<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">AS</font> be   <a href="diraf.html#NM1">AffinSpace</a>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font>;<br/><b>let </b><font color="Maroon">X</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font>;<br/><b>let </b><font color="Maroon">Y</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font>;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:32"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">X</font> <a href="aff_4.html#NR1" target="_self">is_plane</a>  &amp; <font color="Maroon">Y</font> <a href="aff_4.html#NR1" target="_self">is_plane</a>  &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 <b>and </b><br/><a NAME="E2:32"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">Y</font>
 <b>and </b><br/><a NAME="E3:32"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font>
 ;<br/>

<b>set </b><font color="Maroon">Q</font> =  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>;<br/>
<b>set </b><font color="Maroon">Z</font> = <font color="Maroon">X</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Y</font>;<br/>
<a NAME="E4:32"/><i><font color="Green">E37</font></i>: 
(  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &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">b</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">a</font>,<font color="Maroon">b</font> )
 
<b>by </b><i>, <a class="ref" href="aff_1.html#T26">AFF_1:26</a>, <a class="ref" href="aff_1.html#D3">AFF_1:def 3</a></i>;<br/>
<a NAME="E5:32"/><i><font color="Green">E43</font></i>: 
(  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> &amp;  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font> )
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E6:32"/><b>then </b><i><font color="Green">E44</font></i>: 
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T19">XBOOLE_1:19</a></i>;<br/>
<a NAME="E7:32"/>
<font color="Maroon">X</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:32_1"/>
not <font color="Maroon">X</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Y</font> <a href="tarski.html#R1">c=</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:32_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Y</font>
 <b>and </b><br/><a NAME="E4:32_1"/><i><font color="Green">E47</font></i>: 
not <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E5:32_1"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 
<b>by </b><i><a class="ref" href="aff_4.html#T1" target="_self">Th1</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">a'</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font> <b>by </b><i><a class="ref" href="aff_4.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E7:32_1"/>
for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> iff <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<div><i><font color="Green">E57</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:32_1_1_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:32_1_1_1_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font>
 ;<br/><b>reconsider </b><font color="Maroon">b'</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font> <b>by </b><i><a class="txt" href="aff_4.html#E4:32"><i><font color="Green">E37</font></i></a></i>;<br/><b>set </b><font color="Maroon">M</font> =  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a'</font>,<font color="Maroon">b'</font>;<br/><a NAME="E3:32_1_1_1_1"/><i><font color="Green">E61</font></i>: 
(  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a'</font>,<font color="Maroon">b'</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &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">b'</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">a'</font>,<font color="Maroon">b'</font> )
 <b>by </b><i><a class="ref" href="aff_4.html#T2" target="_self">Th2</a>, <a class="ref" href="aff_1.html#T26">AFF_1:26</a>, <a class="ref" href="aff_1.html#D3">AFF_1:def 3</a></i>;<br/><a NAME="E4:32_1_1_1_1"/><i><font color="Green">E62</font></i>: 
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">a'</font>,<font color="Maroon">b'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="txt" href="aff_4.html#E2:32"><i><font color="Green">E35</font></i></a>, <a class="txt" href="aff_4.html#E4:32"><i><font color="Green">E37</font></i></a>, <a class="ref" href="aff_4.html#T2" target="_self">Th2</a>, </i>;<br/><div><i><font color="Green">E63</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:32_1_1_1_1_1"/>
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">a'</font>,<font color="Maroon">b'</font> <a href="aff_1.html#R4">//</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 ;<br/><a NAME="E2:32_1_1_1_1_1"/><b>then </b>
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">a'</font>,<font color="Maroon">b'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font>
 <b>by </b><i>, , <a class="txt" href="aff_4.html#E2:32"><i><font color="Green">E35</font></i></a>, <a class="ref" href="aff_4.html#T3" target="_self">Th3</a>, </i>;<br/><b>hence </b><a NAME="E3:32_1_1_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="aff_4.html#T3" target="_self">Th3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:32_1_1_1_1_2"/>
not  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a'</font>,<font color="Maroon">b'</font> <a href="aff_1.html#R4">//</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 ;<br/><b>then consider </b><font color="Maroon">q</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font><b> such that </b><br/><a NAME="E3:32_1_1_1_1_2"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a'</font>,<font color="Maroon">b'</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">a</font>,<font color="Maroon">b</font> )
 <b>by </b><i>, , , <a class="ref" href="aff_4.html#T3" target="_self">Th3</a>, <a class="ref" href="aff_4.html#T4" target="_self">Th4</a>, </i>;<br/><a NAME="E4:32_1_1_1_1_2"/>
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">a'</font>,<font color="Maroon">b'</font> <a href="hidden.html#R1">=</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a'</font>,<font color="Maroon">q</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:32"><i><font color="Green">E34</font></i></a>, <a class="ref" href="aff_4.html#T3" target="_self">Th3</a>, <a class="txt" href="aff_4.html#E9"><i><font color="Green">Lemma42</font></i></a>, <a class="ref" href="aff_1.html#T71">AFF_1:71</a></i>;<br/><a NAME="E5:32_1_1_1_1_2"/><b>then </b>
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">a'</font>,<font color="Maroon">b'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font>
 <b>by </b><i>, , <a class="txt" href="aff_4.html#E1:32"><i><font color="Green">E34</font></i></a>, <a class="txt" href="aff_4.html#E2:32"><i><font color="Green">E35</font></i></a>, <a class="txt" href="aff_4.html#E9"><i><font color="Green">Lemma42</font></i></a>, </i>;<br/><b>hence </b><a NAME="E6:32_1_1_1_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="aff_4.html#T3" target="_self">Th3</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E7:32_1_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="aff_4.html#T7" target="_self">Th7</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:32_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="aff_4.html#T1" target="_self">Th1</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:32_1_1_2"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:32_1_1_2_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">x</font>
 ;<br/><b>reconsider </b><font color="Maroon">b'</font> = <font color="Maroon">y</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font> <b>by </b><i><a class="txt" href="aff_4.html#E4:32"><i><font color="Green">E37</font></i></a></i>;<br/><b>set </b><font color="Maroon">M</font> =  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a'</font>,<font color="Maroon">b'</font>;<br/><a NAME="E3:32_1_1_2_1"/><i><font color="Green">E70</font></i>: 
(  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a'</font>,<font color="Maroon">b'</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &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">b'</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">a'</font>,<font color="Maroon">b'</font> )
 <b>by </b><i><a class="ref" href="aff_4.html#T2" target="_self">Th2</a>, <a class="ref" href="aff_1.html#T26">AFF_1:26</a>, <a class="ref" href="aff_1.html#D3">AFF_1:def 3</a></i>;<br/><a NAME="E4:32_1_1_2_1"/><i><font color="Green">E71</font></i>: 
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">a'</font>,<font color="Maroon">b'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font>
 <b>by </b><i>, <a class="txt" href="aff_4.html#E2:32"><i><font color="Green">E35</font></i></a>, <a class="txt" href="aff_4.html#E4:32"><i><font color="Green">E37</font></i></a>, <a class="ref" href="aff_4.html#T2" target="_self">Th2</a>, </i>;<br/><div><i><font color="Green">E72</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:32_1_1_2_1_1"/>
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">a'</font>,<font color="Maroon">b'</font> <a href="aff_1.html#R4">//</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 ;<br/><a NAME="E2:32_1_1_2_1_1"/><b>then </b>
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">a'</font>,<font color="Maroon">b'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 <b>by </b><i>, , <a class="txt" href="aff_4.html#E2:32"><i><font color="Green">E35</font></i></a>, <a class="ref" href="aff_4.html#T3" target="_self">Th3</a>, </i>;<br/><b>hence </b><a NAME="E3:32_1_1_2_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="aff_4.html#T3" target="_self">Th3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1_2_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:32_1_1_2_1_2"/>
not  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a'</font>,<font color="Maroon">b'</font> <a href="aff_1.html#R4">//</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 ;<br/><b>then consider </b><font color="Maroon">q</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font><b> such that </b><br/><a NAME="E3:32_1_1_2_1_2"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a'</font>,<font color="Maroon">b'</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">a</font>,<font color="Maroon">b</font> )
 <b>by </b><i>, , , <a class="ref" href="aff_4.html#T3" target="_self">Th3</a>, <a class="ref" href="aff_4.html#T4" target="_self">Th4</a>, </i>;<br/><a NAME="E4:32_1_1_2_1_2"/>
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">a'</font>,<font color="Maroon">b'</font> <a href="hidden.html#R1">=</a>  <a href="aff_1.html#K2">Line</a> <font color="Maroon">a'</font>,<font color="Maroon">q</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:32"><i><font color="Green">E34</font></i></a>, <a class="ref" href="aff_4.html#T3" target="_self">Th3</a>, <a class="txt" href="aff_4.html#E9"><i><font color="Green">Lemma42</font></i></a>, <a class="ref" href="aff_1.html#T71">AFF_1:71</a></i>;<br/><a NAME="E5:32_1_1_2_1_2"/><b>then </b>
 <a href="aff_1.html#K2">Line</a> <font color="Maroon">a'</font>,<font color="Maroon">b'</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 <b>by </b><i>, , <a class="txt" href="aff_4.html#E1:32"><i><font color="Green">E34</font></i></a>, <a class="txt" href="aff_4.html#E2:32"><i><font color="Green">E35</font></i></a>, <a class="txt" href="aff_4.html#E9"><i><font color="Green">Lemma42</font></i></a>, </i>;<br/><b>hence </b><a NAME="E6:32_1_1_2_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="aff_4.html#T3" target="_self">Th3</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E7:32_1_1_2_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="aff_4.html#T7" target="_self">Th7</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:32_1_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="ref" href="aff_4.html#T1" target="_self">Th1</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:32_1_1"/>
( <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> iff <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 <b>by </b><i><a class="txt" href="aff_4.html#E3:32"><i><font color="Green">E36</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:32_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:32"/>
<font color="Maroon">X</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">Y</font> <a href="aff_1.html#NR2" target="_self">is_line</a> 
 <b>by </b><i>, <a class="txt" href="aff_4.html#E1"><i><font color="Green">Lemma32</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div>
