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<b>let </b><font color="Maroon">AS</font> be   <a href="diraf.html#NM1">AffinSpace</a>;<br/><b>let </b><font color="Maroon">K</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font>;<br/><b>let </b><font color="Maroon">P</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">AS</font>;<br/>





<b>assume </b><a NAME="E1:20"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">K</font> <a href="aff_1.html#R4">//</a> <font color="Maroon">P</font>
 ;<br/>

<a NAME="E2:20"/><b>then </b><i><font color="Green">E35</font></i>: 
( <font color="Maroon">K</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  &amp; <font color="Maroon">P</font> <a href="aff_1.html#NR2" target="_self">is_line</a>  )
 
<b>by </b><i><a class="ref" href="aff_1.html#T50">AFF_1:50</a></i>;<br/>
<b>set </b><font color="Maroon">X</font> =  <a href="aff_4.html#K1" target="_self">Plane</a> <font color="Maroon">K</font>,<font color="Maroon">P</font>;<br/>
<a NAME="E3:20"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">P</font> <a href="tarski.html#R1">c=</a>  <a href="aff_4.html#K1" target="_self">Plane</a> <font color="Maroon">K</font>,<font color="Maroon">P</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:20"/>
 <a href="aff_4.html#K1" target="_self">Plane</a> <font color="Maroon">K</font>,<font color="Maroon">P</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:20_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="aff_4.html#K1" target="_self">Plane</a> <font color="Maroon">K</font>,<font color="Maroon">P</font>
 ;<br/><a NAME="E2:20_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">a</font> where <font color="Olive">a</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font> :  ex <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font> st <br/>( <font color="Olive">a</font>,<font color="Olive">b</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">K</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font><b> such that </b><br/><a NAME="E4:20_1_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>and </b><br/><a NAME="E5:20_1_1"/><i><font color="Green">E43</font></i>: 
 ex <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font> st <br/>( <font color="Maroon">a</font>,<font color="Olive">b</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">K</font> &amp; <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> )
 ;<br/><b>consider </b><font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">AS</font><b> such that </b><br/><a NAME="E7:20_1_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">K</font>
 <b>and </b><br/><a NAME="E8:20_1_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i/>;<br/><a NAME="E9:20_1_1"/>
<font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">P</font>
 <b>by </b><i>, , <a class="ref" href="aff_1.html#T57">AFF_1:57</a></i>;<br/><a NAME="E10:20_1_1"/><b>then </b>
<font color="Maroon">b</font>,<font color="Maroon">a</font> <a href="aff_1.html#R2">//</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="aff_1.html#T48">AFF_1:48</a></i>;<br/><b>hence </b><a NAME="E11:20_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font>
 <b>by </b><i>, , , <a class="ref" href="aff_1.html#T37">AFF_1:37</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:20_1"/>
 <a href="aff_4.html#K1" target="_self">Plane</a> <font color="Maroon">K</font>,<font color="Maroon">P</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:20"/>
 <a href="aff_4.html#K1" target="_self">Plane</a> <font color="Maroon">K</font>,<font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


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