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

<b>let </b><font color="Maroon" title="c1">N</font> be  <a href="setfam_1.html#V1" title="SETFAM_1:attr.1">with_non-empty_elements</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">S</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> non <a href="ami_1.html#V2" title="AMI_1:attr.2">void</a> <a href="ami_1.html#V5" title="AMI_1:attr.5">IC-Ins-separated</a> <a href="ami_1.html#V8" title="AMI_1:attr.8">definite</a> <a href="amistd_1.html#V4" title="AMISTD_1:attr.4">standard</a>  <a href="ami_1.html#L1" title="AMI_1:struct.1">AMI-Struct</a> of <font color="Maroon" title="c1">N</font>;<br/><b>let </b><font color="Maroon" title="c3">F</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="ami_1.html#U2" title="AMI_1:sel.2">Instruction-Locations</a> of <font color="Maroon" title="c2">S</font>;<br/>





<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:46_1"/>
( <font color="Maroon" title="c3">F</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> or  not <font color="Maroon" title="c3">F</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_1"><b>suppose </b></a><a NAME="E1:46_1_1"/>
<font color="Maroon" title="c3">F</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c4">F</font> = <font color="Maroon" title="c3">F</font> as  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="ami_1.html#U2" title="AMI_1:sel.2">Instruction-Locations</a> of <font color="Maroon" title="c2">S</font> ;<br/><a NAME="E3:46_1_1"/>
 <a href="amistd_3.html#K3" title="AMISTD_3:func.3">LocNums</a> <font color="Maroon" title="c4">F</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><b>hence </b><a NAME="E4:46_1_1"/>
<font color="Maroon" title="c3">F</font>, <a href="amistd_3.html#K3" title="AMISTD_3:func.3">LocNums</a> <font color="Maroon" title="c3">F</font> <a href="wellord2.html#R2" title="WELLORD2:pred.2">are_equipotent</a> 
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_2"><b>suppose </b></a><a NAME="E1:46_1_2"/><i><font color="Green" title="E24">A1</font></i>: 
not <font color="Maroon" title="c3">F</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 ;<br/><div class="add"><a NAME="E2:46_1_2"/>
 <a href="amistd_3.html#K3" title="AMISTD_3:func.3">LocNums</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c3">F</font> <a href="wellord2.html#R2" title="WELLORD2:pred.2">are_equipotent</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_2_1"><b>proof </b></a><div class="add">

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">n</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 <br/>( <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="amistd_1.html#K5" title="AMISTD_1:func.5">il.</a> <font color="Maroon" title="c2">S</font>,<font color="Olive" title="b1">n</font> );<br/>
<a NAME="E1:46_1_2_1"/><i><font color="Green" title="E25">A2</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="amistd_3.html#K3" title="AMISTD_3:func.3">LocNums</a> <font color="Maroon" title="c3">F</font> holds <br/> ex <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:46_1_2_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="amistd_3.html#K3" title="AMISTD_3:func.3">LocNums</a> <font color="Maroon" title="c3">F</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">l</font> being   <a href="ami_1.html#NM2" title="AMI_1:NM.2">Instruction-Location</a> of <font color="Maroon" title="c2">S</font><b> such that </b><br/><a NAME="E3:46_1_2_1_1"/><i><font color="Green" title="E26">A3</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="amistd_1.html#K7" title="AMISTD_1:func.7">locnum</a> <font color="Maroon" title="c5">l</font> &amp; <font color="Maroon" title="c5">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font> )
 ;<br/>
<b>take </b>
<font color="Maroon" title="c5">l</font>
;<br/>

<b>thus </b><a NAME="E4:46_1_2_1_1"/>
<font color="Maroon" title="c5">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font>
 <b>by </b><i><a class="txt" href="amistd_3.html#E3:46_1_2_1_1"><i><font color="Green" title="E26">A3</font></i></a></i>;<br/>

<b>take </b>
 <a href="amistd_1.html#K7" title="AMISTD_1:func.7">locnum</a> <font color="Maroon" title="c5">l</font>
;<br/>

<b>thus </b><a NAME="E5:46_1_2_1_1"/>
(  <a href="amistd_1.html#K7" title="AMISTD_1:func.7">locnum</a> <font color="Maroon" title="c5">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font> &amp; <font color="Maroon" title="c5">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="amistd_1.html#K5" title="AMISTD_1:func.5">il.</a> <font color="Maroon" title="c2">S</font>,<span class="p1">(<span class="default"><a href="amistd_1.html#K7" title="AMISTD_1:func.7">locnum</a> <font color="Maroon" title="c5">l</font></span>)</span> )
 <b>by </b><i><a class="txt" href="amistd_3.html#E3:46_1_2_1_1"><i><font color="Green" title="E26">A3</font></i></a>, <a class="ref" href="amistd_1.html#D13" title="AMISTD_1:def.13">AMISTD_1:def 13</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c4">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="amistd_3.html#K3" title="AMISTD_3:func.3">LocNums</a> <font color="Maroon" title="c3">F</font>,<font color="Maroon" title="c3">F</font><b> such that </b><br/><a NAME="E3:46_1_2_1"/><i><font color="Green" title="E26">A4</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="amistd_3.html#K3" title="AMISTD_3:func.3">LocNums</a> <font color="Maroon" title="c3">F</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S1" title="FUNCT_2:sch.1">FUNCT_2:sch 1</a>(<a class="txt" href="amistd_3.html#E1:46_1_2_1"><i><font color="Green" title="E25">A2</font></i></a>);<br/></i>
<b>take </b>
<font color="Maroon" title="c4">f</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="wellord2.html#D4" title="WELLORD2:def.4">WELLORD2:def 4</a><br/>

<b>thus </b><a NAME="E4:46_1_2_1"/>
<font color="Maroon" title="c4">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_2_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font>, <font color="Maroon" title="c6">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:46_1_2_1_2"/><i><font color="Green" title="E27">A5</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">f</font>
 <b>and </b><br/><a NAME="E2:46_1_2_1_2"/><i><font color="Green" title="E28">A6</font></i>: 
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">f</font>
 <b>and </b><br/><a NAME="E3:46_1_2_1_2"/><i><font color="Green" title="E29">A7</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y</font>
 ;<br/>

<a NAME="E4:46_1_2_1_2"/><i><font color="Green" title="E30">A8</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="amistd_3.html#K3" title="AMISTD_3:func.3">LocNums</a> <font color="Maroon" title="c3">F</font>
 
<b>by </b><i><a class="txt" href="amistd_3.html#E1:46_1_2"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:46_1_2_1_2"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">x</font>,<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">x</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">y</font>,<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">y</font>] )
 
<b>by </b><i><a class="txt" href="amistd_3.html#E3:46_1_2_1"><i><font color="Green" title="E26">A4</font></i></a>, <a class="txt" href="amistd_3.html#E1:46_1_2_1_2"><i><font color="Green" title="E27">A5</font></i></a>, <a class="txt" href="amistd_3.html#E2:46_1_2_1_2"><i><font color="Green" title="E28">A6</font></i></a>, <a class="txt" href="amistd_3.html#E4:46_1_2_1_2"><i><font color="Green" title="E30">A8</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:46_1_2_1_2"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y</font>
 <b>by </b><i><a class="txt" href="amistd_3.html#E3:46_1_2_1_2"><i><font color="Green" title="E29">A7</font></i></a>, <a class="ref" href="amistd_1.html#T25" title="AMISTD_1:th.25">AMISTD_1:25</a></i>;<br/>


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

<b>thus </b><a NAME="E5:46_1_2_1"/><i><font color="Green" title="E27">A9</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="amistd_3.html#K3" title="AMISTD_3:func.3">LocNums</a> <font color="Maroon" title="c3">F</font>
 <b>by </b><i><a class="txt" href="amistd_3.html#E1:46_1_2"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>

<b>thus </b><a NAME="E6:46_1_2_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">F</font>
 <b>by </b><i><a class="ref" href="relset_1.html#T12" title="RELSET_1:th.12">RELSET_1:12</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><br/>

<b>let </b><font color="Maroon" title="c5">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E7:46_1_2_1"/><i><font color="Green" title="E28">A10</font></i>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c6">l</font> = <font color="Maroon" title="c5">a</font> as   <a href="ami_1.html#NM2" title="AMI_1:NM.2">Instruction-Location</a> of <font color="Maroon" title="c2">S</font> ;<br/>
<a NAME="E9:46_1_2_1"/><i><font color="Green" title="E29">A11</font></i>: 
 <a href="amistd_1.html#K7" title="AMISTD_1:func.7">locnum</a> <font color="Maroon" title="c6">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">f</font>
 
<b>by </b><i><a class="txt" href="amistd_3.html#E5:46_1_2_1"><i><font color="Green" title="E27">A9</font></i></a>, <a class="txt" href="amistd_3.html#E7:46_1_2_1"><i><font color="Green" title="E28">A10</font></i></a></i>;<br/>
<a NAME="E10:46_1_2_1"/><b>then </b>
<font color="Maroon">S<sub>1</sub></font>[ <a href="amistd_1.html#K7" title="AMISTD_1:func.7">locnum</a> <font color="Maroon" title="c6">l</font>,<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="amistd_1.html#K7" title="AMISTD_1:func.7">locnum</a> <font color="Maroon" title="c6">l</font></span>)</span>]
 
<b>by </b><i><a class="txt" href="amistd_3.html#E3:46_1_2_1"><i><font color="Green" title="E26">A4</font></i></a>, <a class="txt" href="amistd_3.html#E5:46_1_2_1"><i><font color="Green" title="E27">A9</font></i></a></i>;<br/>
<a NAME="E11:46_1_2_1"/><b>then </b>
<font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="amistd_1.html#K7" title="AMISTD_1:func.7">locnum</a> <font color="Maroon" title="c6">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a</font>
 
<b>by </b><i><a class="ref" href="amistd_1.html#D13" title="AMISTD_1:def.13">AMISTD_1:def 13</a></i>;<br/>
<b>hence </b><a NAME="E12:46_1_2_1"/>
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">f</font>
 <b>by </b><i><a class="txt" href="amistd_3.html#E9:46_1_2_1"><i><font color="Green" title="E29">A11</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:46_1_2"/>
<font color="Maroon" title="c3">F</font>, <a href="amistd_3.html#K3" title="AMISTD_3:func.3">LocNums</a> <font color="Maroon" title="c3">F</font> <a href="wellord2.html#R2" title="WELLORD2:pred.2">are_equipotent</a> 
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
