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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">Y</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font>;<br/><b>let </b><font color="Maroon">X</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font>;<br/><b>let </b><font color="Maroon">a</font> be   <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">AS</font></span>)</span>;<br/><b>let </b><font color="Maroon">A</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">AS</font></span>)</span>;<br/>









<b>assume </b><a NAME="E1:38"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="afproj.html#K5" target="_self">LDir</a> <font color="Maroon">Y</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">X</font>,1</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> &amp; <font color="Maroon">Y</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">X</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  )
 ;<br/>

<div><i><font color="Green">E26</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:38_1"/>
<font color="Maroon">a</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A</font>
 ;<br/><a NAME="E2:38_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">A</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#K13" target="_self">IncProjSp_of</a> <font color="Maroon">AS</font></span>)</span>
 <b>by </b><i><a class="ref" href="incsp_1.html#D1">INCSP_1:def 1</a></i>;<br/><a NAME="E3:38_1"/><b>then </b><i><font color="Green">E30</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="afproj.html#K5" target="_self">LDir</a> <font color="Maroon">Y</font></span>)</span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">X</font>,1</span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="afproj.html#K11" target="_self">Proj_Inc</a> <font color="Maroon">AS</font>
 <b>by </b><i/>;<br/><div><i><font color="Green">E31</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">K</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font><b> such that </b><a NAME="E1:38_1_1"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">K</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">X</font>,1</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">K</font>,1</span><a href="tarski.html#K4">]</a></span> )
 <b>and </b><br/><a NAME="E2:38_1_1"/><i><font color="Green">E34</font></i>: 
( (  <a href="afproj.html#K5" target="_self">LDir</a> <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">AS</font> &amp;  <a href="afproj.html#K5" target="_self">LDir</a> <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> ) or  <a href="afproj.html#K5" target="_self">LDir</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="afproj.html#K5" target="_self">LDir</a> <font color="Maroon">K</font> )
 ;<br/><a NAME="E3:38_1_1"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">K</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">K</font> )
 <b>by </b><i><a class="txt" href="afproj.html#E1:38_1_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><a NAME="E4:38_1_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">K</font> <a href="hidden.html#R2">in</a>  <a href="afproj.html#K1" target="_self">AfLines</a> <font color="Maroon">AS</font>
 <b>by </b><i><a class="txt" href="afproj.html#E1:38_1_1"><i><font color="Green">E33</font></i></a></i>;<br/><div><i><font color="Green">E38</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:38_1_1_1"/><i><font color="Green">E47</font></i>: 
(  <a href="afproj.html#K5" target="_self">LDir</a> <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">AS</font> &amp;  <a href="afproj.html#K5" target="_self">LDir</a> <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">K</font> )
 ;<br/><a NAME="E2:38_1_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="afproj.html#K7" target="_self">Dir_of_Lines</a> <font color="Maroon">AS</font>
 <b>by </b><i>, </i>;<br/><a NAME="E3:38_1_1_1"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">AS</font> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="afproj.html#K7" target="_self">Dir_of_Lines</a> <font color="Maroon">AS</font></span>)</span> <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#E5:38_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E4:38_1_1_1"/><b>then </b>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">AS</font> <a href="subset_1.html#R2">meets</a>  <a href="afproj.html#K7" target="_self">Dir_of_Lines</a> <font color="Maroon">AS</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><b>hence </b><a NAME="E5:38_1_1_1"/>
contradiction
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:38_1_1_2"/><i><font color="Green">E48</font></i>: 
 <a href="afproj.html#K5" target="_self">LDir</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="afproj.html#K5" target="_self">LDir</a> <font color="Maroon">K</font>
 ;<br/><a NAME="E2:38_1_1_2"/><i><font color="Green">E49</font></i>: 
(  <a href="afproj.html#K5" target="_self">LDir</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="afproj.html#K3" target="_self">LinesParallelity</a> <font color="Maroon">AS</font></span>)</span>,<font color="Maroon">Y</font> &amp;  <a href="afproj.html#K5" target="_self">LDir</a> <font color="Maroon">K</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="afproj.html#K3" target="_self">LinesParallelity</a> <font color="Maroon">AS</font></span>)</span>,<font color="Maroon">K</font> )
 <b>by </b><i>, <a class="txt" href="afproj.html#E1:38_1_1"><i><font color="Green">E33</font></i></a>, </i>;<br/><a NAME="E3:38_1_1_2"/><b>then </b>
<font color="Maroon">Y</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="afproj.html#K3" target="_self">LinesParallelity</a> <font color="Maroon">AS</font></span>)</span>,<font color="Maroon">K</font>
 <b>by </b><i>, <a class="txt" href="afproj.html#E5:38_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="eqrel_1.html#T31">EQREL_1:31</a></i>;<br/><b>then consider </b><font color="Maroon">K'</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font><b> such that </b><br/><a NAME="E5:38_1_1_2"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">K'</font> &amp; <font color="Maroon">K'</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">K</font> <a href="aff_4.html#R1">'||'</a> <font color="Maroon">K'</font> )
 <b>by </b><i><a class="txt" href="afproj.html#E1:38_1_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="afproj.html#T2" target="_self">Th2</a>, </i>;<br/><a NAME="E6:38_1_1_2"/>
<font color="Maroon">K</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">K'</font>
 <b>by </b><i><a class="txt" href="afproj.html#E1:38_1_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="afproj.html#T3" target="_self">Th3</a>, <a class="ref" href="aff_4.html#T40">AFF_4:40</a></i>;<br/><b>hence </b><a NAME="E7:38_1_1_2"/>
<font color="Maroon">Y</font> <a href="aff_4.html#R1">'||'</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="txt" href="afproj.html#E3:38_1_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="afproj.html#T3" target="_self">Th3</a>, <a class="ref" href="aff_4.html#T40">AFF_4:40</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E7:38_1_1"/>
<font color="Maroon">Y</font> <a href="aff_4.html#R1">'||'</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="txt" href="afproj.html#E2:38_1_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="afproj.html#E4:38_1_1"><i><font color="Green">E37</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E5:38_1"/>
for <font color="Olive">K</font>, <font color="Olive">X'</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font>  holds <br/>(  not <font color="Olive">K</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  or  not <font color="Olive">X'</font> <a href="aff_4.html#NR1" target="_self">is_plane</a>  or  not  <a href="afproj.html#K5" target="_self">LDir</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="afproj.html#K5" target="_self">LDir</a> <font color="Olive">K</font> or  not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">X</font>,1</span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="afproj.html#K6" target="_self">PDir</a> <font color="Olive">X'</font></span>)</span>,2</span><a href="tarski.html#K4">]</a></span> or  not <font color="Olive">K</font> <a href="aff_4.html#R1">'||'</a> <font color="Olive">X'</font> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T33">ZFMISC_1:33</a></i>;<br/><b>hence </b><a NAME="E6:38_1"/>
<font color="Maroon">Y</font> <a href="aff_4.html#R1">'||'</a> <font color="Maroon">X</font>
 <b>by </b><i><a class="txt" href="afproj.html#E4:38_1"><i><font color="Green">E31</font></i></a>, , </i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:38_2"/>
<font color="Maroon">Y</font> <a href="aff_4.html#R1">'||'</a> <font color="Maroon">X</font>
 ;<br/><a NAME="E2:38_2"/><b>then </b>
<font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="afproj.html#K5" target="_self">LDir</a> <font color="Maroon">Y</font>
 <b>by </b><i>, </i>;<br/><a NAME="E3:38_2"/><b>then </b>
( <font color="Maroon">X</font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="afproj.html#K3" target="_self">LinesParallelity</a> <font color="Maroon">AS</font></span>)</span>,<font color="Maroon">Y</font> &amp; <font color="Maroon">Y</font> <a href="hidden.html#R2">in</a>  <a href="afproj.html#K1" target="_self">AfLines</a> <font color="Maroon">AS</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E4:38_2"/><b>then </b><i><font color="Green">E52</font></i>: 
 <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="afproj.html#K3" target="_self">LinesParallelity</a> <font color="Maroon">AS</font></span>)</span>,<font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="afproj.html#K3" target="_self">LinesParallelity</a> <font color="Maroon">AS</font></span>)</span>,<font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#T31">EQREL_1:31</a></i>;<br/><a NAME="E5:38_2"/>
(  <a href="afproj.html#K5" target="_self">LDir</a> <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="afproj.html#K3" target="_self">LinesParallelity</a> <font color="Maroon">AS</font></span>)</span>,<font color="Maroon">X</font> &amp;  <a href="afproj.html#K5" target="_self">LDir</a> <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="afproj.html#K3" target="_self">LinesParallelity</a> <font color="Maroon">AS</font></span>)</span>,<font color="Maroon">Y</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E6:38_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">A</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="afproj.html#K11" target="_self">Proj_Inc</a> <font color="Maroon">AS</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E7:38_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">A</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#K13" target="_self">IncProjSp_of</a> <font color="Maroon">AS</font></span>)</span>
 ;<br/><b>hence </b><a NAME="E8:38_2"/>
<font color="Maroon">a</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A</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>
<b>hence </b><a NAME="E4:38"/>
( <font color="Maroon">a</font> <a href="incsp_1.html#R1">on</a> <font color="Maroon">A</font> iff <font color="Maroon">Y</font> <a href="aff_4.html#R1">'||'</a> <font color="Maroon">X</font> )
 <b>by </b><i/>;<br/>


</div>
