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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">K</font>, <font color="Maroon" title="c3">M</font>, <font color="Maroon" title="c4">P</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><a NAME="E1:21"/><i><font color="Green" title="E17">A1</font></i>: 
<font color="Maroon" title="c2">K</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c3">M</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>thus </b><a NAME="E1:21_1"/>
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_4.html#K1" title="AFF_4:func.1">Plane</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c4">P</font> iff <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_4.html#K1" title="AFF_4:func.1">Plane</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">P</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1"><b>proof </b></a><div class="add">

<div><i><font color="Green" title="E18">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:21_1_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_4.html#K1" title="AFF_4:func.1">Plane</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c4">P</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c6">a</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="E3:21_1_1_1"/><i><font color="Green" title="E19">A3</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font>
 <b>and </b><br/><a NAME="E4:21_1_1_1"/><i><font color="Green" title="E20">A4</font></i>: 
 ex <font color="Olive" title="b1">b</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="c6">a</font>,<font color="Olive" title="b1">b</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c2">K</font> &amp; <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font> )
 ;<br/><b>consider </b><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="E6:21_1_1_1"/><i><font color="Green" title="E21">A5</font></i>: 
( <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c2">K</font> &amp; <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font> )
 <b>by </b><i><a class="txt" href="aff_4.html#E4:21_1_1_1"><i><font color="Green" title="E20">A4</font></i></a></i>;<br/><a NAME="E7:21_1_1_1"/>
<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c3">M</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:21"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="aff_4.html#E6:21_1_1_1"><i><font color="Green" title="E21">A5</font></i></a>, <a class="ref" href="aff_1.html#T57" title="AFF_1:th.57">AFF_1:57</a></i>;<br/><b>hence </b><a NAME="E8:21_1_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_4.html#K1" title="AFF_4:func.1">Plane</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">P</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E3:21_1_1_1"><i><font color="Green" title="E19">A3</font></i></a>, <a class="txt" href="aff_4.html#E6:21_1_1_1"><i><font color="Green" title="E21">A5</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:21_1_1_2"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_4.html#K1" title="AFF_4:func.1">Plane</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">P</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c6">a</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="E3:21_1_1_2"/><i><font color="Green" title="E19">A6</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font>
 <b>and </b><br/><a NAME="E4:21_1_1_2"/><i><font color="Green" title="E20">A7</font></i>: 
 ex <font color="Olive" title="b1">b</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="c6">a</font>,<font color="Olive" title="b1">b</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font> )
 ;<br/><b>consider </b><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="E6:21_1_1_2"/><i><font color="Green" title="E21">A8</font></i>: 
( <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c3">M</font> &amp; <font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">P</font> )
 <b>by </b><i><a class="txt" href="aff_4.html#E4:21_1_1_2"><i><font color="Green" title="E20">A7</font></i></a></i>;<br/><a NAME="E7:21_1_1_2"/>
<font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c2">K</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:21"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="aff_4.html#E6:21_1_1_2"><i><font color="Green" title="E21">A8</font></i></a>, <a class="ref" href="aff_1.html#T57" title="AFF_1:th.57">AFF_1:57</a></i>;<br/><b>hence </b><a NAME="E8:21_1_1_2"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_4.html#K1" title="AFF_4:func.1">Plane</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c4">P</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E3:21_1_1_2"><i><font color="Green" title="E19">A6</font></i></a>, <a class="txt" href="aff_4.html#E6:21_1_1_2"><i><font color="Green" title="E21">A8</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:21_1_1"/>
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_4.html#K1" title="AFF_4:func.1">Plane</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c4">P</font> iff <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="aff_4.html#K1" title="AFF_4:func.1">Plane</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">P</font> )
 <b>by </b><i><a class="txt" href="aff_4.html#E1:21_1_1"><i><font color="Green" title="E18">A2</font></i></a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:21"/>
 <a href="aff_4.html#K1" title="AFF_4:func.1">Plane</a> <font color="Maroon" title="c2">K</font>,<font color="Maroon" title="c4">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aff_4.html#K1" title="AFF_4:func.1">Plane</a> <font color="Maroon" title="c3">M</font>,<font color="Maroon" title="c4">P</font>
 <b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/>


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