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

<b>let </b><font color="Maroon" title="c1">f</font> be    <a href="ami_1.html#M4" title="AMI_1:mode.4">IL-Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> ;<br/>

<b>assume </b><a NAME="E1:59"/><i><font color="Green" title="E51">A1</font></i>: 
for <font color="Olive" title="b1">k</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>  holds  <font color="Maroon" title="c1">f</font> <a href="ami_1.html#K24" title="AMI_1:func.24">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_3.html#K13" title="AMI_3:func.13">il.</a> <font color="Olive" title="b1">k</font>
 ;<br/>

<b>thus </b><a NAME="E2:59"/><i><font color="Green" title="E52">A2</font></i>: 
<font color="Maroon" title="c1">f</font> is <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:59_1"/>
<font color="Maroon" title="c1">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
  <i><font color="Red">:: according to </font></i><a class="ref" href="funct_2.html#D4" title="FUNCT_2:def.4">FUNCT_2:def 4</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">x1</font>, <font color="Maroon" title="c3">x2</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="E1:59_1_1"/><i><font color="Green" title="E53">A3</font></i>: 
( <font color="Maroon" title="c2">x1</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="c1">f</font> &amp; <font color="Maroon" title="c3">x2</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="c1">f</font> )
 <b>and </b><br/><a NAME="E2:59_1_1"/><i><font color="Green" title="E54">A4</font></i>: 
<font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x2</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c4">k1</font> = <font color="Maroon" title="c2">x1</font>, <font color="Maroon" title="c5">k2</font> = <font color="Maroon" title="c3">x2</font> as    <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>  <b>by </b><i><a class="txt" href="ami_6.html#E1:59_1_1"><i><font color="Green" title="E53">A3</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="E4:59_1_1"/>
( <font color="Maroon" title="c1">f</font> <a href="ami_1.html#K24" title="AMI_1:func.24">.</a> <font color="Maroon" title="c4">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_3.html#K13" title="AMI_3:func.13">il.</a> <font color="Maroon" title="c4">k1</font> &amp; <font color="Maroon" title="c1">f</font> <a href="ami_1.html#K24" title="AMI_1:func.24">.</a> <font color="Maroon" title="c5">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_3.html#K13" title="AMI_3:func.13">il.</a> <font color="Maroon" title="c5">k2</font> )
 
<b>by </b><i><a class="txt" href="ami_6.html#E1:59"><i><font color="Green" title="E51">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:59_1_1"/>
<font color="Maroon" title="c2">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x2</font>
 <b>by </b><i><a class="txt" href="ami_6.html#E2:59_1_1"><i><font color="Green" title="E54">A4</font></i></a></i>;<br/>


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

<b>thus </b><a NAME="E2:59_1"/>
<font color="Maroon" title="c1">f</font> is <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:59_1_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="ami_1.html#U2" title="AMI_1:sel.2">Instruction-Locations</a> of <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> 
 <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="funct_2.html#D3" title="FUNCT_2:def.3">FUNCT_2:def 3</a>,<a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><br/>

<b>thus </b><a NAME="E2:59_1_2"/>
the <a href="ami_1.html#U2" title="AMI_1:sel.2">Instruction-Locations</a> of <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c1">f</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">x</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="E1:59_1_2_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="ami_1.html#U2" title="AMI_1:sel.2">Instruction-Locations</a> of <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> 
 ;<br/>

<a NAME="E2:59_1_2_1"/><b>then </b>
<font color="Maroon" title="c2">x</font> is    <a href="ami_1.html#M1" title="AMI_1:mode.1">Instruction-Location</a> of  <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> 
 
<b>by </b><i><a class="ref" href="ami_1.html#D4" title="AMI_1:def.4">AMI_1:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c3">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> <b> such that </b><br/><a NAME="E4:59_1_2_1"/><i><font color="Green" title="E53">A5</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_3.html#K13" title="AMI_3:func.13">il.</a> <font color="Maroon" title="c3">i</font>
 <b>by </b><i><a class="ref" href="ami_5.html#T19" title="AMI_5:th.19">AMI_5:19</a></i>;<br/>
<a NAME="E5:59_1_2_1"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E6:59_1_2_1"/><b>then </b>
(  <a href="ami_3.html#K13" title="AMI_3:func.13">il.</a> <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="ami_1.html#K24" title="AMI_1:func.24">.</a> <font color="Maroon" title="c3">i</font> &amp; <font color="Maroon" title="c3">i</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="c1">f</font> )
 
<b>by </b><i><a class="txt" href="ami_6.html#E1:59"><i><font color="Green" title="E51">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:59_1_2_1"/>
<font color="Maroon" title="c2">x</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="c1">f</font>
 <b>by </b><i><a class="txt" href="ami_6.html#E4:59_1_2_1"><i><font color="Green" title="E53">A5</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>


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


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

<b>let </b><font color="Maroon" title="c2">k</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> ;<br/>

<a NAME="E3:59"/><i><font color="Green" title="E53">A6</font></i>: 
 <a href="amistd_1.html#K3" title="AMISTD_1:func.3">SUCC</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="ami_1.html#K24" title="AMI_1:func.24">.</a> <font color="Maroon" title="c2">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="ami_1.html#K24" title="AMI_1:func.24">.</a> <font color="Maroon" title="c2">k</font></span>)</span>,<span class="p2">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">Next</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="ami_1.html#K24" title="AMI_1:func.24">.</a> <font color="Maroon" title="c2">k</font></span>)</span></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<b>by </b><i><a class="ref" href="ami_6.html#T52" target="_self" title="AMI_6:th.52">Th52</a></i>;<br/>
<a NAME="E4:59"/><i><font color="Green" title="E54">A7</font></i>: 
( <font color="Maroon" title="c1">f</font> <a href="ami_1.html#K24" title="AMI_1:func.24">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_3.html#K13" title="AMI_3:func.13">il.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c1">f</font> <a href="ami_1.html#K24" title="AMI_1:func.24">.</a> <font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_3.html#K13" title="AMI_3:func.13">il.</a> <font color="Maroon" title="c2">k</font> )
 
<b>by </b><i><a class="txt" href="ami_6.html#E1:59"><i><font color="Green" title="E51">A1</font></i></a></i>;<br/>
<i><font color="Green" title="E55">A8</font></i>: <font color="Maroon" title="c1">f</font> <a href="ami_1.html#K24" title="AMI_1:func.24">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
 <a href="ami_3.html#K13" title="AMI_3:func.13">il.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="ami_6.html#E1:59"><i><font color="Green" title="E51">A1</font></i></a></i>
<br/>.= 
 <a href="ami_3.html#K11" title="AMI_3:func.11">Next</a> <span class="p1">(<span class="default"><a href="ami_3.html#K13" title="AMI_3:func.13">il.</a> <font color="Maroon" title="c2">k</font></span>)</span>
<b>by </b><i><a class="ref" href="ami_3.html#T54" title="AMI_3:th.54">AMI_3:54</a></i>
;<br/>
<b>hence </b><a NAME="E6:59"/>
<font color="Maroon" title="c1">f</font> <a href="ami_1.html#K24" title="AMI_1:func.24">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="amistd_1.html#K3" title="AMISTD_1:func.3">SUCC</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="ami_1.html#K24" title="AMI_1:func.24">.</a> <font color="Maroon" title="c2">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="ami_6.html#E3:59"><i><font color="Green" title="E53">A6</font></i></a>, <a class="txt" href="ami_6.html#E4:59"><i><font color="Green" title="E54">A7</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>

<b>let </b><font color="Maroon" title="c3">j</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> ;<br/>

<b>assume </b><a NAME="E7:59"/><i><font color="Green" title="E56">A9</font></i>: 
<font color="Maroon" title="c1">f</font> <a href="ami_1.html#K24" title="AMI_1:func.24">.</a> <font color="Maroon" title="c3">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="amistd_1.html#K3" title="AMISTD_1:func.3">SUCC</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="ami_1.html#K24" title="AMI_1:func.24">.</a> <font color="Maroon" title="c2">k</font></span>)</span>
 ;<br/>

<a NAME="E8:59"/><i><font color="Green" title="E57">A10</font></i>: 
<font color="Maroon" title="c1">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="ami_6.html#E2:59"><i><font color="Green" title="E52">A2</font></i></a>, <a class="ref" href="funct_2.html#D4" title="FUNCT_2:def.4">FUNCT_2:def 4</a></i>;<br/>
<a NAME="E9:59"/><i><font color="Green" title="E58">A11</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:59_3"/>
( <font color="Maroon" title="c1">f</font> <a href="ami_1.html#K24" title="AMI_1:func.24">.</a> <font color="Maroon" title="c3">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="ami_1.html#K24" title="AMI_1:func.24">.</a> <font color="Maroon" title="c2">k</font> or <font color="Maroon" title="c1">f</font> <a href="ami_1.html#K24" title="AMI_1:func.24">.</a> <font color="Maroon" title="c3">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_3.html#K11" title="AMI_3:func.11">Next</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="ami_1.html#K24" title="AMI_1:func.24">.</a> <font color="Maroon" title="c2">k</font></span>)</span> )
 </span><b>by </b><i><a class="txt" href="ami_6.html#E3:59"><i><font color="Green" title="E53">A6</font></i></a>, <a class="txt" href="ami_6.html#E7:59"><i><font color="Green" title="E56">A9</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_3_1"><b>suppose </b></a><a NAME="E1:59_3_1"/>
<font color="Maroon" title="c1">f</font> <a href="ami_1.html#K24" title="AMI_1:func.24">.</a> <font color="Maroon" title="c3">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">f</font> <a href="ami_1.html#K24" title="AMI_1:func.24">.</a> <font color="Maroon" title="c2">k</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:59_3_1"/>
<font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">j</font>
 <b>by </b><i><a class="txt" href="ami_6.html#E8:59"><i><font color="Green" title="E57">A10</font></i></a>, <a class="txt" href="ami_6.html#E9:59"><i><font color="Green" title="E58">A11</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_3_2"><b>suppose </b></a><a NAME="E1:59_3_2"/>
<font color="Maroon" title="c1">f</font> <a href="ami_1.html#K24" title="AMI_1:func.24">.</a> <font color="Maroon" title="c3">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_3.html#K11" title="AMI_3:func.11">Next</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">f</font> <a href="ami_1.html#K24" title="AMI_1:func.24">.</a> <font color="Maroon" title="c2">k</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:59_3_2"/><b>then </b>
<font color="Maroon" title="c3">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="ami_6.html#E4:59"><i><font color="Green" title="E54">A7</font></i></a>, <a class="txt" href="ami_6.html#E5:59"><i><font color="Green" title="E55">A8</font></i></a>, <a class="txt" href="ami_6.html#E8:59"><i><font color="Green" title="E57">A10</font></i></a>, <a class="txt" href="ami_6.html#E9:59"><i><font color="Green" title="E58">A11</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/><b>hence </b><a NAME="E3:59_3_2"/>
<font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">j</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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