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<b>let </b><font color="Maroon" title="c1">AS</font> be   <a href="diraf.html#NM1" title="DIRAF:NM.1">AffinSpace</a>;<br/>

<b>assume </b><a NAME="E1:69"/><i><font color="Green" title="E54">A1</font></i>: 
 <a href="afproj.html#K13" title="AFPROJ:func.13">IncProjSp_of</a> <font color="Maroon" title="c1">AS</font> is <a href="incproj.html#V5" title="INCPROJ:attr.5">2-dimensional</a>
 ;<br/>

<b>assume </b><a NAME="E2:69"/><i><font color="Green" title="E55">A2</font></i>: 
<font color="Maroon" title="c1">AS</font> is  not   <a href="diraf.html#NM2" title="DIRAF:NM.2">AffinPlane</a>
 ;<br/>

<b>consider </b><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font>;<br/>
<b>consider </b><font color="Maroon" title="c4">X</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font><b> such that </b><br/><a NAME="E5:69"/><i><font color="Green" title="E56">A3</font></i>: 
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> &amp; <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> &amp; <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font> )
 <b>and </b><br/><a NAME="E6:69"/><i><font color="Green" title="E57">A4</font></i>: 
<font color="Maroon" title="c4">X</font> <a href="aff_4.html#NR1" title="AFF_4:NR.1">is_plane</a> 
 <b>by </b><i><a class="ref" href="aff_4.html#T37" title="AFF_4:th.37">AFF_4:37</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">z</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font><b> such that </b><br/><a NAME="E8:69"/><i><font color="Green" title="E58">A5</font></i>: 
not <font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">X</font>
 <b>by </b><i><a class="txt" href="afproj.html#E2:69"><i><font color="Green" title="E55">A2</font></i></a>, <a class="txt" href="afproj.html#E6:69"><i><font color="Green" title="E57">A4</font></i></a>, <a class="ref" href="aff_4.html#T48" title="AFF_4:th.48">AFF_4:48</a></i>;<br/>
<b>set </b><font color="Maroon" title="c6">Y</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c5">z</font>;<br/>
<a NAME="E9:69"/><i><font color="Green" title="E59">A6</font></i>: 
(  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c5">z</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c5">z</font> &amp; <font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c5">z</font> )
 
<b>by </b><i><a class="txt" href="afproj.html#E5:69"><i><font color="Green" title="E56">A3</font></i></a>, <a class="txt" href="afproj.html#E8:69"><i><font color="Green" title="E58">A5</font></i></a>, <a class="ref" href="aff_1.html#T26" title="AFF_1:th.26">AFF_1:26</a>, <a class="ref" href="aff_1.html#D3" title="AFF_1:def.3">AFF_1:def 3</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c7">A</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="afproj.html#K6" title="AFPROJ:func.6">PDir</a> <font color="Maroon" title="c4">X</font></span>)</span>,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>, <font color="Maroon" title="c8">K</font> = <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c5">z</font></span>)</span>,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> as   <a href="incsp_1.html#NM2" title="INCSP_1:NM.2">LINE</a> of <span class="p1">(<span class="default"><a href="afproj.html#K13" title="AFPROJ:func.13">IncProjSp_of</a> <font color="Maroon" title="c1">AS</font></span>)</span> <b>by </b><i><a class="txt" href="afproj.html#E6:69"><i><font color="Green" title="E57">A4</font></i></a>, <a class="ref" href="afproj.html#T23" target="_self" title="AFPROJ:th.23">Th23</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c9">a</font> being   <a href="incsp_1.html#NM1" title="INCSP_1:NM.1">POINT</a> of <span class="p1">(<span class="default"><a href="afproj.html#K13" title="AFPROJ:func.13">IncProjSp_of</a> <font color="Maroon" title="c1">AS</font></span>)</span><b> such that </b><br/><a NAME="E12:69"/><i><font color="Green" title="E60">A7</font></i>: 
( <font color="Maroon" title="c9">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c7">A</font> &amp; <font color="Maroon" title="c9">a</font> <a href="incsp_1.html#R1" title="INCSP_1:pred.1">on</a> <font color="Maroon" title="c8">K</font> )
 <b>by </b><i><a class="txt" href="afproj.html#E1:69"><i><font color="Green" title="E54">A1</font></i></a>, <a class="ref" href="incproj.html#D14" title="INCPROJ:def.14">INCPROJ:def 14</a></i>;<br/>
<a NAME="E13:69"/>
<font color="Maroon" title="c9">a</font> is  not   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font>
 
<b>by </b><i><a class="txt" href="afproj.html#E6:69"><i><font color="Green" title="E57">A4</font></i></a>, <a class="txt" href="afproj.html#E12:69"><i><font color="Green" title="E60">A7</font></i></a>, <a class="ref" href="afproj.html#T27" target="_self" title="AFPROJ:th.27">Th27</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c10">Y'</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font><b> such that </b><br/><a NAME="E15:69"/><i><font color="Green" title="E61">A8</font></i>: 
( <font color="Maroon" title="c9">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afproj.html#K5" title="AFPROJ:func.5">LDir</a> <font color="Maroon" title="c10">Y'</font> &amp; <font color="Maroon" title="c10">Y'</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  )
 <b>by </b><i><a class="ref" href="afproj.html#T20" target="_self" title="AFPROJ:th.20">Th20</a></i>;<br/>
<a NAME="E16:69"/><i><font color="Green" title="E62">A9</font></i>: 
( <font color="Maroon" title="c10">Y'</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c4">X</font> &amp; <font color="Maroon" title="c10">Y'</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c5">z</font> )
 
<b>by </b><i><a class="txt" href="afproj.html#E6:69"><i><font color="Green" title="E57">A4</font></i></a>, <a class="txt" href="afproj.html#E9:69"><i><font color="Green" title="E59">A6</font></i></a>, <a class="txt" href="afproj.html#E12:69"><i><font color="Green" title="E60">A7</font></i></a>, <a class="txt" href="afproj.html#E15:69"><i><font color="Green" title="E61">A8</font></i></a>, <a class="ref" href="afproj.html#T28" target="_self" title="AFPROJ:th.28">Th28</a>, <a class="ref" href="afproj.html#T29" target="_self" title="AFPROJ:th.29">Th29</a></i>;<br/>
<a NAME="E17:69"/><b>then </b>
<font color="Maroon" title="c10">Y'</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c5">z</font>
 
<b>by </b><i><a class="txt" href="afproj.html#E9:69"><i><font color="Green" title="E59">A6</font></i></a>, <a class="txt" href="afproj.html#E15:69"><i><font color="Green" title="E61">A8</font></i></a>, <a class="ref" href="aff_4.html#T40" title="AFF_4:th.40">AFF_4:40</a></i>;<br/>
<a NAME="E18:69"/><b>then </b>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c5">z</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c4">X</font>
 
<b>by </b><i><a class="txt" href="afproj.html#E16:69"><i><font color="Green" title="E62">A9</font></i></a>, <a class="ref" href="aff_4.html#T56" title="AFF_4:th.56">AFF_4:56</a></i>;<br/>
<a NAME="E19:69"/><b>then </b>
 <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c5">z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">X</font>
 
<b>by </b><i><a class="txt" href="afproj.html#E5:69"><i><font color="Green" title="E56">A3</font></i></a>, <a class="txt" href="afproj.html#E6:69"><i><font color="Green" title="E57">A4</font></i></a>, <a class="txt" href="afproj.html#E9:69"><i><font color="Green" title="E59">A6</font></i></a>, <a class="ref" href="aff_4.html#T43" title="AFF_4:th.43">AFF_4:43</a></i>;<br/>
<b>hence </b><a NAME="E20:69"/>
contradiction
 <b>by </b><i><a class="txt" href="afproj.html#E8:69"><i><font color="Green" title="E58">A5</font></i></a>, <a class="txt" href="afproj.html#E9:69"><i><font color="Green" title="E59">A6</font></i></a></i>;<br/>


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