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

<b>let </b><font color="Maroon" title="c2">s</font> be   <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <a href="scmfsa_2.html#K1" title="SCMFSA_2:func.1">SCM+FSA</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <font color="Maroon" title="c2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies for <font color="Olive" title="b1">a</font> being   <a href="scmfsa_2.html#M1" title="SCMFSA_2:mode.1">Int-Location</a> <br/> for <font color="Olive" title="b2">k</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st  <a href="scmfsa_7.html#K1" title="SCMFSA_7:func.1">Load</a> <span class="p1">(<span class="default"><a href="scmfsa_7.html#K3" title="SCMFSA_7:func.3">aSeq</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">k</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">s</font> holds <br/>for <font color="Olive" title="b3">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b3">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="scmfsa_7.html#K3" title="SCMFSA_7:func.3">aSeq</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">k</font></span>)</span> holds <br/> <a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c2">s</font>,<font color="Olive" title="b3">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> <font color="Olive" title="b3">i</font> )</span><br/>

<b>assume </b><a NAME="E1:18"/><i><font color="Green" title="E12">A1</font></i>: 
 <a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <font color="Maroon" title="c2">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a</font> being   <a href="scmfsa_2.html#M1" title="SCMFSA_2:mode.1">Int-Location</a> <br/> for <font color="Olive" title="b2">k</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st  <a href="scmfsa_7.html#K1" title="SCMFSA_7:func.1">Load</a> <span class="p1">(<span class="default"><a href="scmfsa_7.html#K3" title="SCMFSA_7:func.3">aSeq</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">k</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">s</font> holds <br/>for <font color="Olive" title="b3">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b3">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="scmfsa_7.html#K3" title="SCMFSA_7:func.3">aSeq</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">k</font></span>)</span> holds <br/> <a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c2">s</font>,<font color="Olive" title="b3">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> <font color="Olive" title="b3">i</font></span><br/>

<b>let </b><font color="Maroon" title="c3">a</font> be    <a href="scmfsa_2.html#M1" title="SCMFSA_2:mode.1">Int-Location</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">k</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st  <a href="scmfsa_7.html#K1" title="SCMFSA_7:func.1">Load</a> <span class="p1">(<span class="default"><a href="scmfsa_7.html#K3" title="SCMFSA_7:func.3">aSeq</a> <font color="Maroon" title="c3">a</font>,<font color="Olive" title="b1">k</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">s</font> holds <br/>for <font color="Olive" title="b2">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="scmfsa_7.html#K3" title="SCMFSA_7:func.3">aSeq</a> <font color="Maroon" title="c3">a</font>,<font color="Olive" title="b1">k</font></span>)</span> holds <br/> <a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c2">s</font>,<font color="Olive" title="b2">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> <font color="Olive" title="b2">i</font></span><br/><b>let </b><font color="Maroon" title="c4">k</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="scmfsa_7.html#K1" title="SCMFSA_7:func.1">Load</a> <span class="p1">(<span class="default"><a href="scmfsa_7.html#K3" title="SCMFSA_7:func.3">aSeq</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">k</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">s</font> implies for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="scmfsa_7.html#K3" title="SCMFSA_7:func.3">aSeq</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">k</font></span>)</span> holds <br/> <a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c2">s</font>,<font color="Olive" title="b1">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> <font color="Olive" title="b1">i</font> )</span><br/>



<b>assume </b><a NAME="E2:18"/><i><font color="Green" title="E13">A2</font></i>: 
 <a href="scmfsa_7.html#K1" title="SCMFSA_7:func.1">Load</a> <span class="p1">(<span class="default"><a href="scmfsa_7.html#K3" title="SCMFSA_7:func.3">aSeq</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">k</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">s</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="scmfsa_7.html#K3" title="SCMFSA_7:func.3">aSeq</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">k</font></span>)</span> holds <br/> <a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c2">s</font>,<font color="Olive" title="b1">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> <font color="Olive" title="b1">i</font></span><br/>

<b>let </b><font color="Maroon" title="c5">i</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="scmfsa_7.html#K3" title="SCMFSA_7:func.3">aSeq</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">k</font></span>)</span> implies  <a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c2">s</font>,<font color="Maroon" title="c5">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> <font color="Maroon" title="c5">i</font> )</span><br/>

<b>assume </b><a NAME="E3:18"/><i><font color="Green" title="E14">A3</font></i>: 
<font color="Maroon" title="c5">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="scmfsa_7.html#K3" title="SCMFSA_7:func.3">aSeq</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">k</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c2">s</font>,<font color="Maroon" title="c5">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> <font color="Maroon" title="c5">i</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">c</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">c</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="scmfsa_7.html#K3" title="SCMFSA_7:func.3">aSeq</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">k</font></span>)</span> implies <font color="Maroon" title="c2">s</font> <a href="ami_1.html#K13" title="AMI_1:func.13">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> <span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">c</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="scmfsa_7.html#K3" title="SCMFSA_7:func.3">aSeq</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">c</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )</span><br/><b>assume </b><a NAME="E1:18_1"/>
<font color="Maroon" title="c6">c</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="scmfsa_7.html#K3" title="SCMFSA_7:func.3">aSeq</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">k</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">s</font> <a href="ami_1.html#K13" title="AMI_1:func.13">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> <span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">c</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="scmfsa_7.html#K3" title="SCMFSA_7:func.3">aSeq</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">c</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><br/><a NAME="E2:18_1"/><b>then </b><i><font color="Green" title="E15">A4</font></i>: 
 <a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> <font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="scmfsa_7.html#K1" title="SCMFSA_7:func.1">Load</a> <span class="p2">(<span class="default"><a href="scmfsa_7.html#K3" title="SCMFSA_7:func.3">aSeq</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">k</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="scmfsa_7.html#T29" title="SCMFSA_7:th.29">SCMFSA_7:29</a></i>;<br/><a NAME="E3:18_1"/><b>then </b><i><font color="Green" title="E16">A5</font></i>: 
<font color="Maroon" title="c6">c</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="scmfsa_7.html#K3" title="SCMFSA_7:func.3">aSeq</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="scmfsa_7.html#T26" title="SCMFSA_7:th.26">SCMFSA_7:26</a></i>;<br/><font color="Maroon" title="c2">s</font> <a href="ami_1.html#K13" title="AMI_1:func.13">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> <font color="Maroon" title="c6">c</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="scmfsa_7.html#K1" title="SCMFSA_7:func.1">Load</a> <span class="p2">(<span class="default"><a href="scmfsa_7.html#K3" title="SCMFSA_7:func.3">aSeq</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">k</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> <font color="Maroon" title="c6">c</font></span>)</span>
<b>by </b><i><a class="txt" href="scmfsa7b.html#E2:18"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="scmfsa7b.html#E2:18_1"><i><font color="Green" title="E15">A4</font></i></a>, <a class="ref" href="grfunc_1.html#T8" title="GRFUNC_1:th.8">GRFUNC_1:8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="scmfsa_7.html#K3" title="SCMFSA_7:func.3">aSeq</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">c</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="scmfsa7b.html#E2:18_1"><i><font color="Green" title="E15">A4</font></i></a>, <a class="ref" href="scmfsa_7.html#D1" title="SCMFSA_7:def.1">SCMFSA_7:def 1</a></i>
;<br/><b>hence </b><a NAME="E5:18_1"/>
<font color="Maroon" title="c2">s</font> <a href="ami_1.html#K13" title="AMI_1:func.13">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> <span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">c</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="scmfsa_7.html#K3" title="SCMFSA_7:func.3">aSeq</a> <font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">c</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="scmfsa7b.html#E3:18_1"><i><font color="Green" title="E16">A5</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E5:18"/><b>then </b>
 <a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c2">s</font>,<font color="Maroon" title="c5">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">i</font></span>)</span>
 
<b>by </b><i><a class="txt" href="scmfsa7b.html#E1:18"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="scmfsa7b.html#E3:18"><i><font color="Green" title="E14">A3</font></i></a>, <a class="txt" href="scmfsa7b.html#E14"><i><font color="Green" title="E10">Lm5</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:18"/>
 <a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c2">s</font>,<font color="Maroon" title="c5">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmfsa_2.html#K5" title="SCMFSA_2:func.5">insloc</a> <font color="Maroon" title="c5">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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