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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>let </b><font color="Maroon" title="c2">A</font>, <font color="Maroon" title="c3">K</font>, <font color="Maroon" title="c4">X</font>, <font color="Maroon" title="c5">Y</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font>;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:5"/><i><font color="Green" title="E5">A1</font></i>: 
not <font color="Maroon" title="c2">A</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c3">K</font>
 <b>and </b><br/><a NAME="E2:5"/><i><font color="Green" title="E6">A2</font></i>: 
( <font color="Maroon" title="c2">A</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="c2">A</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c5">Y</font> &amp; <font color="Maroon" title="c3">K</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="c3">K</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c5">Y</font> )
 <b>and </b><br/><a NAME="E3:5"/><i><font color="Green" title="E7">A3</font></i>: 
( <font color="Maroon" title="c2">A</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Maroon" title="c3">K</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Maroon" title="c4">X</font> <a href="aff_4.html#NR1" title="AFF_4:NR.1">is_plane</a>  &amp; <font color="Maroon" title="c5">Y</font> <a href="aff_4.html#NR1" title="AFF_4:NR.1">is_plane</a>  )
 ;<br/>

<a NAME="E4:5"/><i><font color="Green" title="E8">A4</font></i>: 
( <font color="Maroon" title="c4">X</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Maroon" title="c5">Y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="txt" href="afproj.html#E3:5"><i><font color="Green" title="E7">A3</font></i></a>, <a class="ref" href="aff_4.html#T59" title="AFF_4:th.59">AFF_4:59</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">x</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c4">X</font>;<br/>
<b>consider </b><font color="Maroon" title="c7">y</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c5">Y</font>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">a</font> = <font color="Maroon" title="c6">x</font>, <font color="Maroon" title="c9">b</font> = <font color="Maroon" title="c7">y</font> as   <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#E4:5"><i><font color="Green" title="E8">A4</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E8:5"/>
( <font color="Maroon" title="c8">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">a</font> <a href="aff_4.html#K3" title="AFF_4:func.3">+</a> <font color="Maroon" title="c4">X</font> &amp; <font color="Maroon" title="c8">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c3">K</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c8">a</font> <a href="aff_4.html#K3" title="AFF_4:func.3">+</a> <font color="Maroon" title="c4">X</font> &amp; <font color="Maroon" title="c9">b</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">b</font> <a href="aff_4.html#K3" title="AFF_4:func.3">+</a> <font color="Maroon" title="c5">Y</font> &amp; <font color="Maroon" title="c9">b</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c3">K</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">b</font> <a href="aff_4.html#K3" title="AFF_4:func.3">+</a> <font color="Maroon" title="c5">Y</font> )
 
<b>by </b><i><a class="txt" href="afproj.html#E2:5"><i><font color="Green" title="E6">A2</font></i></a>, <a class="txt" href="afproj.html#E3:5"><i><font color="Green" title="E7">A3</font></i></a>, <a class="ref" href="aff_4.html#T68" title="AFF_4:th.68">AFF_4:68</a></i>;<br/>
<a NAME="E9:5"/><b>then </b><i><font color="Green" title="E9">A5</font></i>: 
( <font color="Maroon" title="c8">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">X</font> &amp; <font color="Maroon" title="c8">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c3">K</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">X</font> &amp; <font color="Maroon" title="c9">b</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c2">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">Y</font> &amp; <font color="Maroon" title="c9">b</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c3">K</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">Y</font> )
 
<b>by </b><i><a class="txt" href="afproj.html#E3:5"><i><font color="Green" title="E7">A3</font></i></a>, <a class="txt" href="afproj.html#E4:5"><i><font color="Green" title="E8">A4</font></i></a>, <a class="ref" href="aff_4.html#T66" title="AFF_4:th.66">AFF_4:66</a></i>;<br/>
<a NAME="E10:5"/><i><font color="Green" title="E10">A6</font></i>: 
( <font color="Maroon" title="c2">A</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c8">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c2">A</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c9">b</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c3">K</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c8">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c3">K</font> &amp; <font color="Maroon" title="c3">K</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c9">b</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c3">K</font> )
 
<b>by </b><i><a class="txt" href="afproj.html#E3:5"><i><font color="Green" title="E7">A3</font></i></a>, <a class="ref" href="aff_4.html#D3" title="AFF_4:def.3">AFF_4:def 3</a></i>;<br/>
<a NAME="E11:5"/><i><font color="Green" title="E11">A7</font></i>: 
not <font color="Maroon" title="c8">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c2">A</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c8">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c3">K</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:5_1"/>
<font color="Maroon" title="c8">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c2">A</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c8">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c3">K</font>
 ;<br/>

<a NAME="E2:5_1"/><b>then </b>
<font color="Maroon" title="c8">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c2">A</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c3">K</font>
 
<b>by </b><i><a class="txt" href="afproj.html#E10:5"><i><font color="Green" title="E10">A6</font></i></a>, <a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></i>;<br/>
<b>hence </b><a NAME="E3:5_1"/>
contradiction
 <b>by </b><i><a class="txt" href="afproj.html#E1:5"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="afproj.html#E10:5"><i><font color="Green" title="E10">A6</font></i></a>, <a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:5"/>
( <font color="Maroon" title="c8">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c2">A</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c9">b</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c8">a</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c3">K</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c9">b</font> <a href="aff_4.html#K2" title="AFF_4:func.2">*</a> <font color="Maroon" title="c3">K</font> )
 
<b>by </b><i><a class="txt" href="afproj.html#E10:5"><i><font color="Green" title="E10">A6</font></i></a>, <a class="ref" href="aff_1.html#T58" title="AFF_1:th.58">AFF_1:58</a></i>;<br/>
<b>hence </b><a NAME="E13:5"/>
<font color="Maroon" title="c4">X</font> <a href="aff_4.html#R1" title="AFF_4:pred.1">'||'</a> <font color="Maroon" title="c5">Y</font>
 <b>by </b><i><a class="txt" href="afproj.html#E3:5"><i><font color="Green" title="E7">A3</font></i></a>, <a class="txt" href="afproj.html#E9:5"><i><font color="Green" title="E9">A5</font></i></a>, <a class="txt" href="afproj.html#E11:5"><i><font color="Green" title="E11">A7</font></i></a>, <a class="ref" href="aff_4.html#T55" title="AFF_4:th.55">AFF_4:55</a></i>;<br/>


</div>
