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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">X</font>, <font color="Olive" title="b2">Y</font>, <font color="Olive" title="b3">M</font>, <font color="Olive" title="b4">N</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">X</font> is  <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a>  &amp; <font color="Olive" title="b2">Y</font> is  <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a>  &amp; <font color="Olive" title="b3">M</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Olive" title="b4">N</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Olive" title="b3">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">X</font> &amp; <font color="Olive" title="b4">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">X</font> &amp; <font color="Olive" title="b3">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">Y</font> &amp; <font color="Olive" title="b4">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">Y</font> &amp; <font color="Olive" title="b3">M</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b4">N</font> holds <br/><font color="Olive" title="b1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">Y</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">X</font>, <font color="Maroon" title="c3">Y</font>, <font color="Maroon" title="c4">M</font>, <font color="Maroon" title="c5">N</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">X</font> is  <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a>  &amp; <font color="Maroon" title="c3">Y</font> is  <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a>  &amp; <font color="Maroon" title="c4">M</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c5">N</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a>  &amp; <font color="Maroon" title="c4">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c5">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c4">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Y</font> &amp; <font color="Maroon" title="c5">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Y</font> &amp; <font color="Maroon" title="c4">M</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">N</font> implies <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Y</font> )</span><br/>









<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:34"/><span class="lab"><font color="Green" title="E30">A1</font></span>: 
<font color="Maroon" title="c2">X</font> is  <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a> 
 <span class="kw">and </span><br/><a NAME="E2:34"/><span class="lab"><font color="Green" title="E31">A2</font></span>: 
<font color="Maroon" title="c3">Y</font> is  <a href="aff_4.html#V1" title="AFF_4:attr.1">being_plane</a> 
 <span class="kw">and </span><br/><a NAME="E3:34"/><span class="lab"><font color="Green" title="E32">A3</font></span>: 
<font color="Maroon" title="c4">M</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">and </span><br/><a NAME="E4:34"/><span class="lab"><font color="Green" title="E33">A4</font></span>: 
<font color="Maroon" title="c5">N</font> is  <a href="aff_1.html#V1" title="AFF_1:attr.1">being_line</a> 
 <span class="kw">and </span><br/><a NAME="E5:34"/><span class="lab"><font color="Green" title="E34">A5</font></span>: 
( <font color="Maroon" title="c4">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c5">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font> )
 <span class="kw">and </span><br/><a NAME="E6:34"/><span class="lab"><font color="Green" title="E35">A6</font></span>: 
( <font color="Maroon" title="c4">M</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Y</font> &amp; <font color="Maroon" title="c5">N</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">Y</font> )
 <span class="kw">and </span><br/><a NAME="E7:34"/><span class="lab"><font color="Green" title="E36">A7</font></span>: 
<font color="Maroon" title="c4">M</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">N</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Y</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c6">a</font>, <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><span class="kw"> such that </span><br/><a NAME="E9:34"/><span class="lab"><font color="Green" title="E37">A8</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M</font>
 <span class="kw">and </span><br/><a NAME="E10:34"/><span class="lab"><font color="Green" title="E38">A9</font></span>: 
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M</font>
 <span class="kw">and </span><br/><a NAME="E11:34"/><span class="lab"><font color="Green" title="E39">A10</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34"><span class="lab"><font color="Green" title="E32">A3</font></span></a>, <a class="ref" href="aff_1.html#T31" title="AFF_1:th.31">AFF_1:31</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E40">A11</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1"><span class="kw">now </span></a><div class="add"><span class="kw">given </span><font color="Maroon" title="c8">q</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font><span class="kw"> such that </span><a NAME="E2:34_1"/><span class="lab"><font color="Green" title="E41">A12</font></span>: 
<font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M</font>
 <span class="kw">and </span><br/><a NAME="E3:34_1"/><span class="lab"><font color="Green" title="E42">A13</font></span>: 
<font color="Maroon" title="c8">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">N</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Y</font></span><br/><span class="kw">consider </span><font color="Maroon" title="c9">p</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font><span class="kw"> such that </span><br/><a NAME="E5:34_1"/><span class="lab"><font color="Green" title="E43">A14</font></span>: 
<font color="Maroon" title="c8">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">p</font>
 <span class="kw">and </span><br/><a NAME="E6:34_1"/><span class="lab"><font color="Green" title="E44">A15</font></span>: 
<font color="Maroon" title="c9">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">N</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:34"><span class="lab"><font color="Green" title="E33">A4</font></span></a>, <a class="ref" href="aff_1.html#T32" title="AFF_1:th.32">AFF_1:32</a></span>;<br/><a NAME="E7:34_1"/><span class="lab"><font color="Green" title="E45">A16</font></span>: 
not <font color="Maroon" title="c9">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34"><span class="lab"><font color="Green" title="E32">A3</font></span></a>, <a class="txt" href="#E4:34"><span class="lab"><font color="Green" title="E33">A4</font></span></a>, <a class="txt" href="#E7:34"><span class="lab"><font color="Green" title="E36">A7</font></span></a>, <a class="txt" href="#E2:34_1"><span class="lab"><font color="Green" title="E41">A12</font></span></a>, <a class="txt" href="#E3:34_1"><span class="lab"><font color="Green" title="E42">A13</font></span></a>, <a class="txt" href="#E5:34_1"><span class="lab"><font color="Green" title="E43">A14</font></span></a>, <a class="txt" href="#E6:34_1"><span class="lab"><font color="Green" title="E44">A15</font></span></a>, <a class="ref" href="aff_1.html#T30" title="AFF_1:th.30">AFF_1:30</a></span>;<br/><div><span class="lab"><font color="Green" title="E46">A17</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:34_1_1"/>
<font color="Maroon" title="c7">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Y</font></span><br/><a NAME="E2:34_1_1"/><span class="kw">then </span>
not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">q</font>,<font color="Maroon" title="c9">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34"><span class="lab"><font color="Green" title="E32">A3</font></span></a>, <a class="txt" href="#E10:34"><span class="lab"><font color="Green" title="E38">A9</font></span></a>, <a class="txt" href="#E2:34_1"><span class="lab"><font color="Green" title="E41">A12</font></span></a>, <a class="txt" href="#E7:34_1"><span class="lab"><font color="Green" title="E45">A16</font></span></a>, <a class="txt" href="aff_4.html#E34"><span class="lab"><font color="Green" title="E28">Lm7</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:34_1_1"/>
<font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34"><span class="lab"><font color="Green" title="E30">A1</font></span></a>, <a class="txt" href="#E2:34"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="txt" href="#E5:34"><span class="lab"><font color="Green" title="E34">A5</font></span></a>, <a class="txt" href="#E6:34"><span class="lab"><font color="Green" title="E35">A6</font></span></a>, <a class="txt" href="#E10:34"><span class="lab"><font color="Green" title="E38">A9</font></span></a>, <a class="txt" href="#E2:34_1"><span class="lab"><font color="Green" title="E41">A12</font></span></a>, <a class="txt" href="#E6:34_1"><span class="lab"><font color="Green" title="E44">A15</font></span></a>, <a class="ref" href="aff_4.html#T25" target="_self" title="AFF_4:th.25">Th25</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:34_1_2"/>
<font color="Maroon" title="c6">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Y</font></span><br/><a NAME="E2:34_1_2"/><span class="kw">then </span>
not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c8">q</font>,<font color="Maroon" title="c9">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34"><span class="lab"><font color="Green" title="E32">A3</font></span></a>, <a class="txt" href="#E9:34"><span class="lab"><font color="Green" title="E37">A8</font></span></a>, <a class="txt" href="#E2:34_1"><span class="lab"><font color="Green" title="E41">A12</font></span></a>, <a class="txt" href="#E7:34_1"><span class="lab"><font color="Green" title="E45">A16</font></span></a>, <a class="txt" href="aff_4.html#E34"><span class="lab"><font color="Green" title="E28">Lm7</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:34_1_2"/>
<font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34"><span class="lab"><font color="Green" title="E30">A1</font></span></a>, <a class="txt" href="#E2:34"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="txt" href="#E5:34"><span class="lab"><font color="Green" title="E34">A5</font></span></a>, <a class="txt" href="#E6:34"><span class="lab"><font color="Green" title="E35">A6</font></span></a>, <a class="txt" href="#E9:34"><span class="lab"><font color="Green" title="E37">A8</font></span></a>, <a class="txt" href="#E2:34_1"><span class="lab"><font color="Green" title="E41">A12</font></span></a>, <a class="txt" href="#E6:34_1"><span class="lab"><font color="Green" title="E44">A15</font></span></a>, <a class="ref" href="aff_4.html#T25" target="_self" title="AFF_4:th.25">Th25</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E10:34_1"/>
<font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:34"><span class="lab"><font color="Green" title="E39">A10</font></span></a>, <a class="txt" href="#E8:34_1"><span class="lab"><font color="Green" title="E46">A17</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c8">c</font>, <font color="Maroon" title="c9">d</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">AS</font><span class="kw"> such that </span><br/><a NAME="E14:34"/><span class="lab"><font color="Green" title="E41">A18</font></span>: 
<font color="Maroon" title="c8">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">N</font>
 <span class="kw">and </span><br/><a NAME="E15:34"/>
<font color="Maroon" title="c9">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">N</font>
 <span class="kw">and </span><br/><a NAME="E16:34"/>
<font color="Maroon" title="c8">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:34"><span class="lab"><font color="Green" title="E33">A4</font></span></a>, <a class="ref" href="aff_1.html#T31" title="AFF_1:th.31">AFF_1:31</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:34_2"/>
<font color="Maroon" title="c4">M</font> <a href="aff_1.html#R4" title="AFF_1:pred.4">//</a> <font color="Maroon" title="c5">N</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Y</font></span><br/><a NAME="E2:34_2"/><span class="kw">then </span>
not <font color="Maroon" title="c8">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">M</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:34"><span class="lab"><font color="Green" title="E36">A7</font></span></a>, <a class="txt" href="#E14:34"><span class="lab"><font color="Green" title="E41">A18</font></span></a>, <a class="ref" href="aff_1.html#T59" title="AFF_1:th.59">AFF_1:59</a></span>;<br/><a NAME="E3:34_2"/><span class="kw">then </span>
not  <a href="aff_1.html#R1" title="AFF_1:pred.1">LIN</a> <font color="Maroon" title="c6">a</font>,<font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c8">c</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34"><span class="lab"><font color="Green" title="E32">A3</font></span></a>, <a class="txt" href="#E9:34"><span class="lab"><font color="Green" title="E37">A8</font></span></a>, <a class="txt" href="#E10:34"><span class="lab"><font color="Green" title="E38">A9</font></span></a>, <a class="txt" href="#E11:34"><span class="lab"><font color="Green" title="E39">A10</font></span></a>, <a class="txt" href="aff_4.html#E34"><span class="lab"><font color="Green" title="E28">Lm7</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:34_2"/>
<font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34"><span class="lab"><font color="Green" title="E30">A1</font></span></a>, <a class="txt" href="#E2:34"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="txt" href="#E5:34"><span class="lab"><font color="Green" title="E34">A5</font></span></a>, <a class="txt" href="#E6:34"><span class="lab"><font color="Green" title="E35">A6</font></span></a>, <a class="txt" href="#E9:34"><span class="lab"><font color="Green" title="E37">A8</font></span></a>, <a class="txt" href="#E10:34"><span class="lab"><font color="Green" title="E38">A9</font></span></a>, <a class="txt" href="#E14:34"><span class="lab"><font color="Green" title="E41">A18</font></span></a>, <a class="ref" href="aff_4.html#T25" target="_self" title="AFF_4:th.25">Th25</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E18:34"/>
<font color="Maroon" title="c2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">Y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34"><span class="lab"><font color="Green" title="E30">A1</font></span></a>, <a class="txt" href="#E3:34"><span class="lab"><font color="Green" title="E32">A3</font></span></a>, <a class="txt" href="#E4:34"><span class="lab"><font color="Green" title="E33">A4</font></span></a>, <a class="txt" href="#E5:34"><span class="lab"><font color="Green" title="E34">A5</font></span></a>, <a class="txt" href="#E12:34"><span class="lab"><font color="Green" title="E40">A11</font></span></a>, <a class="ref" href="aff_4.html#T22" target="_self" title="AFF_4:th.22">Th22</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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