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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">p</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>, <font color="Maroon" title="c4">C</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="E1:58"/><i><font color="Green" title="E51">A1</font></i>: 
( <font color="Maroon" title="c3">A</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Maroon" title="c2">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="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">C</font> )
 ;<br/>

<a NAME="E2:58"/>
for <font color="Olive" title="b1">A</font>, <font color="Olive" title="b2">C</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">AS</font><br/> for <font color="Olive" title="b3">p</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font>  st <font color="Olive" title="b1">A</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Olive" title="b2">C</font> &amp; <font color="Olive" title="b3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">A</font> &amp; <font color="Olive" title="b3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">C</font> holds <br/><font color="Olive" title="b1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">C</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">A</font>, <font color="Maroon" title="c6">C</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>let </b><font color="Maroon" title="c7">p</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>assume </b><a NAME="E1:58_1"/><i><font color="Green" title="E52">A2</font></i>: 
( <font color="Maroon" title="c5">A</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c6">C</font> &amp; <font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font> &amp; <font color="Maroon" title="c7">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">C</font> )
 ;<br/>

<a NAME="E2:58_1"/><b>then </b><i><font color="Green" title="E53">A3</font></i>: 
( <font color="Maroon" title="c5">A</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  &amp; <font color="Maroon" title="c6">C</font> <a href="aff_1.html#NR2" title="AFF_1:NR.2">is_line</a>  )
 
<b>by </b><i><a class="ref" href="aff_1.html#T50" target="_self" title="AFF_1:th.50">Th50</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:58_1_1"/><i><font color="Green" title="E54">A4</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">A</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c9">x'</font> = <font color="Maroon" title="c8">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/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:58_1_1_1"/><i><font color="Green" title="E55">A5</font></i>: 
<font color="Maroon" title="c9">x'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">p</font>
 ;<br/><a NAME="E2:58_1_1_1"/><b>then </b>
<font color="Maroon" title="c5">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c7">p</font>,<font color="Maroon" title="c9">x'</font>
 <b>by </b><i><a class="txt" href="aff_1.html#E1:58_1"><i><font color="Green" title="E52">A2</font></i></a>, <a class="txt" href="aff_1.html#E2:58_1"><i><font color="Green" title="E53">A3</font></i></a>, <a class="txt" href="aff_1.html#E1:58_1_1"><i><font color="Green" title="E54">A4</font></i></a>, <a class="txt" href="aff_1.html#E37"><i><font color="Green" title="E25">Lm6</font></i></a></i>;<br/><a NAME="E3:58_1_1_1"/><b>then </b><i><font color="Green" title="E56">A6</font></i>: 
<font color="Maroon" title="c7">p</font>,<font color="Maroon" title="c9">x'</font> <a href="aff_1.html#R2" title="AFF_1:pred.2">//</a> <font color="Maroon" title="c6">C</font>
 <b>by </b><i><a class="txt" href="aff_1.html#E1:58_1"><i><font color="Green" title="E52">A2</font></i></a>, <a class="txt" href="aff_1.html#E1:58_1_1_1"><i><font color="Green" title="E55">A5</font></i></a>, <a class="ref" href="aff_1.html#T43" target="_self" title="AFF_1:th.43">Th43</a>, <a class="ref" href="aff_1.html#T57" target="_self" title="AFF_1:th.57">Th57</a></i>;<br/><b>consider </b><font color="Maroon" title="c10">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="E5:58_1_1_1"/><i><font color="Green" title="E57">A7</font></i>: 
( <font color="Maroon" title="c7">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">q</font> &amp; <font color="Maroon" title="c10">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">C</font> )
 <b>by </b><i><a class="txt" href="aff_1.html#E2:58_1"><i><font color="Green" title="E53">A3</font></i></a>, <a class="ref" href="aff_1.html#T32" target="_self" title="AFF_1:th.32">Th32</a></i>;<br/><a NAME="E6:58_1_1_1"/><i><font color="Green" title="E58">A8</font></i>: 
<font color="Maroon" title="c6">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="aff_1.html#K2" title="AFF_1:func.2">Line</a> <font color="Maroon" title="c7">p</font>,<font color="Maroon" title="c10">q</font>
 <b>by </b><i><a class="txt" href="aff_1.html#E1:58_1"><i><font color="Green" title="E52">A2</font></i></a>, <a class="txt" href="aff_1.html#E2:58_1"><i><font color="Green" title="E53">A3</font></i></a>, <a class="txt" href="aff_1.html#E5:58_1_1_1"><i><font color="Green" title="E57">A7</font></i></a>, <a class="txt" href="aff_1.html#E37"><i><font color="Green" title="E25">Lm6</font></i></a></i>;<br/><a NAME="E7:58_1_1_1"/>
<font color="Maroon" title="c7">p</font>,<font color="Maroon" title="c9">x'</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">p</font>,<font color="Maroon" title="c10">q</font>
 <b>by </b><i><a class="txt" href="aff_1.html#E1:58_1"><i><font color="Green" title="E52">A2</font></i></a>, <a class="txt" href="aff_1.html#E2:58_1"><i><font color="Green" title="E53">A3</font></i></a>, <a class="txt" href="aff_1.html#E3:58_1_1_1"><i><font color="Green" title="E56">A6</font></i></a>, <a class="txt" href="aff_1.html#E5:58_1_1_1"><i><font color="Green" title="E57">A7</font></i></a>, <a class="ref" href="aff_1.html#T41" target="_self" title="AFF_1:th.41">Th41</a></i>;<br/><a NAME="E8:58_1_1_1"/><b>then </b>
<font color="Maroon" title="c7">p</font>,<font color="Maroon" title="c10">q</font> <a href="analoaf.html#R2" title="ANALOAF:pred.2">//</a> <font color="Maroon" title="c7">p</font>,<font color="Maroon" title="c9">x'</font>
 <b>by </b><i><a class="ref" href="aff_1.html#T13" target="_self" title="AFF_1:th.13">Th13</a></i>;<br/><a NAME="E9:58_1_1_1"/><b>then </b>
 <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">p</font>,<font color="Maroon" title="c10">q</font>,<font color="Maroon" title="c9">x'</font>
 <b>by </b><i><a class="ref" href="aff_1.html#D1" target="_self" title="AFF_1:def.1">Def1</a></i>;<br/><b>hence </b><a NAME="E10:58_1_1_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">C</font>
 <b>by </b><i><a class="txt" href="aff_1.html#E6:58_1_1_1"><i><font color="Green" title="E58">A8</font></i></a>, <a class="ref" href="aff_1.html#D2" target="_self" title="AFF_1:def.2">Def2</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:58_1_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">C</font>
 <b>by </b><i><a class="txt" href="aff_1.html#E1:58_1"><i><font color="Green" title="E52">A2</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:58_1"/>
<font color="Maroon" title="c5">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">C</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:58"/><b>then </b>
( <font color="Maroon" title="c3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">C</font> &amp; <font color="Maroon" title="c4">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">A</font> )
 
<b>by </b><i><a class="txt" href="aff_1.html#E1:58"><i><font color="Green" title="E51">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:58"/>
<font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">C</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>


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