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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> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font>;<br/><b>let </b><font color="Maroon" title="c3">A</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="E1:49"/><i><font color="Green" title="E42">A1</font></i>: 
<font color="Maroon" title="c3">A</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">p</font>, <font color="Maroon" title="c5">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="E3:49"/><i><font color="Green" title="E43">A2</font></i>: 
( <font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Maroon" title="c5">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font> &amp; <font color="Maroon" title="c4">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">q</font> )
 <b>by </b><i><a class="ref" href="aff_1.html#T31" title="AFF_1:th.31">AFF_1:31</a></i>;<br/>
<div><i><font color="Green" title="E44">A3</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:49_1"/><i><font color="Green" title="E45">A4</font></i>: 
not <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 ;<br/><b>consider </b><font color="Maroon" title="c6">P</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="E3:49_1"/><i><font color="Green" title="E46">A5</font></i>: 
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P</font> &amp; <font color="Maroon" title="c6">P</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  )
 <b>by </b><i><a class="ref" href="aff_4.html#T11" target="_self" title="AFF_4:th.11">Th11</a></i>;<br/><b>set </b><font color="Maroon" title="c7">X</font> =  <a href="aff_4.html#K1" title="AFF_4:func.1">Plane</a> <font color="Maroon" title="c6">P</font>,<font color="Maroon" title="c3">A</font>;<br/><a NAME="E4:49_1"/>
not <font color="Maroon" title="c6">P</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c3">A</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E3:49"><i><font color="Green" title="E43">A2</font></i></a>, <a class="txt" href="aff_4.html#E1:49_1"><i><font color="Green" title="E45">A4</font></i></a>, <a class="txt" href="aff_4.html#E3:49_1"><i><font color="Green" title="E46">A5</font></i></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></i>;<br/><a NAME="E5:49_1"/><b>then </b><i><font color="Green" title="E47">A6</font></i>: 
 <a href="aff_4.html#K1" title="AFF_4:func.1">Plane</a> <font color="Maroon" title="c6">P</font>,<font color="Maroon" title="c3">A</font> <a href="aff_4.html#NR1" title="AFF_4:NR.1">is_plane</a> 
 <b>by </b><i><a class="txt" href="aff_4.html#E1:49"><i><font color="Green" title="E42">A1</font></i></a>, <a class="txt" href="aff_4.html#E3:49_1"><i><font color="Green" title="E46">A5</font></i></a>, <a class="ref" href="aff_4.html#D2" target="_self" title="AFF_4:def.2">Def2</a></i>;<br/><a NAME="E6:49_1"/><i><font color="Green" title="E48">A7</font></i>: 
<font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="aff_4.html#K1" title="AFF_4:func.1">Plane</a> <font color="Maroon" title="c6">P</font>,<font color="Maroon" title="c3">A</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E3:49_1"><i><font color="Green" title="E46">A5</font></i></a>, <a class="ref" href="aff_4.html#T14" target="_self" title="AFF_4:th.14">Th14</a></i>;<br/><a NAME="E7:49_1"/>
<font color="Maroon" title="c6">P</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c6">P</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E3:49_1"><i><font color="Green" title="E46">A5</font></i></a>, <a class="ref" href="aff_1.html#T55" title="AFF_1:th.55">AFF_1:55</a></i>;<br/><a NAME="E8:49_1"/><b>then </b>
<font color="Maroon" title="c6">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="aff_4.html#K1" title="AFF_4:func.1">Plane</a> <font color="Maroon" title="c6">P</font>,<font color="Maroon" title="c3">A</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:49"><i><font color="Green" title="E42">A1</font></i></a>, <a class="txt" href="aff_4.html#E3:49"><i><font color="Green" title="E43">A2</font></i></a>, <a class="txt" href="aff_4.html#E3:49_1"><i><font color="Green" title="E46">A5</font></i></a>, <a class="txt" href="aff_4.html#E6:49_1"><i><font color="Green" title="E48">A7</font></i></a>, <a class="txt" href="aff_4.html#E24"><i><font color="Green" title="E18">Lm4</font></i></a></i>;<br/><b>hence </b><a NAME="E9:49_1"/>
 ex <font color="Olive" title="b1">X</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font> st <br/>( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">X</font> &amp; <font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">X</font> &amp; <font color="Olive" title="b1">X</font> <a href="aff_4.html#NR1" title="AFF_4:NR.1">is_plane</a>  )
 <b>by </b><i><a class="txt" href="aff_4.html#E3:49_1"><i><font color="Green" title="E46">A5</font></i></a>, <a class="txt" href="aff_4.html#E5:49_1"><i><font color="Green" title="E47">A6</font></i></a>, <a class="txt" href="aff_4.html#E6:49_1"><i><font color="Green" title="E48">A7</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:49_2"/><i><font color="Green" title="E45">A8</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 ;<br/><b>consider </b><font color="Maroon" title="c6">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="E3:49_2"/><i><font color="Green" title="E46">A9</font></i>: 
not <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">A</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:49"><i><font color="Green" title="E42">A1</font></i></a>, <a class="ref" href="aff_4.html#T12" target="_self" title="AFF_4:th.12">Th12</a></i>;<br/><b>consider </b><font color="Maroon" title="c7">P</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="E5:49_2"/><i><font color="Green" title="E47">A10</font></i>: 
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">P</font> &amp; <font color="Maroon" title="c6">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">P</font> &amp; <font color="Maroon" title="c7">P</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  )
 <b>by </b><i><a class="ref" href="aff_4.html#T11" target="_self" title="AFF_4:th.11">Th11</a></i>;<br/><b>set </b><font color="Maroon" title="c8">X</font> =  <a href="aff_4.html#K1" title="AFF_4:func.1">Plane</a> <font color="Maroon" title="c7">P</font>,<font color="Maroon" title="c3">A</font>;<br/><a NAME="E6:49_2"/>
not <font color="Maroon" title="c7">P</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c3">A</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:49_2"><i><font color="Green" title="E45">A8</font></i></a>, <a class="txt" href="aff_4.html#E3:49_2"><i><font color="Green" title="E46">A9</font></i></a>, <a class="txt" href="aff_4.html#E5:49_2"><i><font color="Green" title="E47">A10</font></i></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></i>;<br/><a NAME="E7:49_2"/><b>then </b><i><font color="Green" title="E48">A11</font></i>: 
 <a href="aff_4.html#K1" title="AFF_4:func.1">Plane</a> <font color="Maroon" title="c7">P</font>,<font color="Maroon" title="c3">A</font> <a href="aff_4.html#NR1" title="AFF_4:NR.1">is_plane</a> 
 <b>by </b><i><a class="txt" href="aff_4.html#E1:49"><i><font color="Green" title="E42">A1</font></i></a>, <a class="txt" href="aff_4.html#E5:49_2"><i><font color="Green" title="E47">A10</font></i></a>, <a class="ref" href="aff_4.html#D2" target="_self" title="AFF_4:def.2">Def2</a></i>;<br/><a NAME="E8:49_2"/>
<font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="aff_4.html#K1" title="AFF_4:func.1">Plane</a> <font color="Maroon" title="c7">P</font>,<font color="Maroon" title="c3">A</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E5:49_2"><i><font color="Green" title="E47">A10</font></i></a>, <a class="ref" href="aff_4.html#T14" target="_self" title="AFF_4:th.14">Th14</a></i>;<br/><b>hence </b><a NAME="E9:49_2"/>
 ex <font color="Olive" title="b1">X</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font> st <br/>( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">X</font> &amp; <font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">X</font> &amp; <font color="Olive" title="b1">X</font> <a href="aff_4.html#NR1" title="AFF_4:NR.1">is_plane</a>  )
 <b>by </b><i><a class="txt" href="aff_4.html#E1:49_2"><i><font color="Green" title="E45">A8</font></i></a>, <a class="txt" href="aff_4.html#E7:49_2"><i><font color="Green" title="E48">A11</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:49"/>
 ex <font color="Olive" title="b1">X</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font> st <br/>( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">X</font> &amp; <font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">X</font> &amp; <font color="Olive" title="b1">X</font> <a href="aff_4.html#NR1" title="AFF_4:NR.1">is_plane</a>  )
 <b>by </b><i><a class="txt" href="aff_4.html#E4:49"><i><font color="Green" title="E44">A3</font></i></a></i>;<br/>


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