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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="diraf.html#NM1">AffinSpace</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>











<b>assume </b><a NAME="E1:25"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">c<sub>5</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">c<sub>6</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_4.html#K1" target="_self">Plane</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_4.html#K1" target="_self">Plane</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></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">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:25_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><b>consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:25_1"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="aff_1.html#R2">//</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="aff_4.html#E1:25"><i><font color="Green">E20</font></i></a>, <a class="txt" href="aff_4.html#E17"><i><font color="Green">Lemma14</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E6:25_1"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="aff_1.html#R2">//</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="txt" href="aff_4.html#E1:25"><i><font color="Green">E20</font></i></a>, <a class="txt" href="aff_4.html#E17"><i><font color="Green">Lemma14</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E8:25_1"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>11</sub></font> )
 <b>by </b><i><a class="txt" href="aff_4.html#E1:25"><i><font color="Green">E20</font></i></a>, <a class="ref" href="aff_1.html#T63">AFF_1:63</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E10:25_1"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="txt" href="aff_4.html#E1:25"><i><font color="Green">E20</font></i></a>, <a class="ref" href="aff_1.html#T63">AFF_1:63</a></i>;<br/><a NAME="E11:25_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E8:25_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="aff_4.html#E10:25_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="aff_1.html#T58">AFF_1:58</a></i>;<br/><a NAME="E12:25_1"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="aff_1.html#R2">//</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="aff_1.html#R2">//</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="txt" href="aff_4.html#E4:25_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="aff_4.html#E6:25_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="aff_4.html#E8:25_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="aff_4.html#E10:25_1"><i><font color="Green">E25</font></i></a>, <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">E28</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="txt" href="aff_4.html#E8:25_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="aff_4.html#E10:25_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="aff_4.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E14:25_1"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">c<sub>12</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  )
 <b>by </b><i><a class="txt" href="aff_4.html#E12:25_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="aff_1.html#T40">AFF_1:40</a></i>;<br/><div><i><font color="Green">E30</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">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/><a NAME="E2:25_1_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:25"><i><font color="Green">E20</font></i></a>, <a class="txt" href="aff_4.html#E8:25_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="aff_4.html#E10:25_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="aff_4.html#E14:25_1"><i><font color="Green">E29</font></i></a>, <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">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a>  <a href="aff_4.html#K1" target="_self">Plane</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:25"><i><font color="Green">E20</font></i></a>, <a class="txt" href="aff_4.html#E8:25_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="aff_4.html#E24"><i><font color="Green">Lemma18</font></i></a></i>;<br/><b>hence </b><a NAME="E4:25_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_4.html#K1" target="_self">Plane</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:25_1"><i><font color="Green">E21</font></i></a></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">c<sub>11</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/><a NAME="E2:25_1_2"/><b>then </b><i><font color="Green">E31</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>6</sub></font> or ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> ) )
 <b>by </b><i><a class="txt" href="aff_4.html#E1:25"><i><font color="Green">E20</font></i></a>, <a class="txt" href="aff_4.html#E4:25_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="aff_4.html#E6:25_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="aff_4.html#E8:25_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="aff_4.html#E10:25_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="aff_4.html#E11:25_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="aff_4.html#E13:25_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="aff_4.html#T18" target="_self">Th18</a></i>;<br/><div><i><font color="Green">E32</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">E33</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><div><i><font color="Green">E34</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">E35</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><a NAME="E2:25_1_2_1_1"/>
<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="aff_1.html#R2">//</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:25"><i><font color="Green">E20</font></i></a>, <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">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_4.html#K1" target="_self">Plane</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:25_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="aff_4.html#E1:25_1_2_1_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">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><a NAME="E2:25_1_2_1_2"/><b>then </b><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:25"><i><font color="Green">E20</font></i></a>, <a class="txt" href="aff_4.html#E6:25_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="aff_4.html#E1:25_1_2_1"><i><font color="Green">E33</font></i></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">E36</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><b>consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:25_1_2_1_2_1"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>13</sub></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">c<sub>3</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="aff_1.html#R2">//</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:25"><i><font color="Green">E20</font></i></a>, <a class="txt" href="aff_4.html#E1:25_1"><i><font color="Green">E21</font></i></a>, <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">c<sub>10</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="aff_1.html#R2">//</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:25_1_2_1_2_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="aff_4.html#E3:25_1_2_1_2_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">c<sub>10</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="aff_1.html#R2">//</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:25_1_2_1"><i><font color="Green">E33</font></i></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">E38</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E6:25_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="aff_4.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E8:25_1_2_1_2_1"/>
<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="aff_1.html#R2">//</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E6:25_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="aff_4.html#E2:25_1_2_1_2"><i><font color="Green">E35</font></i></a>, <a class="txt" href="aff_4.html#E3:25_1_2_1_2_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">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_4.html#K1" target="_self">Plane</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E7:25_1_2_1_2_1"><i><font color="Green">E38</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:25_1_2_1_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_4.html#K1" target="_self">Plane</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:25"><i><font color="Green">E20</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:25_1_2_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_4.html#K1" target="_self">Plane</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E2:25_1_2_1"><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">c<sub>13</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><a NAME="E1:25_1_2_2"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>and </b><br/><a NAME="E2:25_1_2_2"/><i><font color="Green">E34</font></i>: 
not <font color="Maroon">c<sub>5</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><a NAME="E3:25_1_2_2"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:25"><i><font color="Green">E20</font></i></a>, <a class="txt" href="aff_4.html#E2:25_1_2_2"><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">E36</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">E37</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><a NAME="E2:25_1_2_2_1"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:25"><i><font color="Green">E20</font></i></a>, <a class="txt" href="aff_4.html#E4:25_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="aff_4.html#E1:25_1_2_2"><i><font color="Green">E33</font></i></a>, <a class="txt" href="aff_4.html#E3:25_1_2_2"><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">E39</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">E40</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><a NAME="E2:25_1_2_2_1_1"/>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="aff_1.html#R2">//</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:25"><i><font color="Green">E20</font></i></a>, <a class="txt" href="aff_4.html#E1:25_1_2_2"><i><font color="Green">E33</font></i></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">c<sub>4</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E4:25_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="aff_4.html#E1:25_1_2_2_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="aff_4.html#E1:25_1_2_2_1_1"><i><font color="Green">E40</font></i></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">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a>  <a href="aff_4.html#K1" target="_self">Plane</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:25"><i><font color="Green">E20</font></i></a>, <a class="txt" href="aff_4.html#E24"><i><font color="Green">Lemma18</font></i></a></i>;<br/><b>hence </b><a NAME="E5:25_1_2_2_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_4.html#K1" target="_self">Plane</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:25_1"><i><font color="Green">E21</font></i></a></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">E40</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><a NAME="E2:25_1_2_2_1_2"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:25"><i><font color="Green">E20</font></i></a>, <a class="txt" href="aff_4.html#E1:25_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="aff_4.html#E1:25_1_2_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="aff_1.html#T33">AFF_1:33</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:25_1_2_2_1_2"/><i><font color="Green">E41</font></i>: 
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>14</sub></font> )
 <b>by </b><i><a class="txt" href="aff_4.html#E1:25_1_2_2_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="aff_4.html#T1" target="_self">Th1</a></i>;<br/><a NAME="E5:25_1_2_2_1_2"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E4:25_1_2_2_1_2"><i><font color="Green">E41</font></i></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">c<sub>13</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="aff_1.html#R2">//</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:25"><i><font color="Green">E20</font></i></a>, <a class="txt" href="aff_4.html#E4:25_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="aff_4.html#E1:25_1_2_2"><i><font color="Green">E33</font></i></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">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="aff_1.html#R2">//</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:25_1_2_2_1_2"><i><font color="Green">E40</font></i></a>, <a class="txt" href="aff_4.html#E5:25_1_2_2_1_2"><i><font color="Green">E42</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">E43</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:25_1_2_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="aff_4.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E9:25_1_2_2_1_2"/>
<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="aff_1.html#R2">//</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E4:25_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="aff_4.html#E2:25_1_2_2_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="aff_4.html#E4:25_1_2_2_1_2"><i><font color="Green">E41</font></i></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">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_4.html#K1" target="_self">Plane</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E8:25_1_2_2_1_2"><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">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_4.html#K1" target="_self">Plane</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E3:25_1_2_2_1"><i><font color="Green">E39</font></i></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">E37</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E2:25_1_2_2_2"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:25"><i><font color="Green">E20</font></i></a>, <a class="txt" href="aff_4.html#E6:25_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="aff_4.html#E1:25_1_2_2"><i><font color="Green">E33</font></i></a>, <a class="txt" href="aff_4.html#E3:25_1_2_2"><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">E39</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">E40</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/><a NAME="E2:25_1_2_2_2_1"/>
<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>13</sub></font> <a href="aff_1.html#R2">//</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:25"><i><font color="Green">E20</font></i></a>, <a class="txt" href="aff_4.html#E1:25_1_2_2"><i><font color="Green">E33</font></i></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">c<sub>4</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E6:25_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="aff_4.html#E1:25_1_2_2_2"><i><font color="Green">E37</font></i></a>, <a class="txt" href="aff_4.html#E1:25_1_2_2_2_1"><i><font color="Green">E40</font></i></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">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a>  <a href="aff_4.html#K1" target="_self">Plane</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:25"><i><font color="Green">E20</font></i></a>, <a class="txt" href="aff_4.html#E24"><i><font color="Green">Lemma18</font></i></a></i>;<br/><b>hence </b><a NAME="E5:25_1_2_2_2_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_4.html#K1" target="_self">Plane</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:25_1"><i><font color="Green">E21</font></i></a></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">E40</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/><a NAME="E2:25_1_2_2_2_2"/>
 <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:25"><i><font color="Green">E20</font></i></a>, <a class="txt" href="aff_4.html#E1:25_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="aff_4.html#E1:25_1_2_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="aff_1.html#T33">AFF_1:33</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E4:25_1_2_2_2_2"/><i><font color="Green">E41</font></i>: 
(  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>14</sub></font> )
 <b>by </b><i><a class="txt" href="aff_4.html#E1:25_1_2_2_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="aff_4.html#T1" target="_self">Th1</a></i>;<br/><a NAME="E5:25_1_2_2_2_2"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="analoaf.html#R2">//</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E4:25_1_2_2_2_2"><i><font color="Green">E41</font></i></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">c<sub>13</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="aff_1.html#R2">//</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:25"><i><font color="Green">E20</font></i></a>, <a class="txt" href="aff_4.html#E6:25_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="aff_4.html#E1:25_1_2_2"><i><font color="Green">E33</font></i></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">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="aff_1.html#R2">//</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:25_1_2_2_2_2"><i><font color="Green">E40</font></i></a>, <a class="txt" href="aff_4.html#E5:25_1_2_2_2_2"><i><font color="Green">E42</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">E43</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:25_1_2_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="aff_4.html#T2" target="_self">Th2</a></i>;<br/><a NAME="E9:25_1_2_2_2_2"/>
<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="aff_1.html#R2">//</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E6:25_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="aff_4.html#E2:25_1_2_2_2"><i><font color="Green">E38</font></i></a>, <a class="txt" href="aff_4.html#E4:25_1_2_2_2_2"><i><font color="Green">E41</font></i></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">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_4.html#K1" target="_self">Plane</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E8:25_1_2_2_2_2"><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">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_4.html#K1" target="_self">Plane</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E3:25_1_2_2_2"><i><font color="Green">E39</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E6:25_1_2_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_4.html#K1" target="_self">Plane</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:25"><i><font color="Green">E20</font></i></a>, <a class="txt" href="aff_4.html#E4:25_1_2_2"><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">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_4.html#K1" target="_self">Plane</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E2:25_1_2"><i><font color="Green">E31</font></i></a>, <a class="txt" href="aff_4.html#E3:25_1_2"><i><font color="Green">E32</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E17:25_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="aff_4.html#K1" target="_self">Plane</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E15:25_1"><i><font color="Green">E30</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:25"/>
<font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">c=</a>  <a href="aff_4.html#K1" target="_self">Plane</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


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