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

<b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">I</font> be    <a href="ami_1.html#M1">FinPartState</a> of  <a href="scmfsa_2.html#K1">SCM+FSA</a> ;<br/>





<b>set </b><font color="Maroon">A</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <span class="p3">(<span class="default"><font color="Olive">l</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">l</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  :  <a href="scmfsa_2.html#K5">insloc</a> <font color="Olive">l</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">I</font> </span>}</span> ;<br/>
<a NAME="E1:45"/><i><font color="Green">E27</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="scmfsa_4.html#K8" target="_self">Shift</a> <font color="Maroon">I</font>,<font color="Maroon">m</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <span class="p3">(<span class="default"><font color="Olive">l</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">l</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  :  <a href="scmfsa_2.html#K5">insloc</a> <font color="Olive">l</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">I</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="scmfsa_4.html#E1:45"><i><font color="Green">E27</font></i></a></i>;<br/>
<a NAME="E2:45"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <span class="p3">(<span class="default"><font color="Olive">p</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span> where <font color="Olive">p</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  :  <a href="scmfsa_2.html#K5">insloc</a> <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <span class="p3">(<span class="default"><font color="Olive">l</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">l</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  :  <a href="scmfsa_2.html#K5">insloc</a> <font color="Olive">l</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">I</font> </span>}</span>  </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <span class="p3">(<span class="default"><font color="Olive">q</font> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> where <font color="Olive">q</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  :  <a href="scmfsa_2.html#K5">insloc</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">I</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:45_1"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <span class="p3">(<span class="default"><font color="Olive">p</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span> where <font color="Olive">p</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  :  <a href="scmfsa_2.html#K5">insloc</a> <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <span class="p3">(<span class="default"><font color="Olive">l</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">l</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  :  <a href="scmfsa_2.html#K5">insloc</a> <font color="Olive">l</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">I</font> </span>}</span>  </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <span class="p3">(<span class="default"><font color="Olive">q</font> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> where <font color="Olive">q</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  :  <a href="scmfsa_2.html#K5">insloc</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">I</font> </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:45_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <span class="p3">(<span class="default"><font color="Olive">p</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span> where <font color="Olive">p</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  :  <a href="scmfsa_2.html#K5">insloc</a> <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <span class="p3">(<span class="default"><font color="Olive">l</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">l</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  :  <a href="scmfsa_2.html#K5">insloc</a> <font color="Olive">l</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">I</font> </span>}</span>  </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">p</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:45_1_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="scmfsa_2.html#K5">insloc</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span>
 <b>and </b><br/><a NAME="E4:45_1_1"/><i><font color="Green">E31</font></i>: 
 <a href="scmfsa_2.html#K5">insloc</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <span class="p3">(<span class="default"><font color="Olive">l</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">l</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  :  <a href="scmfsa_2.html#K5">insloc</a> <font color="Olive">l</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">I</font> </span>}</span> 
 ;<br/>
<b>consider </b><font color="Maroon">l</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:45_1_1"/><i><font color="Green">E32</font></i>: 
 <a href="scmfsa_2.html#K5">insloc</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="scmfsa_2.html#K5">insloc</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span>
 <b>and </b><br/><a NAME="E7:45_1_1"/><i><font color="Green">E68</font></i>: 
 <a href="scmfsa_2.html#K5">insloc</a> <font color="Maroon">l</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">I</font>
 <b>by </b><i><a class="txt" href="scmfsa_4.html#E4:45_1_1"><i><font color="Green">E31</font></i></a></i>;<br/>
<a NAME="E8:45_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="txt" href="scmfsa_4.html#E6:45_1_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="scmfsa_2.html#T18">SCMFSA_2:18</a></i>;<br/>
<a NAME="E9:45_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="scmfsa_2.html#K5">insloc</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="scmfsa_4.html#E3:45_1_1"><i><font color="Green">E30</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:45_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <span class="p3">(<span class="default"><font color="Olive">q</font> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> where <font color="Olive">q</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  :  <a href="scmfsa_2.html#K5">insloc</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">I</font> </span>}</span> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:45_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <span class="p3">(<span class="default"><font color="Olive">q</font> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> where <font color="Olive">q</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  :  <a href="scmfsa_2.html#K5">insloc</a> <font color="Olive">q</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">I</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">q</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:45_1"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="scmfsa_2.html#K5">insloc</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E5:45_1"/><i><font color="Green">E79</font></i>: 
 <a href="scmfsa_2.html#K5">insloc</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">I</font>
 ;<br/>
<a NAME="E6:45_1"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="scmfsa_2.html#K5">insloc</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E7:45_1"/>
 <a href="scmfsa_2.html#K5">insloc</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <span class="p3">(<span class="default"><font color="Olive">l</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">l</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  :  <a href="scmfsa_2.html#K5">insloc</a> <font color="Olive">l</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">I</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E8:45_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <span class="p3">(<span class="default"><font color="Olive">p</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span> where <font color="Olive">p</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  :  <a href="scmfsa_2.html#K5">insloc</a> <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <span class="p3">(<span class="default"><font color="Olive">l</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span></span>)</span> where <font color="Olive">l</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  :  <a href="scmfsa_2.html#K5">insloc</a> <font color="Olive">l</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">I</font> </span>}</span>  </span>}</span> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E3:45"/><b>then </b><i><font color="Green">E87</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="scmfsa_4.html#K8" target="_self">Shift</a> <span class="p2">(<span class="default"><a href="scmfsa_4.html#K8" target="_self">Shift</a> <font color="Maroon">I</font>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <span class="p3">(<span class="default"><font color="Olive">l</font> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> where <font color="Olive">l</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  :  <a href="scmfsa_2.html#K5">insloc</a> <font color="Olive">l</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">I</font> </span>}</span> 
 
<b>by </b><i>, <a class="txt" href="scmfsa_4.html#E1:45"><i><font color="Green">E27</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">l</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:45_2"/><i><font color="Green">E88</font></i>: 
 <a href="scmfsa_2.html#K5">insloc</a> <font color="Maroon">l</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">I</font>
 ;<br/><a NAME="E2:45_2"/><b>then </b><i><font color="Green">E89</font></i>: 
 <a href="scmfsa_2.html#K5">insloc</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="scmfsa_4.html#K8" target="_self">Shift</a> <font color="Maroon">I</font>,<font color="Maroon">m</font></span>)</span>
 <b>by </b><i/>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="scmfsa_4.html#K8" target="_self">Shift</a> <span class="p2">(<span class="default"><a href="scmfsa_4.html#K8" target="_self">Shift</a> <font color="Maroon">I</font>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="scmfsa_4.html#K8" target="_self">Shift</a> <span class="p2">(<span class="default"><a href="scmfsa_4.html#K8" target="_self">Shift</a> <font color="Maroon">I</font>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">n</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="scmfsa_4.html#K8" target="_self">Shift</a> <font color="Maroon">I</font>,<font color="Maroon">m</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="txt" href="scmfsa_4.html#E1:45"><i><font color="Green">E27</font></i></a></i>
<br/>.= 
<font color="Maroon">I</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="scmfsa_2.html#K5">insloc</a> <font color="Maroon">l</font></span>)</span>
<b>by </b><i>, <a class="txt" href="scmfsa_4.html#E1:45"><i><font color="Green">E27</font></i></a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:45"/>
 <a href="scmfsa_4.html#K8" target="_self">Shift</a> <span class="p1">(<span class="default"><a href="scmfsa_4.html#K8" target="_self">Shift</a> <font color="Maroon">I</font>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="scmfsa_4.html#K8" target="_self">Shift</a> <font color="Maroon">I</font>,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="scmfsa_4.html#E1:45"><i><font color="Green">E27</font></i></a></i>;<br/>


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