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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">X</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="E1:62"/><i><font color="Green" title="E51">A1</font></i>: 
<font color="Maroon" title="c1">AS</font> is  not   <a href="diraf.html#NM2" title="DIRAF:NM.2">AffinPlane</a>
 <b>and </b><br/><a NAME="E2:62"/><i><font color="Green" title="E52">A2</font></i>: 
<font color="Maroon" title="c2">X</font> <a href="aff_4.html#NR1" title="AFF_4:NR.1">is_plane</a> 
 ;<br/>

<b>assume </b><a NAME="E3:62"/><i><font color="Green" title="E53">A3</font></i>: 
for <font color="Olive" title="b1">q</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font> holds  <font color="Olive" title="b1">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 ;<br/>

<a NAME="E4:62"/><i><font color="Green" title="E54">A4</font></i>: 
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> <font color="Maroon" title="c2">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_1"><b>proof </b></a><div class="add">

<a NAME="E1:62_1"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">AS</font> holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 
<b>by </b><i><a class="txt" href="aff_4.html#E3:62"><i><font color="Green" title="E53">A3</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:62_1"/>
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> <font color="Maroon" title="c2">X</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:62"/>
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font>, <font color="Olive" title="b4">d</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font>  st not <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b3">c</font>,<font color="Olive" title="b4">d</font> holds <br/> ex <font color="Olive" title="b5">q</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font> st <br/>( <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b5">q</font> &amp; <font color="Olive" title="b3">c</font>,<font color="Olive" title="b4">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Olive" title="b3">c</font>,<font color="Olive" title="b5">q</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="62_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</font>, <font color="Maroon" title="c5">c</font>, <font color="Maroon" title="c6">d</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font>;<br/>







<b>assume </b><a NAME="E1:62_2"/><i><font color="Green" title="E55">A5</font></i>: 
not <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">d</font>
 ;<br/>

<a NAME="E2:62_2"/><b>then </b><i><font color="Green" title="E56">A6</font></i>: 
( <font color="Maroon" title="c3">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">b</font> &amp; <font color="Maroon" title="c5">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">d</font> )
 
<b>by </b><i><a class="ref" href="aff_1.html#T12" title="AFF_1:th.12">AFF_1:12</a></i>;<br/>
<b>set </b><font color="Maroon" title="c7">M</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>;<br/>
<b>set </b><font color="Maroon" title="c8">N</font> =  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">d</font>;<br/>
<a NAME="E3:62_2"/>
( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c5">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c6">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> )
 
<b>by </b><i><a class="txt" href="aff_4.html#E4:62"><i><font color="Green" title="E54">A4</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<a NAME="E4:62_2"/><b>then </b><i><font color="Green" title="E57">A7</font></i>: 
(  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font> &amp;  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">d</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font> )
 
<b>by </b><i><a class="txt" href="aff_4.html#E2:62"><i><font color="Green" title="E52">A2</font></i></a>, <a class="txt" href="aff_4.html#E2:62_2"><i><font color="Green" title="E56">A6</font></i></a>, <a class="ref" href="aff_4.html#T19" target="_self" title="AFF_4:th.19">Th19</a></i>;<br/>
<a NAME="E5:62_2"/><i><font color="Green" title="E58">A8</font></i>: 
(  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp;  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">d</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  )
 
<b>by </b><i><a class="txt" href="aff_4.html#E2:62_2"><i><font color="Green" title="E56">A6</font></i></a>, <a class="ref" href="aff_1.html#D3" title="AFF_1:def.3">AFF_1:def 3</a></i>;<br/>
<a NAME="E6:62_2"/><b>then </b><i><font color="Green" title="E59">A9</font></i>: 
(  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</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="c5">c</font>,<font color="Maroon" title="c6">d</font> or  ex <font color="Olive" title="b1">q</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font> st <br/>( <font color="Olive" title="b1">q</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="c3">a</font>,<font color="Maroon" title="c4">b</font> &amp; <font color="Olive" title="b1">q</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="c5">c</font>,<font color="Maroon" title="c6">d</font> ) )
 
<b>by </b><i><a class="txt" href="aff_4.html#E2:62"><i><font color="Green" title="E52">A2</font></i></a>, <a class="txt" href="aff_4.html#E4:62_2"><i><font color="Green" title="E57">A7</font></i></a>, <a class="ref" href="aff_4.html#T22" target="_self" title="AFF_4:th.22">Th22</a></i>;<br/>
<a NAME="E7:62_2"/><i><font color="Green" title="E60">A10</font></i>: 
( <font color="Maroon" title="c3">a</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="c3">a</font>,<font color="Maroon" title="c4">b</font> &amp; <font color="Maroon" title="c4">b</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="c3">a</font>,<font color="Maroon" title="c4">b</font> &amp; <font color="Maroon" title="c5">c</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="c5">c</font>,<font color="Maroon" title="c6">d</font> &amp; <font color="Maroon" title="c6">d</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="c5">c</font>,<font color="Maroon" title="c6">d</font> )
 
<b>by </b><i><a class="ref" href="aff_1.html#T26" title="AFF_1:th.26">AFF_1:26</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c9">q</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="E9:62_2"/><i><font color="Green" title="E61">A11</font></i>: 
( <font color="Maroon" title="c9">q</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="c3">a</font>,<font color="Maroon" title="c4">b</font> &amp; <font color="Maroon" title="c9">q</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="c5">c</font>,<font color="Maroon" title="c6">d</font> )
 <b>by </b><i><a class="txt" href="aff_4.html#E1:62_2"><i><font color="Green" title="E55">A5</font></i></a>, <a class="txt" href="aff_4.html#E6:62_2"><i><font color="Green" title="E59">A9</font></i></a>, <a class="ref" href="aff_1.html#T53" title="AFF_1:th.53">AFF_1:53</a></i>;<br/>
<a NAME="E10:62_2"/>
(  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font>,<font color="Maroon" title="c9">q</font> &amp;  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">d</font>,<font color="Maroon" title="c9">q</font> )
 
<b>by </b><i><a class="txt" href="aff_4.html#E5:62_2"><i><font color="Green" title="E58">A8</font></i></a>, <a class="txt" href="aff_4.html#E7:62_2"><i><font color="Green" title="E60">A10</font></i></a>, <a class="txt" href="aff_4.html#E9:62_2"><i><font color="Green" title="E61">A11</font></i></a>, <a class="ref" href="aff_1.html#T33" title="AFF_1:th.33">AFF_1:33</a></i>;<br/>
<a NAME="E11:62_2"/><b>then </b>
( <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c9">q</font> &amp; <font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c9">q</font> )
 
<b>by </b><i><a class="ref" href="aff_1.html#D1" title="AFF_1:def.1">AFF_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E12:62_2"/>
 ex <font color="Olive" title="b1">q</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font> st <br/>( <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c3">a</font>,<font color="Olive" title="b1">q</font> &amp; <font color="Maroon" title="c5">c</font>,<font color="Maroon" title="c6">d</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c5">c</font>,<font color="Olive" title="b1">q</font> )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:62"/>
contradiction
 <b>by </b><i><a class="txt" href="aff_4.html#E1:62"><i><font color="Green" title="E51">A1</font></i></a>, <a class="ref" href="diraf.html#D8" title="DIRAF:def.8">DIRAF:def 8</a></i>;<br/>


</div>
