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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> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be   <a href="incsp_1.html#NM2">LINE</a> of <span class="p1">(<span class="default"><a href="afproj.html#K13" target="_self">IncProjSp_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>











<b>assume </b><b>that </b><br/><a NAME="E1:60"/><i><font color="Green">E53</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_1.html#NR2" target="_self">is_line</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>3</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>2</sub></font> )
 <b>and </b><br/><a NAME="E2:60"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>3</sub></font>,1</span><a href="tarski.html#K4">]</a></span> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,1</span><a href="tarski.html#K4">]</a></span> )
 ;<br/>

<div><i><font color="Green">E55</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:60_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><a NAME="E2:60_1"/><b>then </b><i><font color="Green">E56</font></i>: 
 <a href="afproj.html#K5" target="_self">LDir</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="afproj.html#K5" target="_self">LDir</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="afproj.html#E1:60"><i><font color="Green">E53</font></i></a>, <a class="ref" href="afproj.html#T11" target="_self">Th11</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>7</sub></font> =  <a href="afproj.html#K5" target="_self">LDir</a> <font color="Maroon">c<sub>3</sub></font> as   <a href="incsp_1.html#NM1">POINT</a> of <span class="p1">(<span class="default"><a href="afproj.html#K13" target="_self">IncProjSp_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="txt" href="afproj.html#E1:60"><i><font color="Green">E53</font></i></a>, <a class="ref" href="afproj.html#T20" target="_self">Th20</a></i>;<br/><b>take </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font>;<br/><b>thus </b><a NAME="E4:60_1"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="afproj.html#E1:60"><i><font color="Green">E53</font></i></a>, <a class="txt" href="afproj.html#E2:60"><i><font color="Green">E54</font></i></a>, <a class="txt" href="afproj.html#E2:60_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="afproj.html#T30" target="_self">Th30</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_2"><b>now </b></a><div class="add"><b>given </b><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><a NAME="E1:60_2"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</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> )
 ;<br/><b>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="incsp_1.html#U1">Points</a> of <span class="p1">(<span class="default"><a href="afproj.html#K13" target="_self">IncProjSp_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="ref" href="afproj.html#T20" target="_self">Th20</a></i>;<br/><b>take </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font>;<br/><b>thus </b><a NAME="E3:60_2"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="afproj.html#E1:60"><i><font color="Green">E53</font></i></a>, <a class="txt" href="afproj.html#E2:60"><i><font color="Green">E54</font></i></a>, <a class="txt" href="afproj.html#E1:60_2"><i><font color="Green">E56</font></i></a>, <a class="ref" href="afproj.html#T26" target="_self">Th26</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:60"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="incsp_1.html#NM1">POINT</a> of <span class="p1">(<span class="default"><a href="afproj.html#K13" target="_self">IncProjSp_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="afproj.html#E1:60"><i><font color="Green">E53</font></i></a>, <a class="txt" href="afproj.html#E3:60"><i><font color="Green">E55</font></i></a>, <a class="ref" href="aff_4.html#T22">AFF_4:22</a></i>;<br/>


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