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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="lattice6.html#V2" target="_self">co-noetherian</a> <a href="lattices.html#NM1">Lattice</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<b>assume </b><a NAME="E1:22"/><i><font color="Green">E14</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lattice3.html#K7">~</a> </span>)</span>] means (  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> implies ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>(  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="lattices.html#R3">[=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="lattice6.html#NR2" target="_self">is-lower-neighbour-of</a> <font color="Maroon">c<sub>2</sub></font> ) );<br/>
<a NAME="E2:22"/><i><font color="Green">E15</font></i>: 
<span class="p1">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lattice3.html#K7">~</a>  is <a href="wellfnd1.html#V1">well_founded</a>
 
<b>by </b><i><a class="ref" href="lattice6.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E3:22"/><i><font color="Green">E16</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lattice3.html#K7">~</a> </span>)</span> st ( for <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lattice3.html#K7">~</a> </span>)</span> st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lattice3.html#K7">~</a> </span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lattice3.html#K7">~</a> </span>)</span>;<br/>

<b>assume </b><a NAME="E1:22_1"/><i><font color="Green">E17</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lattice3.html#K7">~</a> </span>)</span> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lattice3.html#K7">~</a> </span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 ;<br/>

<a NAME="E2:22_1"/>
(  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> implies ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>(  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="lattices.html#R3">[=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="lattice6.html#NR2" target="_self">is-lower-neighbour-of</a> <font color="Maroon">c<sub>2</sub></font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:22_1_1"/><i><font color="Green">E18</font></i>: 
(  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> )
 ;<br/>

<i><font color="Green">E19</font></i>: <span class="p1">(<span class="default"><a href="lattice3.html#K5">%</a> <span class="p2">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="lattice3.html#K4">%</a>  = 
 <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#D3">LATTICE3:def 3</a></i>
<br/>.= 
 <a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="lattice3.html#D4">LATTICE3:def 4</a></i>
;<br/>
<i><font color="Green">E20</font></i>: <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="lattice3.html#K8">~</a>  = 
 <a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="lattice3.html#D6">LATTICE3:def 6</a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="lattice3.html#D7">LATTICE3:def 7</a></i>
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_1_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:22_1_1_3"/>
(  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="lattice6.html#NR2" target="_self">is-lower-neighbour-of</a> <font color="Maroon">c<sub>2</sub></font> or  not  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="lattice6.html#NR2" target="_self">is-lower-neighbour-of</a> <font color="Maroon">c<sub>2</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_1_3_1"><b>suppose </b></a><a NAME="E1:22_1_1_3_1"/>
 <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="lattice6.html#NR2" target="_self">is-lower-neighbour-of</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:22_1_1_3_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>(  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="lattices.html#R3">[=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="lattice6.html#NR2" target="_self">is-lower-neighbour-of</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_1_1_3_2"><b>suppose </b></a><a NAME="E1:22_1_1_3_2"/>
not  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="lattice6.html#NR2" target="_self">is-lower-neighbour-of</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:22_1_1_3_2"/><i><font color="Green">E21</font></i>: 
(  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>2</sub></font> &amp; not <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; not <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="lattice6.html#E1:22_1_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="lattice6.html#D5" target="_self">Def5</a></i>;<br/><i><font color="Green">E22</font></i>:  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="lattice3.html#K4">%</a> </span>)</span> <a href="lattice3.html#K8">~</a> </span>)</span></span>)</span> = 
 <a href="lattice3.html#K9">~</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="lattice3.html#K4">%</a> </span>)</span> <a href="lattice3.html#K8">~</a> </span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#D4">LATTICE3:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="lattice3.html#K4">%</a> </span>)</span> <a href="lattice3.html#K8">~</a> 
<b>by </b><i><a class="ref" href="lattice3.html#D7">LATTICE3:def 7</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="lattice3.html#K4">%</a> 
<b>by </b><i><a class="ref" href="lattice3.html#D6">LATTICE3:def 6</a></i>
<br/>.= 
<font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="ref" href="lattice3.html#D3">LATTICE3:def 3</a></i>
;<br/><a NAME="E5:22_1_1_3_2"/>
 <a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>4</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> <a href="lattice3.html#K4">%</a> 
 <b>by </b><i><a class="txt" href="lattice6.html#E2:22_1_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="lattice6.html#E3:22_1_1_3_2"><i><font color="Green">E21</font></i></a>, <a class="ref" href="lattice3.html#T7">LATTICE3:7</a></i>;<br/><a NAME="E6:22_1_1_3_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="lattice3.html#K4">%</a> </span>)</span> <a href="lattice3.html#K8">~</a>  <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="lattice6.html#E3:22_1_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="lattice3.html#T9">LATTICE3:9</a></i>;<br/><a NAME="E7:22_1_1_3_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="lattice3.html#K4">%</a> </span>)</span> <a href="lattice3.html#K8">~</a> </span>)</span>,<font color="Maroon">c<sub>4</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lattice3.html#K7">~</a> </span>)</span>
 <b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E9:22_1_1_3_2"/><i><font color="Green">E23</font></i>: 
(  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="lattice3.html#K4">%</a> </span>)</span> <a href="lattice3.html#K8">~</a> </span>)</span></span>)</span> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="lattice6.html#NR2" target="_self">is-lower-neighbour-of</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="lattice6.html#E1:22_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="lattice6.html#E3:22_1_1_3_2"><i><font color="Green">E21</font></i></a>, <a class="txt" href="lattice6.html#E4:22_1_1_3_2"><i><font color="Green">E22</font></i></a></i>;<br/><a NAME="E10:22_1_1_3_2"/>
 <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="lattices.html#R3">[=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="lattice6.html#E3:22_1_1_3_2"><i><font color="Green">E21</font></i></a>, <a class="txt" href="lattice6.html#E4:22_1_1_3_2"><i><font color="Green">E22</font></i></a>, <a class="txt" href="lattice6.html#E9:22_1_1_3_2"><i><font color="Green">E23</font></i></a>, <a class="ref" href="lattices.html#T25">LATTICES:25</a></i>;<br/><b>hence </b><a NAME="E11:22_1_1_3_2"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>(  <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="lattices.html#R3">[=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="lattice6.html#NR2" target="_self">is-lower-neighbour-of</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="lattice6.html#E9:22_1_1_3_2"><i><font color="Green">E23</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E3:22_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:22"/><i><font color="Green">E17</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="lattice3.html#K3">LattPOSet</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lattice3.html#K7">~</a> </span>)</span> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="wellfnd1.html#S3">WELLFND1:sch 3</a>(<a class="txt" href="lattice6.html#E3:22"><i><font color="Green">E16</font></i></a>, <a class="txt" href="lattice6.html#E2:22"><i><font color="Green">E15</font></i></a>);<br/></i>
 <a href="lattice3.html#K5">%</a> <span class="p1">(<span class="default"><a href="lattice3.html#K9">~</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K4">%</a> </span>)</span> <a href="lattice3.html#K8">~</a> </span>)</span></span>)</span> = 
 <a href="lattice3.html#K9">~</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K4">%</a> </span>)</span> <a href="lattice3.html#K8">~</a> </span>)</span>
<b>by </b><i><a class="ref" href="lattice3.html#D4">LATTICE3:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K4">%</a> </span>)</span> <a href="lattice3.html#K8">~</a> 
<b>by </b><i><a class="ref" href="lattice3.html#D7">LATTICE3:def 7</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="lattice3.html#K4">%</a> 
<b>by </b><i><a class="ref" href="lattice3.html#D6">LATTICE3:def 6</a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="lattice3.html#D3">LATTICE3:def 3</a></i>
;<br/>
<b>hence </b><a NAME="E6:22"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#R3">[=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="lattice6.html#NR2" target="_self">is-lower-neighbour-of</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="lattice6.html#E1:22"><i><font color="Green">E14</font></i></a>, <a class="txt" href="lattice6.html#E4:22"><i><font color="Green">E17</font></i></a></i>;<br/>


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