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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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">K</font>, <font color="Olive" title="b2">P</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font>  st <font color="Olive" title="b1">K</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  holds <br/><font color="Olive" title="b2">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="Olive" title="b1">K</font>,<font color="Olive" title="b2">P</font></span><br/><b>let </b><font color="Maroon" title="c2">K</font>, <font color="Maroon" title="c3">P</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">K</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  implies <font color="Maroon" title="c3">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="c2">K</font>,<font color="Maroon" title="c3">P</font> )</span><br/>





<b>assume </b><a NAME="E1:19"/><i><font color="Green" title="E16">A1</font></i>: 
<font color="Maroon" title="c2">K</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">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="c2">K</font>,<font color="Maroon" title="c3">P</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> implies <font color="Maroon" title="c4">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="c3">P</font> )</span><br/><b>assume </b><a NAME="E1:19_1"/><i><font color="Green" title="E17">A2</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">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="c3">P</font></span><br/><b>then reconsider </b><font color="Maroon" title="c5">a</font> = <font color="Maroon" title="c4">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font> ;<br/><a NAME="E3:19_1"/>
( <font color="Maroon" title="c5">a</font>,<font color="Maroon" title="c5">a</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="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> )
 <b>by </b><i><a class="txt" href="aff_4.html#E1:19"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="aff_4.html#E1:19_1"><i><font color="Green" title="E17">A2</font></i></a>, <a class="ref" href="aff_1.html#T47" title="AFF_1:th.47">AFF_1:47</a></i>;<br/><b>hence </b><a NAME="E4:19_1"/>
<font color="Maroon" title="c4">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="c3">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:19"/>
<font color="Maroon" title="c3">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="c2">K</font>,<font color="Maroon" title="c3">P</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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