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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="E1:23"/><i><font color="Green" title="E16">A1</font></i>: 
<font color="Maroon" title="c1">AS</font> is   <a href="diraf.html#NM2" title="DIRAF:NM.2">AffinPlane</a>
 ;<br/>

<b>assume </b><a NAME="E2:23"/>
not  <a href="afproj.html#K1" title="AFPROJ:func.1">AfLines</a> <font color="Maroon" title="c1">AS</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="afproj.html#K8" title="AFPROJ:func.8">Dir_of_Planes</a> <font color="Maroon" title="c1">AS</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c2">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:23"/><i><font color="Green" title="E17">A2</font></i>: 
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afproj.html#K1" title="AFPROJ:func.1">AfLines</a> <font color="Maroon" title="c1">AS</font> &amp; <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afproj.html#K8" title="AFPROJ:func.8">Dir_of_Planes</a> <font color="Maroon" title="c1">AS</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E5:23"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">AS</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">AS</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c3">X'</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">AS</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font> ;<br/>
<a NAME="E7:23"/><i><font color="Green" title="E18">A3</font></i>: 
<font color="Maroon" title="c3">X'</font> <a href="aff_4.html#NR1" title="AFF_4:NR.1">is_plane</a> 
 
<b>by </b><i><a class="txt" href="afproj.html#E1:23"><i><font color="Green" title="E16">A1</font></i></a>, <a class="ref" href="afproj.html#T1" target="_self" title="AFPROJ:th.1">Th1</a></i>;<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="E9:23"/><i><font color="Green" title="E19">A4</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="afproj.html#K6" title="AFPROJ:func.6">PDir</a> <font color="Maroon" title="c4">X</font>
 <b>and </b><br/><a NAME="E10:23"/><i><font color="Green" title="E20">A5</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="txt" href="afproj.html#E4:23"><i><font color="Green" title="E17">A2</font></i></a>, <a class="ref" href="afproj.html#T15" target="_self" title="AFPROJ:th.15">Th15</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">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="E12:23"/><i><font color="Green" title="E21">A6</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">Y</font>
 <b>and </b><br/><a NAME="E13:23"/><i><font color="Green" title="E22">A7</font></i>: 
<font color="Maroon" title="c5">Y</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a> 
 <b>by </b><i><a class="txt" href="afproj.html#E4:23"><i><font color="Green" title="E17">A2</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">a</font>, <font color="Maroon" title="c7">b</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="E15:23"/><i><font color="Green" title="E23">A8</font></i>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font>
 <b>and </b><br/><a NAME="E16:23"/>
( <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Y</font> &amp; <font color="Maroon" title="c6">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">b</font> )
 <b>by </b><i><a class="txt" href="afproj.html#E13:23"><i><font color="Green" title="E22">A7</font></i></a>, <a class="ref" href="aff_1.html#T31" title="AFF_1:th.31">AFF_1:31</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">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="E18:23"/><i><font color="Green" title="E24">A9</font></i>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">Y'</font>
 <b>and </b><br/><a NAME="E19:23"/><i><font color="Green" title="E25">A10</font></i>: 
<font color="Maroon" title="c8">Y'</font> <a href="aff_4.html#NR1" title="AFF_4:NR.1">is_plane</a> 
 <b>and </b><br/><a NAME="E20:23"/>
<font color="Maroon" title="c4">X</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c8">Y'</font>
 <b>by </b><i><a class="txt" href="afproj.html#E9:23"><i><font color="Green" title="E19">A4</font></i></a>, <a class="txt" href="afproj.html#E10:23"><i><font color="Green" title="E20">A5</font></i></a>, <a class="txt" href="afproj.html#E12:23"><i><font color="Green" title="E21">A6</font></i></a>, <a class="txt" href="afproj.html#E15:23"><i><font color="Green" title="E23">A8</font></i></a>, <a class="ref" href="afproj.html#T10" target="_self" title="AFPROJ:th.10">Th10</a></i>;<br/>
<a NAME="E21:23"/><i><font color="Green" title="E26">A11</font></i>: 
<font color="Maroon" title="c8">Y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">X'</font>
 
<b>by </b><i><a class="txt" href="afproj.html#E7:23"><i><font color="Green" title="E18">A3</font></i></a>, <a class="txt" href="afproj.html#E19:23"><i><font color="Green" title="E25">A10</font></i></a>, <a class="ref" href="aff_4.html#T33" title="AFF_4:th.33">AFF_4:33</a></i>;<br/>
<a NAME="E22:23"/>
not <font color="Maroon" title="c8">Y'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">Y'</font>
 
;<br/>
<b>hence </b><a NAME="E23:23"/>
contradiction
 <b>by </b><i><a class="txt" href="afproj.html#E18:23"><i><font color="Green" title="E24">A9</font></i></a>, <a class="txt" href="afproj.html#E21:23"><i><font color="Green" title="E26">A11</font></i></a></i>;<br/>


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