<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">L</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="orders_2.html#V3" title="ORDERS_2:attr.3">reflexive</a> <a href="orders_2.html#V5" title="ORDERS_2:attr.5">antisymmetric</a>  <a href="orders_2.html#L1" title="ORDERS_2:struct.1">RelStr</a> ;<br/><b>let </b><font color="Maroon" title="c2">R</font> be  <a href="waybel_4.html#V1" title="WAYBEL_4:attr.1">auxiliary(i)</a> <a href="alg_1.html#NM1" title="ALG_1:NM.1">Relation</a> of <font color="Maroon" title="c1">L</font>;<br/><b>let </b><font color="Maroon" title="c3">C</font> be    <a href="waybel35.html#M1" title="WAYBEL35:mode.1">strict_chain</a> of <font color="Maroon" title="c2">R</font>;<br/>





<b>assume </b><a NAME="E1:54"/><i><font color="Green" title="E23">A1</font></i>: 
for <font color="Olive" title="b1">c</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b1">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> holds <br/>(  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a>  <a href="waybel35.html#K4" title="WAYBEL35:func.4">SetBelow</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">C</font>,<font color="Olive" title="b1">c</font>,<font color="Maroon" title="c1">L</font> &amp; <font color="Olive" title="b1">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="waybel35.html#K4" title="WAYBEL35:func.4">SetBelow</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">C</font>,<font color="Olive" title="b1">c</font></span>)</span> )
 ;<br/>

<b>let </b><font color="Maroon" title="c4">x</font>, <font color="Maroon" title="c5">z</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel35.html#D6" target="_self" title="WAYBEL35:def.6">WAYBEL35:def 6</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E2:54"/><i><font color="Green" title="E24">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">C</font>
 <b>and </b><br/><a NAME="E3:54"/><i><font color="Green" title="E25">A3</font></i>: 
<font color="Maroon" title="c5">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font>
 <b>and </b><br/><a NAME="E4:54"/><i><font color="Green" title="E26">A4</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font>
 <b>and </b><br/><a NAME="E5:54"/><i><font color="Green" title="E27">A5</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">z</font>
 ;<br/>

<a NAME="E6:54"/><i><font color="Green" title="E28">A6</font></i>: 
<font color="Maroon" title="c5">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><a href="waybel35.html#K4" title="WAYBEL35:func.4">SetBelow</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c5">z</font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel35.html#E1:54"><i><font color="Green" title="E23">A1</font></i></a>, <a class="txt" href="waybel35.html#E3:54"><i><font color="Green" title="E25">A3</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:54_1"/>
( for <font color="Olive" title="b1">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  holds <br/>(  not <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel35.html#K4" title="WAYBEL35:func.4">SetBelow</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c5">z</font> or  not <font color="Maroon" title="c4">x</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Olive" title="b1">y</font> ) or  ex <font color="Olive" title="b1">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel35.html#K4" title="WAYBEL35:func.4">SetBelow</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c5">z</font> &amp; <font color="Maroon" title="c4">x</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Olive" title="b1">y</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_1"><b>suppose </b></a><a NAME="E1:54_1_1"/><i><font color="Green" title="E29">A7</font></i>: 
for <font color="Olive" title="b1">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  holds <br/>(  not <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel35.html#K4" title="WAYBEL35:func.4">SetBelow</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c5">z</font> or  not <font color="Maroon" title="c4">x</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Olive" title="b1">y</font> )
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon" title="c6">x</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">L</font> ;<br/><a NAME="E3:54_1_1"/><i><font color="Green" title="E30">A8</font></i>: 
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a>  <a href="waybel35.html#K4" title="WAYBEL35:func.4">SetBelow</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="txt" href="waybel35.html#E1:54"><i><font color="Green" title="E23">A1</font></i></a>, <a class="txt" href="waybel35.html#E3:54"><i><font color="Green" title="E25">A3</font></i></a></i>;<br/><a NAME="E4:54_1_1"/><i><font color="Green" title="E31">A9</font></i>: 
 <a href="waybel35.html#K4" title="WAYBEL35:func.4">SetBelow</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c5">z</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Maroon" title="c6">x</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D9" title="LATTICE3:def.9">LATTICE3:def 9</a><br/>

<b>assume </b><a NAME="E1:54_1_1_1"/><i><font color="Green" title="E32">A10</font></i>: 
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel35.html#K4" title="WAYBEL35:func.4">SetBelow</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c5">z</font>
 ;<br/>

<a NAME="E2:54_1_1_1"/><b>then </b><i><font color="Green" title="E33">A11</font></i>: 
not <font color="Maroon" title="c6">x</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c7">b</font>
 
<b>by </b><i><a class="txt" href="waybel35.html#E1:54_1_1"><i><font color="Green" title="E29">A7</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:54_1_1_1_1"/>
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">b</font> or <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:54_1_1_1_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">b</font>
 ;<br/><div class="add"><a NAME="E2:54_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E34">A12</font></i>: 
not <font color="Maroon" title="c6">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c7">b</font>
 <b>by </b><i><a class="txt" href="waybel35.html#E2:54_1_1_1"><i><font color="Green" title="E33">A11</font></i></a>, <a class="ref" href="orders_2.html#D10" title="ORDERS_2:def.10">ORDERS_2:def 10</a></i>;<br/><a NAME="E3:54_1_1_1_1_1"/>
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font>
 <b>by </b><i><a class="txt" href="waybel35.html#E1:54_1_1_1"><i><font color="Green" title="E32">A10</font></i></a>, <a class="ref" href="waybel35.html#T18" target="_self" title="WAYBEL35:th.18">Th18</a></i>;<br/><a NAME="E4:54_1_1_1_1_1"/><b>then </b>
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">b</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> or <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b</font> or <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c7">b</font>,<font color="Maroon" title="c6">x</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> )
 <b>by </b><i><a class="txt" href="waybel35.html#E2:54"><i><font color="Green" title="E24">A2</font></i></a>, <a class="ref" href="waybel35.html#D3" target="_self" title="WAYBEL35:def.3">Def3</a></i>;<br/><b>hence </b><a NAME="E5:54_1_1_1_1_1"/>
<font color="Maroon" title="c7">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">x</font>
 <b>by </b><i><a class="txt" href="waybel35.html#E2:54_1_1_1_1_1"><i><font color="Green" title="E34">A12</font></i></a>, <a class="ref" href="waybel_4.html#D4" title="WAYBEL_4:def.4">WAYBEL_4:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:54_1_1_1_1_2"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:54_1_1_1_1_2"/>
<font color="Maroon" title="c7">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c6">x</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E5:54_1_1"/>
for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st  <a href="waybel35.html#K4" title="WAYBEL35:func.4">SetBelow</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c5">z</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Olive" title="b1">a</font> holds <br/><font color="Maroon" title="c6">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Olive" title="b1">a</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>;<br/>

<b>assume </b><a NAME="E1:54_1_1_2"/><i><font color="Green" title="E32">A13</font></i>: 
 <a href="waybel35.html#K4" title="WAYBEL35:func.4">SetBelow</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c5">z</font> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a> <font color="Maroon" title="c7">a</font>
 ;<br/>

<a NAME="E2:54_1_1_2"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel35.html#K4" title="WAYBEL35:func.4">SetBelow</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c5">z</font>
 
<b>by </b><i><a class="txt" href="waybel35.html#E2:54"><i><font color="Green" title="E24">A2</font></i></a>, <a class="txt" href="waybel35.html#E4:54"><i><font color="Green" title="E26">A4</font></i></a>, <a class="ref" href="waybel35.html#T18" target="_self" title="WAYBEL35:th.18">Th18</a></i>;<br/>
<b>hence </b><a NAME="E3:54_1_1_2"/>
<font color="Maroon" title="c6">x</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c7">a</font>
 <b>by </b><i><a class="txt" href="waybel35.html#E1:54_1_1_2"><i><font color="Green" title="E32">A13</font></i></a>, <a class="ref" href="lattice3.html#D9" title="LATTICE3:def.9">LATTICE3:def 9</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E6:54_1_1"/>
 ex <font color="Olive" title="b1">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Olive" title="b1">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">y</font>,<font color="Maroon" title="c5">z</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font> &amp; <font color="Maroon" title="c4">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b1">y</font> )
 <b>by </b><i><a class="txt" href="waybel35.html#E5:54"><i><font color="Green" title="E27">A5</font></i></a>, <a class="txt" href="waybel35.html#E6:54"><i><font color="Green" title="E28">A6</font></i></a>, <a class="txt" href="waybel35.html#E3:54_1_1"><i><font color="Green" title="E30">A8</font></i></a>, <a class="txt" href="waybel35.html#E4:54_1_1"><i><font color="Green" title="E31">A9</font></i></a>, <a class="ref" href="yellow_0.html#D9" title="YELLOW_0:def.9">YELLOW_0:def 9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_2"><b>suppose </b></a><a NAME="E1:54_1_2"/>
 ex <font color="Olive" title="b1">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> st <br/>( <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel35.html#K4" title="WAYBEL35:func.4">SetBelow</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c5">z</font> &amp; <font color="Maroon" title="c4">x</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Olive" title="b1">y</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c6">y</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E3:54_1_2"/><i><font color="Green" title="E29">A14</font></i>: 
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel35.html#K4" title="WAYBEL35:func.4">SetBelow</a> <font color="Maroon" title="c2">R</font>,<font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c5">z</font>
 <b>and </b><br/><a NAME="E4:54_1_2"/><i><font color="Green" title="E30">A15</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="orders_2.html#R2" title="ORDERS_2:pred.2">&lt;</a> <font color="Maroon" title="c6">y</font>
 ;<br/><b>take </b>
<font color="Maroon" title="c6">y</font>
;<br/><b>thus </b><a NAME="E5:54_1_2"/>
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font>
 <b>by </b><i><a class="txt" href="waybel35.html#E3:54_1_2"><i><font color="Green" title="E29">A14</font></i></a>, <a class="ref" href="waybel35.html#T18" target="_self" title="WAYBEL35:th.18">Th18</a></i>;<br/><b>hence </b><a NAME="E6:54_1_2"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c6">y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font>
 <b>by </b><i><a class="txt" href="waybel35.html#E2:54"><i><font color="Green" title="E24">A2</font></i></a>, <a class="txt" href="waybel35.html#E4:54_1_2"><i><font color="Green" title="E30">A15</font></i></a>, <a class="ref" href="waybel35.html#T5" target="_self" title="WAYBEL35:th.5">Th5</a></i>;<br/><b>thus </b><a NAME="E7:54_1_2"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c5">z</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">R</font>
 <b>by </b><i><a class="txt" href="waybel35.html#E3:54_1_2"><i><font color="Green" title="E29">A14</font></i></a>, <a class="ref" href="waybel35.html#T18" target="_self" title="WAYBEL35:th.18">Th18</a></i>;<br/><b>thus </b><a NAME="E8:54_1_2"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">y</font>
 <b>by </b><i><a class="txt" href="waybel35.html#E4:54_1_2"><i><font color="Green" title="E30">A15</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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