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

<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_3.html#V1">programmed</a>  <a href="ami_1.html#M1">FinPartState</a> of <font color="Maroon">S</font>;<br/>







<a NAME="E1:97"/><i><font color="Green">E44</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> the <a href="ami_1.html#U2">Instruction-Locations</a> of <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="ami_3.html#D13">AMI_3:def 13</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">il</font> being  <a href="ami_1.html#NM2">Instruction-Location</a> of <font color="Maroon">S</font> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">il</font> &amp; <font color="Maroon">a<sub>2</sub></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">k</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="amistd_1.html#K7">locnum</a> <font color="Olive">il</font></span>)</span></span>)</span> );<br/>
<a NAME="E2:97"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">e</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> holds <br/> ex <font color="Olive">u</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">e</font>,<font color="Olive">u</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">e</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:97_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">e</font> = <font color="Maroon">e</font> as   <a href="ami_1.html#NM2">Instruction-Location</a> of <font color="Maroon">S</font> <b>by </b><i/>;<br/>
<b>take </b>
 <a href="amistd_1.html#K5">il.</a> <font color="Maroon">S</font>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="amistd_1.html#K7">locnum</a> <font color="Maroon">e</font></span>)</span></span>)</span>
;<br/>

<b>take </b>
<font color="Maroon">e</font>
;<br/>

<b>thus </b><a NAME="E3:97_1"/>
( <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> &amp;  <a href="amistd_1.html#K5">il.</a> <font color="Maroon">S</font>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="amistd_1.html#K7">locnum</a> <font color="Maroon">e</font></span>)</span></span>)</span> <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">k</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="amistd_1.html#K7">locnum</a> <font color="Maroon">e</font></span>)</span></span>)</span> )
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E4:97"/><i><font color="Green">E56</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font>
 <b>and </b><br/><a NAME="E5:97"/><i><font color="Green">E57</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font>]
 <b>from </b><i><a class="ref" href="zfrefle1.html#S1">ZFREFLE1:sch 1</a>(<a class="txt" href="amistd_2.html#E4:97"><i><font color="Green">E56</font></i></a>);<br/></i>
<b>take </b>
<font color="Maroon">f</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="wellord2.html#D4">WELLORD2:def 4</a><br/>

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a>
<div class="add"><b>let </b><font color="Maroon">x1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:97_2"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E2:97_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E3:97_2"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font>
 ;<br/><b>consider </b><font color="Maroon">i1</font> being   <a href="ami_1.html#NM2">Instruction-Location</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E5:97_2"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font>
 <b>and </b><br/><a NAME="E6:97_2"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</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">k</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="amistd_1.html#K7">locnum</a> <font color="Maroon">i1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="amistd_2.html#E5:97"><i><font color="Green">E57</font></i></a>, <a class="txt" href="amistd_2.html#E1:97_2"><i><font color="Green">E58</font></i></a>, <a class="txt" href="amistd_2.html#E2:97_2"><i><font color="Green">E59</font></i></a></i>;<br/><b>consider </b><font color="Maroon">i2</font> being   <a href="ami_1.html#NM2">Instruction-Location</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E8:97_2"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font>
 <b>and </b><br/><a NAME="E9:97_2"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</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">k</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="amistd_1.html#K7">locnum</a> <font color="Maroon">i2</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="amistd_2.html#E5:97"><i><font color="Green">E57</font></i></a>, <a class="txt" href="amistd_2.html#E1:97_2"><i><font color="Green">E58</font></i></a>, </i>;<br/><a NAME="E10:97_2"/>
<font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="amistd_1.html#K7">locnum</a> <font color="Maroon">i1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="amistd_1.html#K7">locnum</a> <font color="Maroon">i2</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="amistd_2.html#E5:97_2"><i><font color="Green">E63</font></i></a>, <a class="txt" href="amistd_2.html#E8:97_2"><i><font color="Green">E65</font></i></a>, <a class="ref" href="amistd_1.html#T25">AMISTD_1:25</a></i>;<br/><a NAME="E11:97_2"/><b>then </b>
 <a href="amistd_1.html#K7">locnum</a> <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a>  <a href="amistd_1.html#K7">locnum</a> <font color="Maroon">i2</font>
 ;<br/><b>hence </b><a NAME="E12:97_2"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 <b>by </b><i><a class="txt" href="amistd_2.html#E3:97_2"><i><font color="Green">E62</font></i></a>, <a class="txt" href="amistd_2.html#E6:97_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="amistd_1.html#T27">AMISTD_1:27</a></i>;<br/></div>
<b>end;</b></div>

<b>thus </b><a NAME="E7:97"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</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#E5:97"><i><font color="Green">E57</font></i></a></i>;<br/>

<a NAME="E8:97"/><i><font color="Green">E67</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">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">m</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k</font></span>)</span></span>)</span> where <font color="Olive">m</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">m</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/>
<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">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:97_3"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 ;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:97_3"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E4:97_3"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>consider </b><font color="Maroon">il</font> being   <a href="ami_1.html#NM2">Instruction-Location</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E6:97_3"/><i><font color="Green">E82</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">il</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <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">k</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="amistd_1.html#K7">locnum</a> <font color="Maroon">il</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="amistd_2.html#E5:97"><i><font color="Green">E57</font></i></a>, , </i>;<br/><b>consider </b><font color="Maroon">a</font> being  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E8:97_3"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">il</font> <a href="hidden.html#R1">=</a>  <a href="amistd_1.html#K5">il.</a> <font color="Maroon">S</font>,<font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="amistd_1.html#T26">AMISTD_1:26</a></i>;<br/><b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font> as    <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="E10:97_3"/>
<font color="Maroon">a</font> <a href="hidden.html#R1">=</a>  <a href="amistd_1.html#K7">locnum</a> <font color="Maroon">il</font>
 <b>by </b><i>, <a class="ref" href="amistd_1.html#D13">AMISTD_1:def 13</a></i>;<br/><b>hence </b><a NAME="E11:97_3"/>
<font color="Maroon">y</font> <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">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="amistd_2.html#E5:97"><i><font color="Green">E57</font></i></a>, , , , , </i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">y</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="E10:97"/>
<font color="Maroon">y</font> <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">k</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">m</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="E12:97"/><i><font color="Green">E132</font></i>: 
<font color="Maroon">y</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">m</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k</font></span>)</span>
 <b>and </b><br/><a NAME="E13:97"/><i><font color="Green">E133</font></i>: 
 <a href="amistd_1.html#K5">il.</a> <font color="Maroon">S</font>,<font color="Maroon">m</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/>;<br/>
<b>consider </b><font color="Maroon">il</font> being   <a href="ami_1.html#NM2">Instruction-Location</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E15:97"/><i><font color="Green">E134</font></i>: 
(  <a href="amistd_1.html#K5">il.</a> <font color="Maroon">S</font>,<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">il</font> &amp; <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">m</font></span>)</span> <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">k</font> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="amistd_1.html#K7">locnum</a> <font color="Maroon">il</font></span>)</span></span>)</span> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E16:97"/>
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a>  <a href="amistd_1.html#K7">locnum</a> <font color="Maroon">il</font>
 
<b>by </b><i>, <a class="ref" href="amistd_1.html#D13">AMISTD_1:def 13</a></i>;<br/>
<b>hence </b><a NAME="E17:97"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="txt" href="amistd_2.html#E5:97"><i><font color="Green">E57</font></i></a>, , , , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div>
