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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#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <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#K14" target="_self">ProjHorizon</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="subset_1.html#M1">Element</a> of the <a href="incsp_1.html#U2">Lines</a> of <span class="p1">(<span class="default"><a href="afproj.html#K14" target="_self">ProjHorizon</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:45"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="afproj.html#K5" target="_self">LDir</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="afproj.html#K6" target="_self">PDir</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>and </b><br/><a NAME="E2:45"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="aff_4.html#NR1" target="_self">is_plane</a>  )
 ;<br/>

<div><i><font color="Green">E40</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:45_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="aff_4.html#R1">'||'</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E2:45_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="afproj.html#K12" target="_self">Inc_of_Dir</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="afproj.html#E1:45"><i><font color="Green">E38</font></i></a>, <a class="txt" href="afproj.html#E2:45"><i><font color="Green">E39</font></i></a>, <a class="ref" href="afproj.html#D12" target="_self">Def12</a></i>;<br/><a NAME="E3:45_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="incsp_1.html#U3">Inc</a> of <span class="p1">(<span class="default"><a href="afproj.html#K14" target="_self">ProjHorizon</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/><b>hence </b><a NAME="E4:45_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="incsp_1.html#D1">INCSP_1:def 1</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:45_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><a NAME="E2:45_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="incsp_1.html#U3">Inc</a> of <span class="p1">(<span class="default"><a href="afproj.html#K14" target="_self">ProjHorizon</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="incsp_1.html#D1">INCSP_1:def 1</a></i>;<br/><a NAME="E3:45_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="afproj.html#K12" target="_self">Inc_of_Dir</a> <font color="Maroon">c<sub>1</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#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E5:45_2"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="afproj.html#K5" target="_self">LDir</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="afproj.html#K6" target="_self">PDir</a> <font color="Maroon">c<sub>7</sub></font> &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>7</sub></font> <a href="aff_4.html#NR1" target="_self">is_plane</a>  &amp; <font color="Maroon">c<sub>6</sub></font> <a href="aff_4.html#R1">'||'</a> <font color="Maroon">c<sub>7</sub></font> )
 <b>by </b><i><a class="ref" href="afproj.html#D12" target="_self">Def12</a></i>;<br/><a NAME="E6:45_2"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="aff_1.html#R4">//</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="aff_4.html#R1">'||'</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="afproj.html#E1:45"><i><font color="Green">E38</font></i></a>, <a class="txt" href="afproj.html#E2:45"><i><font color="Green">E39</font></i></a>, <a class="txt" href="afproj.html#E5:45_2"><i><font color="Green">E41</font></i></a>, <a class="ref" href="afproj.html#T11" target="_self">Th11</a>, <a class="ref" href="afproj.html#T13" target="_self">Th13</a></i>;<br/><a NAME="E7:45_2"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="aff_4.html#R1">'||'</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="afproj.html#E5:45_2"><i><font color="Green">E41</font></i></a>, <a class="ref" href="aff_4.html#T56">AFF_4:56</a></i>;<br/><a NAME="E8:45_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="afproj.html#E5:45_2"><i><font color="Green">E41</font></i></a>, <a class="ref" href="aff_4.html#T59">AFF_4:59</a></i>;<br/><b>hence </b><a NAME="E9:45_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="aff_4.html#R1">'||'</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="afproj.html#E6:45_2"><i><font color="Green">E42</font></i></a>, <a class="txt" href="afproj.html#E7:45_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="aff_4.html#T60">AFF_4:60</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:45"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">c<sub>5</sub></font> iff <font color="Maroon">c<sub>2</sub></font> <a href="aff_4.html#R1">'||'</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="afproj.html#E3:45"><i><font color="Green">E40</font></i></a></i>;<br/>


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