Maroon">q'</font>
 <b>by </b><i>, , , , , , , , , </i>;<br/><a NAME="E6:13_5_1"/><b>then </b><i><font color="Green">E79</font></i>: 
<font color="Maroon">q</font>,<font color="Maroon">x</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">q'</font>,<font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="aff_1.html#T13">AFF_1:13</a></i>;<br/><b>set </b><font color="Maroon">A</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/><a NAME="E7:13_5_1"/><i><font color="Green">E80</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; <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">b</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">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> &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> &amp; <font color="Maroon">r</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#T26">AFF_1:26</a>, <a class="ref" href="aff_1.html#D2">AFF_1:def 2</a>, <a class="ref" href="aff_1.html#D3">AFF_1:def 3</a></i>;<br/><a NAME="E8:13_5_1"/><i><font color="Green">E81</font></i>: 
(  <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; <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">q</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">q'</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> )
 <b>by </b><i>, , , , <a class="ref" href="aff_1.html#T26">AFF_1:26</a>, <a class="ref" href="aff_1.html#D2">AFF_1:def 2</a>, <a class="ref" href="aff_1.html#D3">AFF_1:def 3</a></i>;<br/><a NAME="E9:13_5_1"/><i><font color="Green">E82</font></i>: 
not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">q</font>,<font color="Maroon">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_5_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:13_5_1_1"/><i><font color="Green">E83</font></i>: 
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">q</font>,<font color="Maroon">x</font>
 ;<br/>

<a NAME="E2:13_5_1_1"/>
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">o</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">q</font>
 
<b>by </b><i>, , <a class="ref" href="aff_1.html#T38">AFF_1:38</a></i>;<br/>
<a NAME="E3:13_5_1_1"/><b>then </b>
<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">p</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#D2">AFF_1:def 2</a></i>;<br/>
<a NAME="E4:13_5_1_1"/><b>then </b>
(  <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; <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">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">a</font> )
 
<b>by </b><i>, , , <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/>
<b>hence </b><a NAME="E5:13_5_1_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><a NAME="E10:13_5_1"/>
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">x</font>,<font color="Maroon">r</font> )
 <b>by </b><i>, <a class="ref" href="aff_1.html#T33">AFF_1:33</a></i>;<br/><b>hence </b><a NAME="E11:13_5_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</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="13_5_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:13_5_2"/><i><font color="Green">E84</font></i>: 
not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">p</font>,<font color="Maroon">q</font>
 ;<br/><a NAME="E2:13_5_2"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">p</font>,<font color="Maroon">q</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p'</font>,<font color="Maroon">q'</font>
 <b>by </b><i>, , , , </i>;<br/><a NAME="E3:13_5_2"/><i><font color="Green">E86</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="13_5_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:13_5_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="E2:13_5_2_1"/><b>then </b><i><font color="Green">E87</font></i>: 
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">x</font>,<font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#T15">AFF_1:15</a></i>;<br/>
<a NAME="E3:13_5_2_1"/>
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">a</font>,<font color="Maroon">o</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">a</font>,<font color="Maroon">x</font> )
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T16">AFF_1:16</a></i>;<br/>
<b>hence </b><a NAME="E4:13_5_2_1"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="aff_1.html#T20">AFF_1:20</a></i>;<br/>


</div><b>end;</b></div><b>set </b><font color="Maroon">A</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">q</font>;<br/><a NAME="E4:13_5_2"/><i><font color="Green">E88</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; <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">b</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">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> &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> &amp; <font color="Maroon">r</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#T26">AFF_1:26</a>, <a class="ref" href="aff_1.html#D2">AFF_1:def 2</a>, <a class="ref" href="aff_1.html#D3">AFF_1:def 3</a></i>;<br/><a NAME="E5:13_5_2"/><i><font color="Green">E89</font></i>: 
(  <a href="aff_1.html#K2">Line</a> <font color="Maroon">o</font>,<font color="Maroon">q</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &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">q</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">o</font>,<font color="Maroon">q</font> &amp; <font color="Maroon">q'</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">o</font>,<font color="Maroon">q</font> )
 <b>by </b><i>, , <a class="ref" href="aff_1.html#T26">AFF_1:26</a>, <a class="ref" href="aff_1.html#D2">AFF_1:def 2</a>, <a class="ref" href="aff_1.html#D3">AFF_1:def 3</a></i>;<br/><a NAME="E6:13_5_2"/>
 <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#T33">AFF_1:33</a></i>;<br/><a NAME="E7:13_5_2"/><b>then </b><i><font color="Green">E90</font></i>: 
<font color="Maroon">q</font>,<font color="Maroon">x</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">q'</font>,<font color="Maroon">y</font>
 <b>by </b><i>, , , , , , </i>;<br/><a NAME="E8:13_5_2"/><i><font color="Green">E91</font></i>: 
not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">q</font>,<font color="Maroon">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_5_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:13_5_2_2"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">q</font>,<font color="Maroon">x</font>
 ;<br/>

<a NAME="E2:13_5_2_2"/><b>then </b>
<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">q</font>
 
<b>by </b><i><a class="ref" href="aff_1.html#D2">AFF_1:def 2</a></i>;<br/>
<a NAME="E3:13_5_2_2"/><b>then </b>
(  <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; <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">q</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#T30">AFF_1:30</a></i>;<br/>
<b>hence </b><a NAME="E4:13_5_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><a NAME="E9:13_5_2"/>
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">x</font>,<font color="Maroon">y</font> &amp;  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">o</font>,<font color="Maroon">x</font>,<font color="Maroon">r</font> )
 <b>by </b><i>, <a class="ref" href="aff_1.html#T33">AFF_1:33</a></i>;<br/><b>hence </b><a NAME="E10:13_5_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</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="E4:13_5"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E16:13"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 <b>by </b><i>, , , </i>;<br/>


</div>
