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

<b>let </b><font color="Maroon">m</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">k</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">N</font> be  <a href="setfam_1.html#V1">with_non-empty_elements</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">S</font> be  non <a href="struct_0.html#V3">empty</a> non <a href="ami_1.html#V2">void</a> <a href="ami_1.html#V5">IC-Ins-separated</a> <a href="ami_1.html#V8">definite</a> <a href="amistd_1.html#V4">standard</a>  <a href="ami_1.html#L1">AMI-Struct</a> of <font color="Maroon">N</font>;<br/><b>let </b><font color="Maroon">F</font> be    <a href="ami_1.html#M1">FinPartState</a> of <font color="Maroon">S</font>;<br/>









<b>set </b><font color="Maroon">A</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="amistd_1.html#K5">il.</a> <font color="Maroon">S</font>,<span class="p3">(<span class="default"><font color="Olive">l</font> <a href="xcmplx_0.html#K2">+</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="amistd_1.html#K5">il.</a> <font color="Maroon">S</font>,<font color="Olive">l</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> </span>}</span> ;<br/>
<a NAME="E1:96"/><i><font color="Green">E44</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K10" target="_self">Shift</a> <font color="Maroon">F</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="amistd_1.html#K5">il.</a> <font color="Maroon">S</font>,<span class="p3">(<span class="default"><font color="Olive">l</font> <a href="xcmplx_0.html#K2">+</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="amistd_1.html#K5">il.</a> <font color="Maroon">S</font>,<font color="Olive">l</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E2:96"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="amistd_1.html#K5">il.</a> <font color="Maroon">S</font>,<span class="p3">(<span class="default"><font color="Olive">r</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k</font></span>)</span></span>)</span> where <font color="Olive">r</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  :  <a href="amistd_1.html#K5">il.</a> <font color="Maroon">S</font>,<font color="Olive">r</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="amistd_1.html#K5">il.</a> <font color="Maroon">S</font>,<span class="p3">(<span class="default"><font color="Olive">l</font> <a href="xcmplx_0.html#K2">+</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="amistd_1.html#K5">il.</a> <font color="Maroon">S</font>,<font color="Olive">l</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</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="amistd_1.html#K5">il.</a> <font color="Maroon">S</font>,<span class="p3">(<span class="default"><font color="Olive">q</font> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k</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="amistd_1.html#K5">il.</a> <font color="Maroon">S</font>,<font color="Olive">q</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_1"><b>proof </b></a><div class="add">

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a>
<div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:96_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="amistd_1.html#K5">il.</a> <font color="Maroon">S</font>,<span class="p3">(<span class="default"><font color="Olive">r</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k</font></span>)</span></span>)</span> where <font color="Olive">r</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  :  <a href="amistd_1.html#K5">il.</a> <font color="Maroon">S</font>,<font color="Olive">r</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="amistd_1.html#K5">il.</a> <font color="Maroon">S</font>,<span class="p3">(<span class="default"><font color="Olive">l</font> <a href="xcmplx_0.html#K2">+</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="amistd_1.html#K5">il.</a> <font color="Maroon">S</font>,<font color="Olive">l</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> </span>}</span>  </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">r</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="E3:96_1_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="amistd_1.html#K5">il.</a> <font color="Maroon">S</font>,<span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k</font></span>)</span>
 <b>and </b><br/><a NAME="E4:96_1_1"/><i><font color="Green">E56</font></i>: 
 <a href="amistd_1.html#K5">il.</a> <font color="Maroon">S</font>,<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="amistd_1.html#K5">il.</a> <font color="Maroon">S</font>,<span class="p3">(<span class="default"><font color="Olive">l</font> <a href="xcmplx_0.html#K2">+</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="amistd_1.html#K5">il.</a> <font color="Maroon">S</font>,<font color="Olive">l</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</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="E6:96_1_1"/><i><font color="Green">E57</font></i>: 
 <a href="amistd_1.html#K5">il.</a> <font color="Maroon">S</font>,<font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="amistd_1.html#K5">il.</a> <font color="Maroon">S</font>,<span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">m</font></span>)</span>
 <b>and </b><br/><a NAME="E7:96_1_1"/><i><font color="Green">E58</font></i>: 
 <a href="amistd_1.html#K5">il.</a> <font color="Maroon">S</font>,<font color="Maroon">l</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="txt" href="amistd_2.html#E4:96_1_1"><i><font color="Green">E56</font></i></a></i>;<br/><a NAME="E8:96_1_1"/>
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="txt" href="amistd_2.html#E6:96_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="amistd_1.html#T25">AMISTD_1:25</a></i>;<br/><a NAME="E9:96_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="amistd_1.html#K5">il.</a> <font color="Maroon">S</font>,<span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E10:96_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="amistd_1.html#K5">il.</a> <font color="Maroon">S</font>,<span class="p3">(<span class="default"><font color="Olive">q</font> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k</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="amistd_1.html#K5">il.</a> <font color="Maroon">S</font>,<font color="Olive">q</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="amistd_2.html#E7:96_1_1"><i><font color="Green">E58</font></i></a></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="E2:96_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="amistd_1.html#K5">il.</a> <font color="Maroon">S</font>,<span class="p3">(<span class="default"><font color="Olive">q</font> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k</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="amistd_1.html#K5">il.</a> <font color="Maroon">S</font>,<font color="Olive">q</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</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="E4:96_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="amistd_1.html#K5">il.</a> <font color="Maroon">S</font>,<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E5:96_1"/><i><font color="Green">E62</font></i>: 
 <a href="amistd_1.html#K5">il.</a> <font color="Maroon">S</font>,<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font>
 ;<br/>
<a NAME="E6:96_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="amistd_1.html#K5">il.</a> <font color="Maroon">S</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E7:96_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">q</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">m</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E8:96_1"/>
 <a href="amistd_1.html#K5">il.</a> <font color="Maroon">S</font>,<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="xcmplx_0.html#K2">+</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="amistd_1.html#K5">il.</a> <font color="Maroon">S</font>,<span class="p3">(<span class="default"><font color="Olive">l</font> <a href="xcmplx_0.html#K2">+</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="amistd_1.html#K5">il.</a> <font color="Maroon">S</font>,<font color="Olive">l</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E9:96_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="amistd_1.html#K5">il.</a> <font color="Maroon">S</font>,<span class="p3">(<span class="default"><font color="Olive">r</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k</font></span>)</span></span>)</span> where <font color="Olive">r</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  :  <a href="amistd_1.html#K5">il.</a> <font color="Maroon">S</font>,<font color="Olive">r</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="amistd_1.html#K5">il.</a> <font color="Maroon">S</font>,<span class="p3">(<span class="default"><font color="Olive">l</font> <a href="xcmplx_0.html#K2">+</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="amistd_1.html#K5">il.</a> <font color="Maroon">S</font>,<font color="Olive">l</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> </span>}</span>  </span>}</span> 
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:96"/><b>then </b><i><font color="Green">E65</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K10" target="_self">Shift</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K10" target="_self">Shift</a> <font color="Maroon">F</font>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="amistd_1.html#K5">il.</a> <font color="Maroon">S</font>,<span class="p3">(<span class="default"><font color="Olive">l</font> <a href="xcmplx_0.html#K2">+</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k</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="amistd_1.html#K5">il.</a> <font color="Maroon">S</font>,<font color="Olive">l</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> </span>}</span> 
 
<b>by </b><i>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_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="E1:96_2"/><i><font color="Green">E66</font></i>: 
 <a href="amistd_1.html#K5">il.</a> <font color="Maroon">S</font>,<font color="Maroon">l</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font>
 ;<br/><a NAME="E2:96_2"/><b>then </b><i><font color="Green">E67</font></i>: 
 <a href="amistd_1.html#K5">il.</a> <font color="Maroon">S</font>,<span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="xcmplx_0.html#K2">+</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="amistd_2.html#K10" target="_self">Shift</a> <font color="Maroon">F</font>,<font color="Maroon">m</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E3:96_2"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">l</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">m</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><a href="amistd_2.html#K10" target="_self">Shift</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K10" target="_self">Shift</a> <font color="Maroon">F</font>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">k</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="amistd_1.html#K5">il.</a> <font color="Maroon">S</font>,<span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="amistd_2.html#K10" target="_self">Shift</a> <span class="p2">(<span class="default"><a href="amistd_2.html#K10" target="_self">Shift</a> <font color="Maroon">F</font>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">k</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="amistd_1.html#K5">il.</a> <font color="Maroon">S</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">l</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k</font></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="amistd_2.html#K10" target="_self">Shift</a> <font color="Maroon">F</font>,<font color="Maroon">m</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="amistd_1.html#K5">il.</a> <font color="Maroon">S</font>,<span class="p2">(<span class="default"><font color="Maroon">l</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">m</font></span>)</span></span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="amistd_1.html#K5">il.</a> <font color="Maroon">S</font>,<font color="Maroon">l</font></span>)</span>
<b>by </b><i>, </i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:96"/>
 <a href="amistd_2.html#K10" target="_self">Shift</a> <span class="p1">(<span class="default"><a href="amistd_2.html#K10" target="_self">Shift</a> <font color="Maroon">F</font>,<font color="Maroon">m</font></span>)</span>,<font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="amistd_2.html#K10" target="_self">Shift</a> <font color="Maroon">F</font>,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i>, </i>;<br/>


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