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<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">K</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font>;<br/><b>let </b><font color="Maroon">P</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font>;<br/><b>let </b><font color="Maroon">Q</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font>;<br/>











<b>assume </b><a NAME="E1:25"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">K</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">P</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">Q</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_4.html#K1" target="_self">Plane</a> <font color="Maroon">K</font>,<font color="Maroon">P</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="aff_4.html#K1" target="_self">Plane</a> <font color="Maroon">K</font>,<font color="Maroon">P</font> &amp; <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#R2">in</a> <font color="Maroon">Q</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:25_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 ;<br/><b>then reconsider </b><font color="Maroon">c</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">AS</font> ;<br/><b>consider </b><font color="Maroon">a'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font><b> such that </b><br/><a NAME="E4:25_1"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">a</font>,<font color="Maroon">a'</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">K</font> &amp; <font color="Maroon">a'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> )
 <b>by </b><i>, </i>;<br/><b>consider </b><font color="Maroon">b'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font><b> such that </b><br/><a NAME="E6:25_1"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">b</font>,<font color="Maroon">b'</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">K</font> &amp; <font color="Maroon">b'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> )
 <b>by </b><i>, </i>;<br/><b>consider </b><font color="Maroon">X</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font><b> such that </b><br/><a NAME="E8:25_1"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> &amp; <font color="Maroon">K</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">X</font> )
 <b>by </b><i>, <a class="ref" href="aff_1.html#T63">AFF_1:63</a></i>;<br/><b>consider </b><font color="Maroon">Y</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font><b> such that </b><br/><a NAME="E10:25_1"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">K</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">Y</font> )
 <b>by </b><i>, <a class="ref" href="aff_1.html#T63">AFF_1:63</a></i>;<br/><a NAME="E11:25_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">X</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">Y</font>
 <b>by </b><i>, , <a class="ref" href="aff_1.html#T58">AFF_1:58</a></i>;<br/><a NAME="E12:25_1"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">a</font>,<font color="Maroon">a'</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">X</font> &amp; <font color="Maroon">b</font>,<font color="Maroon">b'</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">Y</font> )
 <b>by </b><i>, , , , <a class="ref" href="aff_1.html#T57">AFF_1:57</a></i>;<br/><a NAME="E13:25_1"/><b>then </b><i><font color="Green">E49</font></i>: 
( <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">Y</font> )
 <b>by </b><i>, , </i>;<br/><a NAME="E14:25_1"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">X</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">Y</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/><div><i><font color="Green">E59</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:25_1_1"/>
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font>
 ;<br/><a NAME="E2:25_1_1"/><b>then </b>
<font color="Maroon">Q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font>
 <b>by </b><i>, , , , <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/><a NAME="E3:25_1_1"/><b>then </b>
<font color="Maroon">Q</font> <a href="tarski.html#R1">c=</a>  <a href="aff_4.html#K1" target="_self">Plane</a> <font color="Maroon">K</font>,<font color="Maroon">P</font>
 <b>by </b><i>, , </i>;<br/><b>hence </b><a NAME="E4:25_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="aff_4.html#K1" target="_self">Plane</a> <font color="Maroon">K</font>,<font color="Maroon">P</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:25_1_2"/>
<font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">Y</font>
 ;<br/><a NAME="E2:25_1_2"/><b>then </b><i><font color="Green">E60</font></i>: 
( <font color="Maroon">P</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">Q</font> or  ex <font color="Olive">q</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font> st <br/>( <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> ) )
 <b>by </b><i>, , , , , , , </i>;<br/><div><i><font color="Green">E61</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:25_1_2_1"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">P</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">Q</font>
 ;<br/><div><i><font color="Green">E63</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:25_1_2_1_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Q</font>
 ;<br/><a NAME="E2:25_1_2_1_1"/>
<font color="Maroon">c</font>,<font color="Maroon">c</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">K</font>
 <b>by </b><i>, <a class="ref" href="aff_1.html#T47">AFF_1:47</a></i>;<br/><b>hence </b><a NAME="E3:25_1_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="aff_4.html#K1" target="_self">Plane</a> <font color="Maroon">K</font>,<font color="Maroon">P</font>
 <b>by </b><i>, <a class="txt" href="aff_4.html#E1:25_1"><i><font color="Green">E35</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_2_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:25_1_2_1_2"/>
<font color="Maroon">P</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">Q</font>
 ;<br/><a NAME="E2:25_1_2_1_2"/><b>then </b><i><font color="Green">E66</font></i>: 
<font color="Maroon">b</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_4.html#T1" target="_self">Th1</a>, <a class="ref" href="aff_1.html#T59">AFF_1:59</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_2_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:25_1_2_1_2_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font>
 ;<br/><b>consider </b><font color="Maroon">c'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font><b> such that </b><br/><a NAME="E3:25_1_2_1_2_1"/><i><font color="Green">E70</font></i>: 
( <font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">b'</font>,<font color="Maroon">c'</font> &amp; <font color="Maroon">b</font>,<font color="Maroon">b'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c</font>,<font color="Maroon">c'</font> )
 <b>by </b><i><a class="ref" href="diraf.html#T47">DIRAF:47</a></i>;<br/><a NAME="E4:25_1_2_1_2_1"/>
<font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">Q</font>
 <b>by </b><i>, , <a class="ref" href="aff_1.html#T37">AFF_1:37</a></i>;<br/><a NAME="E5:25_1_2_1_2_1"/><b>then </b>
<font color="Maroon">b'</font>,<font color="Maroon">c'</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">Q</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E4:25_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="aff_4.html#E6:25_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="aff_1.html#T46">AFF_1:46</a></i>;<br/><a NAME="E6:25_1_2_1_2_1"/><b>then </b>
<font color="Maroon">b'</font>,<font color="Maroon">c'</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="aff_4.html#T1" target="_self">Th1</a>, <a class="ref" href="aff_1.html#T57">AFF_1:57</a></i>;<br/><a NAME="E7:25_1_2_1_2_1"/><b>then </b><i><font color="Green">E71</font></i>: 
<font color="Maroon">c'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i>, </i>;<br/><a NAME="E8:25_1_2_1_2_1"/>
<font color="Maroon">c</font>,<font color="Maroon">c'</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">K</font>
 <b>by </b><i>, <a class="txt" href="aff_4.html#E1:25_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="aff_4.html#E6:25_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="aff_1.html#T46">AFF_1:46</a></i>;<br/><b>hence </b><a NAME="E9:25_1_2_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="aff_4.html#K1" target="_self">Plane</a> <font color="Maroon">K</font>,<font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="aff_4.html#T2" target="_self">Th2</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:25_1_2_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="aff_4.html#K1" target="_self">Plane</a> <font color="Maroon">K</font>,<font color="Maroon">P</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:25_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="aff_4.html#K1" target="_self">Plane</a> <font color="Maroon">K</font>,<font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:25"><i><font color="Green">E34</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_2_2"><b>now </b></a><div class="add"><b>given </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><a NAME="E1:25_1_2_2"/><i><font color="Green">E72</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> )
 <b>and </b><br/><a NAME="E2:25_1_2_2"/><i><font color="Green">E73</font></i>: 
not <font color="Maroon">P</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">Q</font>
 ;<br/><a NAME="E3:25_1_2_2"/><i><font color="Green">E74</font></i>: 
<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="txt" href="aff_4.html#E1:25"><i><font color="Green">E34</font></i></a>, <a class="ref" href="aff_1.html#T55">AFF_1:55</a></i>;<br/><div><i><font color="Green">E75</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_2_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:25_1_2_2_1"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a</font>
 ;<br/><a NAME="E2:25_1_2_2_1"/><b>then </b><i><font color="Green">E77</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a'</font>
 <b>by </b><i>, , <a class="ref" href="aff_4.html#T1" target="_self">Th1</a>, <a class="txt" href="aff_4.html#E1:25_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/><div><i><font color="Green">E78</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_2_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:25_1_2_2_1_1"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a'</font>
 ;<br/><a NAME="E2:25_1_2_2_1_1"/>
<font color="Maroon">a</font>,<font color="Maroon">q</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">Q</font>
 <b>by </b><i>, <a class="ref" href="aff_4.html#T1" target="_self">Th1</a>, <a class="ref" href="aff_1.html#T37">AFF_1:37</a></i>;<br/><a NAME="E3:25_1_2_2_1_1"/><b>then </b>
<font color="Maroon">K</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">Q</font>
 <b>by </b><i>, <a class="txt" href="aff_4.html#E6:25_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="aff_4.html#T4" target="_self">Th4</a>, <a class="ref" href="aff_1.html#T67">AFF_1:67</a></i>;<br/><a NAME="E4:25_1_2_2_1_1"/><b>then </b>
<font color="Maroon">Q</font> <a href="tarski.html#R1">c=</a>  <a href="aff_4.html#K1" target="_self">Plane</a> <font color="Maroon">K</font>,<font color="Maroon">P</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E5:25_1_2_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="aff_4.html#K1" target="_self">Plane</a> <font color="Maroon">K</font>,<font color="Maroon">P</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_2_2_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:25_1_2_2_1_2"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a'</font>
 ;<br/><a NAME="E2:25_1_2_2_1_2"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">q</font>,<font color="Maroon">a</font>,<font color="Maroon">c</font>
 <b>by </b><i>, , <a class="ref" href="aff_4.html#T1" target="_self">Th1</a>, <a class="ref" href="aff_1.html#T33">AFF_1:33</a></i>;<br/><b>then consider </b><font color="Maroon">c'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font><b> such that </b><br/><a NAME="E4:25_1_2_2_1_2"/><i><font color="Green">E81</font></i>: 
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">q</font>,<font color="Maroon">a'</font>,<font color="Maroon">c'</font> &amp; <font color="Maroon">a</font>,<font color="Maroon">a'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c</font>,<font color="Maroon">c'</font> )
 <b>by </b><i><a class="txt" href="aff_4.html#E6:25_1"><i><font color="Green">E37</font></i></a>, </i>;<br/><a NAME="E5:25_1_2_2_1_2"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">q</font>,<font color="Maroon">a'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">q</font>,<font color="Maroon">c'</font>
 <b>by </b><i><a class="ref" href="aff_4.html#T7" target="_self">Th7</a>, <a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/><a NAME="E6:25_1_2_2_1_2"/>
<font color="Maroon">q</font>,<font color="Maroon">a'</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">P</font>
 <b>by </b><i>, , <a class="ref" href="aff_4.html#T1" target="_self">Th1</a>, <a class="ref" href="aff_1.html#T37">AFF_1:37</a></i>;<br/><a NAME="E7:25_1_2_2_1_2"/><b>then </b>
<font color="Maroon">q</font>,<font color="Maroon">c'</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="aff_4.html#T4" target="_self">Th4</a>, <a class="txt" href="aff_4.html#E9"><i><font color="Green">Lemma42</font></i></a>, <a class="ref" href="aff_1.html#T46">AFF_1:46</a></i>;<br/><a NAME="E8:25_1_2_2_1_2"/><b>then </b><i><font color="Green">E83</font></i>: 
<font color="Maroon">c'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="aff_4.html#T1" target="_self">Th1</a>, </i>;<br/><a NAME="E9:25_1_2_2_1_2"/>
<font color="Maroon">c</font>,<font color="Maroon">c'</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">K</font>
 <b>by </b><i>, <a class="ref" href="aff_4.html#T2" target="_self">Th2</a>, <a class="ref" href="aff_4.html#T7" target="_self">Th7</a>, <a class="ref" href="aff_1.html#T46">AFF_1:46</a></i>;<br/><b>hence </b><a NAME="E10:25_1_2_2_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="aff_4.html#K1" target="_self">Plane</a> <font color="Maroon">K</font>,<font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E8:25_1"><i><font color="Green">E43</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:25_1_2_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="aff_4.html#K1" target="_self">Plane</a> <font color="Maroon">K</font>,<font color="Maroon">P</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="25_1_2_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:25_1_2_2_2"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font>
 ;<br/><a NAME="E2:25_1_2_2_2"/><b>then </b><i><font color="Green">E85</font></i>: 
<font color="Maroon">b</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_4.html#T1" target="_self">Th1</a>, <a class="txt" href="aff_4.html#E1:25_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="aff_1.html#T30">AFF_1:30</a></i>;<br/><div><i><font color="Green">E86</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_2_2_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:25_1_2_2_2_1"/><i><font color="Green">E87</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b'</font>
 ;<br/><a NAME="E2:25_1_2_2_2_1"/>
<font color="Maroon">b</font>,<font color="Maroon">q</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">Q</font>
 <b>by </b><i>, <a class="ref" href="aff_4.html#T1" target="_self">Th1</a>, <a class="ref" href="aff_1.html#T37">AFF_1:37</a></i>;<br/><a NAME="E3:25_1_2_2_2_1"/><b>then </b>
<font color="Maroon">K</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">Q</font>
 <b>by </b><i>, <a class="txt" href="aff_4.html#E6:25_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="aff_4.html#T4" target="_self">Th4</a>, <a class="ref" href="aff_1.html#T67">AFF_1:67</a></i>;<br/><a NAME="E4:25_1_2_2_2_1"/><b>then </b>
<font color="Maroon">Q</font> <a href="tarski.html#R1">c=</a>  <a href="aff_4.html#K1" target="_self">Plane</a> <font color="Maroon">K</font>,<font color="Maroon">P</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E5:25_1_2_2_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="aff_4.html#K1" target="_self">Plane</a> <font color="Maroon">K</font>,<font color="Maroon">P</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_2_2_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:25_1_2_2_2_2"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b'</font>
 ;<br/><a NAME="E2:25_1_2_2_2_2"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">q</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font>
 <b>by </b><i>, , <a class="ref" href="aff_4.html#T1" target="_self">Th1</a>, <a class="ref" href="aff_1.html#T33">AFF_1:33</a></i>;<br/><b>then consider </b><font color="Maroon">c'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font><b> such that </b><br/><a NAME="E4:25_1_2_2_2_2"/><i><font color="Green">E89</font></i>: 
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">q</font>,<font color="Maroon">b'</font>,<font color="Maroon">c'</font> &amp; <font color="Maroon">b</font>,<font color="Maroon">b'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c</font>,<font color="Maroon">c'</font> )
 <b>by </b><i><a class="txt" href="aff_4.html#E6:25_1"><i><font color="Green">E37</font></i></a>, </i>;<br/><a NAME="E5:25_1_2_2_2_2"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">q</font>,<font color="Maroon">b'</font> <a href="analoaf.html#R2">//</a> <font color="Maroon">q</font>,<font color="Maroon">c'</font>
 <b>by </b><i><a class="ref" href="aff_4.html#T7" target="_self">Th7</a>, <a class="ref" href="aff_1.html#D1">AFF_1:def 1</a></i>;<br/><a NAME="E6:25_1_2_2_2_2"/>
<font color="Maroon">q</font>,<font color="Maroon">b'</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">P</font>
 <b>by </b><i>, , <a class="ref" href="aff_4.html#T1" target="_self">Th1</a>, <a class="ref" href="aff_1.html#T37">AFF_1:37</a></i>;<br/><a NAME="E7:25_1_2_2_2_2"/><b>then </b>
<font color="Maroon">q</font>,<font color="Maroon">c'</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="aff_4.html#T4" target="_self">Th4</a>, <a class="txt" href="aff_4.html#E9"><i><font color="Green">Lemma42</font></i></a>, <a class="ref" href="aff_1.html#T46">AFF_1:46</a></i>;<br/><a NAME="E8:25_1_2_2_2_2"/><b>then </b><i><font color="Green">E91</font></i>: 
<font color="Maroon">c'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="aff_4.html#T1" target="_self">Th1</a>, </i>;<br/><a NAME="E9:25_1_2_2_2_2"/>
<font color="Maroon">c</font>,<font color="Maroon">c'</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">K</font>
 <b>by </b><i>, <a class="ref" href="aff_4.html#T2" target="_self">Th2</a>, <a class="ref" href="aff_4.html#T7" target="_self">Th7</a>, <a class="ref" href="aff_1.html#T46">AFF_1:46</a></i>;<br/><b>hence </b><a NAME="E10:25_1_2_2_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="aff_4.html#K1" target="_self">Plane</a> <font color="Maroon">K</font>,<font color="Maroon">P</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E8:25_1"><i><font color="Green">E43</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:25_1_2_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="aff_4.html#K1" target="_self">Plane</a> <font color="Maroon">K</font>,<font color="Maroon">P</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="E6:25_1_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="aff_4.html#K1" target="_self">Plane</a> <font color="Maroon">K</font>,<font color="Maroon">P</font>
 <b>by </b><i>, <a class="txt" href="aff_4.html#E4:25_1"><i><font color="Green">E36</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:25_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="aff_4.html#K1" target="_self">Plane</a> <font color="Maroon">K</font>,<font color="Maroon">P</font>
 <b>by </b><i>, <a class="txt" href="aff_4.html#E1"><i><font color="Green">Lemma32</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E17:25_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="aff_4.html#K1" target="_self">Plane</a> <font color="Maroon">K</font>,<font color="Maroon">P</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:25"/>
<font color="Maroon">Q</font> <a href="tarski.html#R1">c=</a>  <a href="aff_4.html#K1" target="_self">Plane</a> <font color="Maroon">K</font>,<font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


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