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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">K</font>, <font color="Maroon" title="c4">P</font>, <font color="Maroon" title="c5">Q</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="E1:24"/><i><font color="Green" title="E19">A1</font></i>: 
( <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">P</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Q</font> &amp; <font color="Maroon" title="c2">a</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">K</font>,<font color="Maroon" title="c4">P</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="c5">Q</font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:24_1"/><i><font color="Green" title="E20">A2</font></i>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Q</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="E3:24_1"/><i><font color="Green" title="E21">A3</font></i>: 
( <font color="Maroon" title="c2">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">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#E1:24"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="aff_4.html#E17"><i><font color="Green" title="E14">Lm3</font></i></a></i>;<br/><a NAME="E4:24_1"/><i><font color="Green" title="E22">A4</font></i>: 
 <a href="aff_4.html#K1" title="AFF_4:func.1">Plane</a> <font color="Maroon" title="c3">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="c5">Q</font>,<font color="Maroon" title="c4">P</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:24"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="aff_4.html#T16" target="_self" title="AFF_4:th.16">Th16</a></i>;<br/><a NAME="E5:24_1"/><i><font color="Green" title="E23">A5</font></i>: 
<font color="Maroon" title="c5">Q</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#E1:24"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="aff_1.html#T50" title="AFF_1:th.50">AFF_1:50</a></i>;<br/><a NAME="E6:24_1"/>
<font color="Maroon" title="c2">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="c5">Q</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:24"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="aff_4.html#E3:24_1"><i><font color="Green" title="E21">A3</font></i></a>, <a class="ref" href="aff_1.html#T57" title="AFF_1:th.57">AFF_1:57</a></i>;<br/><a NAME="E7:24_1"/><b>then </b><i><font color="Green" title="E24">A6</font></i>: 
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">Q</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:24"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="aff_4.html#E5:24_1"><i><font color="Green" title="E23">A5</font></i></a>, <a class="ref" href="aff_1.html#T37" title="AFF_1:th.37">AFF_1:37</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c8">c</font> = <font color="Maroon" title="c6">x</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="aff_4.html#E1:24_1"><i><font color="Green" title="E20">A2</font></i></a></i>;<br/><a NAME="E9:24_1"/>
<font color="Maroon" title="c8">c</font>,<font color="Maroon" title="c7">b</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c5">Q</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E1:24_1"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="aff_4.html#E5:24_1"><i><font color="Green" title="E23">A5</font></i></a>, <a class="txt" href="aff_4.html#E7:24_1"><i><font color="Green" title="E24">A6</font></i></a>, <a class="ref" href="aff_1.html#T37" title="AFF_1:th.37">AFF_1:37</a></i>;<br/><b>hence </b><a NAME="E10:24_1"/>
<font color="Maroon" title="c6">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">K</font>,<font color="Maroon" title="c4">P</font>
 <b>by </b><i><a class="txt" href="aff_4.html#E3:24_1"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="aff_4.html#E4:24_1"><i><font color="Green" title="E22">A4</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:24"/>
<font color="Maroon" title="c5">Q</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="c3">K</font>,<font color="Maroon" title="c4">P</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>


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