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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>, <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</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="E1:35"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="aff_4.html#NR1" target="_self">is_plane</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="aff_4.html#NR1" target="_self">is_plane</a>  &amp; <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>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font> )
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</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="E3:35"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="ref" href="aff_1.html#T31">AFF_1:31</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></font>, <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="E5:35"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</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> &amp; <font color="Maroon">c<sub>8</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#E1:35"><i><font color="Green">E30</font></i></a>, <a class="ref" href="aff_1.html#T31">AFF_1:31</a></i>;<br/>
<div><i><font color="Green">E33</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:35_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><a NAME="E2:35_1"/><b>then </b>
not <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:35"><i><font color="Green">E30</font></i></a>, <a class="txt" href="aff_4.html#E5:35"><i><font color="Green">E32</font></i></a>, <a class="ref" href="aff_1.html#T59">AFF_1:59</a></i>;<br/><a NAME="E3:35_1"/><b>then </b>
not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:35"><i><font color="Green">E30</font></i></a>, <a class="txt" href="aff_4.html#E3:35"><i><font color="Green">E31</font></i></a>, <a class="txt" href="aff_4.html#E34"><i><font color="Green">Lemma28</font></i></a></i>;<br/><b>hence </b><a NAME="E4:35_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:35"><i><font color="Green">E30</font></i></a>, <a class="txt" href="aff_4.html#E3:35"><i><font color="Green">E31</font></i></a>, <a class="txt" href="aff_4.html#E5:35"><i><font color="Green">E32</font></i></a>, <a class="ref" href="aff_4.html#T25" target="_self">Th25</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2"><b>now </b></a><div class="add"><b>given </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><a NAME="E1:35_2"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</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> )
 ;<br/><b>consider </b><font color="Maroon">c<sub>11</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="E3:35_2"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</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#E1:35"><i><font color="Green">E30</font></i></a>, <a class="ref" href="aff_1.html#T32">AFF_1:32</a></i>;<br/><a NAME="E4:35_2"/><i><font color="Green">E36</font></i>: 
not <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:35"><i><font color="Green">E30</font></i></a>, <a class="txt" href="aff_4.html#E1:35_2"><i><font color="Green">E34</font></i></a>, <a class="txt" href="aff_4.html#E3:35_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">E37</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:35_2_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/><a NAME="E2:35_2_1"/><b>then </b>
not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:35"><i><font color="Green">E30</font></i></a>, <a class="txt" href="aff_4.html#E3:35"><i><font color="Green">E31</font></i></a>, <a class="txt" href="aff_4.html#E1:35_2"><i><font color="Green">E34</font></i></a>, <a class="txt" href="aff_4.html#E4:35_2"><i><font color="Green">E36</font></i></a>, <a class="txt" href="aff_4.html#E34"><i><font color="Green">Lemma28</font></i></a></i>;<br/><b>hence </b><a NAME="E3:35_2_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:35"><i><font color="Green">E30</font></i></a>, <a class="txt" href="aff_4.html#E3:35"><i><font color="Green">E31</font></i></a>, <a class="txt" href="aff_4.html#E1:35_2"><i><font color="Green">E34</font></i></a>, <a class="txt" href="aff_4.html#E3:35_2"><i><font color="Green">E35</font></i></a>, <a class="ref" href="aff_4.html#T25" target="_self">Th25</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:35_2_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/><a NAME="E2:35_2_2"/><b>then </b>
not  <a href="aff_1.html#R1">LIN</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:35"><i><font color="Green">E30</font></i></a>, <a class="txt" href="aff_4.html#E3:35"><i><font color="Green">E31</font></i></a>, <a class="txt" href="aff_4.html#E1:35_2"><i><font color="Green">E34</font></i></a>, <a class="txt" href="aff_4.html#E4:35_2"><i><font color="Green">E36</font></i></a>, <a class="txt" href="aff_4.html#E34"><i><font color="Green">Lemma28</font></i></a></i>;<br/><b>hence </b><a NAME="E3:35_2_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:35"><i><font color="Green">E30</font></i></a>, <a class="txt" href="aff_4.html#E3:35"><i><font color="Green">E31</font></i></a>, <a class="txt" href="aff_4.html#E1:35_2"><i><font color="Green">E34</font></i></a>, <a class="txt" href="aff_4.html#E3:35_2"><i><font color="Green">E35</font></i></a>, <a class="ref" href="aff_4.html#T25" target="_self">Th25</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E7:35_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E3:35"><i><font color="Green">E31</font></i></a>, <a class="txt" href="aff_4.html#E5:35_2"><i><font color="Green">E37</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:35"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:35"><i><font color="Green">E30</font></i></a>, <a class="txt" href="aff_4.html#E6:35"><i><font color="Green">E33</font></i></a>, <a class="ref" href="aff_4.html#T22" target="_self">Th22</a></i>;<br/>


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