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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">K</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AFP</font>;<br/>







<b>assume </b><a NAME="E1:25"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">K</font> &amp; not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</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/>

<a NAME="E2:25"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">K</font> <a href="aff_1.html#NR2" target="_self">is_line</a> 
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T40">AFF_1:40</a></i>;<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="E4:25"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> &amp; <font color="Maroon">p</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#T31">AFF_1:31</a></i>;<br/>
<a NAME="E5:25"/><i><font color="Green">E46</font></i>: 
not <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T49">AFF_1:49</a></i>;<br/>
<a NAME="E6:25"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font> )
 
<b>by </b><i>, , <a class="ref" href="aff_1.html#T49">AFF_1:49</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:25_1"/><i><font color="Green">E48</font></i>: 
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font>
 ;<br/><b>set </b><font color="Maroon">A</font> =  <a href="aff_1.html#K2">Line</a> <font color="Maroon">p</font>,<font color="Maroon">a</font>;<br/><b>set </b><font color="Maroon">B'</font> =  <a href="aff_1.html#K2">Line</a> <font color="Maroon">p</font>,<font color="Maroon">b</font>;<br/><a NAME="E2:25_1"/><i><font color="Green">E49</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">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">p</font>,<font color="Maroon">b</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">p</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">p</font>,<font color="Maroon">b</font> )
 <b>by </b><i><a class="ref" href="aff_1.html#T26">AFF_1:26</a></i>;<br/><a NAME="E3:25_1"/><i><font color="Green">E50</font></i>: 
(  <a href="aff_1.html#K2">Line</a> <font color="Maroon">p</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">p</font>,<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#D3">AFF_1:def 3</a></i>;<br/><a NAME="E4:25_1"/><i><font color="Green">E51</font></i>: 
( not  <a href="aff_1.html#K2">Line</a> <font color="Maroon">p</font>,<font color="Maroon">a</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">K</font> &amp; not  <a href="aff_1.html#K2">Line</a> <font color="Maroon">p</font>,<font color="Maroon">b</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">K</font> )
 <b>by </b><i>, , , , <a class="ref" href="aff_1.html#T59">AFF_1:59</a></i>;<br/><b>consider </b><font color="Maroon">M</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AFP</font><b> such that </b><br/><a NAME="E6:25_1"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">M</font> &amp; <font color="Maroon">K</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">M</font> )
 <b>by </b><i>, <a class="ref" href="aff_1.html#T63">AFF_1:63</a></i>;<br/><a NAME="E7:25_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">M</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/><b>consider </b><font color="Maroon">C</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AFP</font><b> such that </b><br/><a NAME="E9:25_1"/><i><font color="Green">E55</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">p</font>,<font color="Maroon">a</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">C</font> )
 <b>by </b><i>, <a class="ref" href="aff_1.html#T63">AFF_1:63</a></i>;<br/><a NAME="E10:25_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">C</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="E11:25_1"/>
not <font color="Maroon">C</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">K</font>
 <b>by </b><i>, , <a class="ref" href="aff_1.html#T58">AFF_1:58</a></i>;<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><b> such that </b><br/><a NAME="E13:25_1"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">p'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <font color="Maroon">p'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> )
 <b>by </b><i>, , <a class="ref" href="aff_1.html#T72">AFF_1:72</a></i>;<br/><b>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="E15:25_1"/><i><font color="Green">E58</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">b</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="E16:25_1"/><i><font color="Green">E59</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="E17:25_1"/>
not <font color="Maroon">D</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">M</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:25_1_1"/>
<font color="Maroon">D</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">M</font>
 ;<br/>

<a NAME="E2:25_1_1"/><b>then </b>
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">p</font>,<font color="Maroon">b</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">M</font>
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T58">AFF_1:58</a></i>;<br/>
<b>hence </b><a NAME="E3:25_1_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="aff_1.html#T58">AFF_1:58</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="E19:25_1"/><i><font color="Green">E60</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> <font color="Maroon">M</font> )
 <b>by </b><i>, , <a class="ref" href="aff_1.html#T72">AFF_1:72</a></i>;<br/><a NAME="E20:25_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">K</font>
 <b>by </b><i>, <a class="txt" href="homothet.html#E1"><i><font color="Green">Lemma42</font></i></a>, <a class="ref" href="aff_1.html#T54">AFF_1:54</a></i>;<br/><a NAME="E21:25_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">p</font>,<font color="Maroon">a</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p'</font>,<font color="Maroon">x</font>
 <b>by </b><i>, , , <a class="ref" href="aff_1.html#T53">AFF_1:53</a></i>;<br/><a NAME="E22:25_1"/>
<font color="Maroon">p</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">p'</font>,<font color="Maroon">y</font>
 <b>by </b><i>, , <a class="txt" href="homothet.html#E1"><i><font color="Green">Lemma42</font></i></a>, <a class="ref" href="aff_1.html#T53">AFF_1:53</a></i>;<br/><b>hence </b><a NAME="E23:25_1"/>
 ex <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font> st <br/>( ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> ) or ( not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</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/>( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> &amp; <font color="Olive">p'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> &amp; <font color="Olive">p</font>,<font color="Maroon">a</font> <a href="analoaf.html#R2">//</a> <font color="Olive">p'</font>,<font color="Maroon">x</font> &amp; <font color="Olive">p</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Olive">p'</font>,<font color="Olive">y</font> &amp; <font color="Maroon">x</font>,<font color="Olive">y</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">K</font> ) ) )
 <b>by </b><i>, , , <a class="txt" href="homothet.html#E1:25"><i><font color="Green">E43</font></i></a>, <a class="txt" href="homothet.html#E2:25"><i><font color="Green">E44</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:25"/>
 ex <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AFP</font> st <br/>( ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">y</font> ) or ( not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</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/>( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> &amp; <font color="Olive">p'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> &amp; <font color="Olive">p</font>,<font color="Maroon">a</font> <a href="analoaf.html#R2">//</a> <font color="Olive">p'</font>,<font color="Maroon">x</font> &amp; <font color="Olive">p</font>,<font color="Maroon">b</font> <a href="analoaf.html#R2">//</a> <font color="Olive">p'</font>,<font color="Olive">y</font> &amp; <font color="Maroon">x</font>,<font color="Olive">y</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">K</font> ) ) )
 ;<br/>


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