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

<b>let </b><font color="Maroon" title="c1">R</font> be  <a href="scmring1.html#V1" title="SCMRING1:attr.1">good</a> <a href="funcsdom.html#NM1" title="FUNCSDOM:NM.1">Ring</a>;<br/><b>let </b><font color="Maroon" title="c2">f</font> be    <a href="ami_1.html#M6" title="AMI_1:mode.6">IL-Function</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a> , <a href="scmring2.html#K1" title="SCMRING2:func.1">SCM</a> <font color="Maroon" title="c1">R</font>;<br/>



<b>assume </b><a NAME="E1:67"/><i><font color="Green" title="E62">A1</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>  holds  <font color="Maroon" title="c2">f</font> <a href="ami_1.html#K25" title="AMI_1:func.25">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_3.html#K12" title="AMI_3:func.12">il.</a> <font color="Olive" title="b1">k</font>
 ;<br/>

<b>thus </b><a NAME="E2:67"/><i><font color="Green" title="E63">A2</font></i>: 
<font color="Maroon" title="c2">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="67_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:67_1"/>
<font color="Maroon" title="c2">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="67_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x1</font>, <font color="Maroon" title="c4">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="E1:67_1_1"/><i><font color="Green" title="E64">A3</font></i>: 
( <font color="Maroon" title="c3">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f</font> &amp; <font color="Maroon" title="c4">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f</font> )
 <b>and </b><br/><a NAME="E2:67_1_1"/><i><font color="Green" title="E65">A4</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">x2</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c5">k1</font> = <font color="Maroon" title="c3">x1</font>, <font color="Maroon" title="c6">k2</font> = <font color="Maroon" title="c4">x2</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>  <b>by </b><i><a class="txt" href="scmring3.html#E1:67_1_1"><i><font color="Green" title="E64">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="E4:67_1_1"/>
( <font color="Maroon" title="c2">f</font> <a href="ami_1.html#K25" title="AMI_1:func.25">.</a> <font color="Maroon" title="c5">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_3.html#K12" title="AMI_3:func.12">il.</a> <font color="Maroon" title="c5">k1</font> &amp; <font color="Maroon" title="c2">f</font> <a href="ami_1.html#K25" title="AMI_1:func.25">.</a> <font color="Maroon" title="c6">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_3.html#K12" title="AMI_3:func.12">il.</a> <font color="Maroon" title="c6">k2</font> )
 
<b>by </b><i><a class="txt" href="scmring3.html#E1:67"><i><font color="Green" title="E62">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:67_1_1"/>
<font color="Maroon" title="c3">x1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x2</font>
 <b>by </b><i><a class="txt" href="scmring3.html#E2:67_1_1"><i><font color="Green" title="E65">A4</font></i></a></i>;<br/>


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

<b>thus </b><a NAME="E2:67_1"/>
<font color="Maroon" title="c2">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="67_1_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:67_1_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c2">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</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="E2:67_1_2"/>
 <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</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="c2">f</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="67_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">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="E1:67_1_2_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c4">l</font> = <font color="Maroon" title="c3">x</font> as    <a href="ami_1.html#M3" title="AMI_1:mode.3">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>reconsider </b><font color="Maroon" title="c5">i</font> = <font color="Maroon" title="c4">l</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E4:67_1_2_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_2.html#K5" title="AMI_2: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="E5:67_1_2_1"/><b>then </b>
(  <a href="ami_3.html#K12" title="AMI_3:func.12">il.</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="ami_1.html#K25" title="AMI_1:func.25">.</a> <font color="Maroon" title="c5">i</font> &amp; <font color="Maroon" title="c5">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f</font> )
 
<b>by </b><i><a class="txt" href="scmring3.html#E1:67"><i><font color="Green" title="E62">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:67_1_2_1"/>
<font color="Maroon" title="c3">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="c2">f</font>
 <b>by </b><i><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="c3">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a> ;<br/>

<a NAME="E3:67"/><i><font color="Green" title="E64">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="c2">f</font> <a href="ami_1.html#K25" title="AMI_1:func.25">.</a> <font color="Maroon" title="c3">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="c2">f</font> <a href="ami_1.html#K25" title="AMI_1:func.25">.</a> <font color="Maroon" title="c3">k</font></span>)</span>,<span class="p2">(<span class="default"><a href="scmnorm.html#K1" title="SCMNORM:func.1">Next</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="ami_1.html#K25" title="AMI_1:func.25">.</a> <font color="Maroon" title="c3">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="scmring3.html#T64" target="_self" title="SCMRING3:th.64">Th64</a></i>;<br/>
<a NAME="E4:67"/><i><font color="Green" title="E65">A7</font></i>: 
( <font color="Maroon" title="c2">f</font> <a href="ami_1.html#K25" title="AMI_1:func.25">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_3.html#K12" title="AMI_3:func.12">il.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c2">f</font> <a href="ami_1.html#K25" title="AMI_1:func.25">.</a> <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_3.html#K12" title="AMI_3:func.12">il.</a> <font color="Maroon" title="c3">k</font> )
 
<b>by </b><i><a class="txt" href="scmring3.html#E1:67"><i><font color="Green" title="E62">A1</font></i></a></i>;<br/>
<b>thus </b><a NAME="E5:67"/>
<font color="Maroon" title="c2">f</font> <a href="ami_1.html#K25" title="AMI_1:func.25">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</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="c2">f</font> <a href="ami_1.html#K25" title="AMI_1:func.25">.</a> <font color="Maroon" title="c3">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="scmring3.html#E3:67"><i><font color="Green" title="E64">A6</font></i></a>, <a class="txt" href="scmring3.html#E4:67"><i><font color="Green" title="E65">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="c4">j</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E6:67"/><i><font color="Green" title="E66">A11</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="ami_1.html#K25" title="AMI_1:func.25">.</a> <font color="Maroon" title="c4">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="c2">f</font> <a href="ami_1.html#K25" title="AMI_1:func.25">.</a> <font color="Maroon" title="c3">k</font></span>)</span>
 ;<br/>

<a NAME="E7:67"/><i><font color="Green" title="E67">A12</font></i>: 
<font color="Maroon" title="c2">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="scmring3.html#E2:67"><i><font color="Green" title="E63">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="E8:67"/><i><font color="Green" title="E68">A13</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_2.html#K5" title="AMI_2: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="67_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:67_2"/>
( <font color="Maroon" title="c2">f</font> <a href="ami_1.html#K25" title="AMI_1:func.25">.</a> <font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="ami_1.html#K25" title="AMI_1:func.25">.</a> <font color="Maroon" title="c3">k</font> or <font color="Maroon" title="c2">f</font> <a href="ami_1.html#K25" title="AMI_1:func.25">.</a> <font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmnorm.html#K1" title="SCMNORM:func.1">Next</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="ami_1.html#K25" title="AMI_1:func.25">.</a> <font color="Maroon" title="c3">k</font></span>)</span> )
 </span><b>by </b><i><a class="txt" href="scmring3.html#E3:67"><i><font color="Green" title="E64">A6</font></i></a>, <a class="txt" href="scmring3.html#E6:67"><i><font color="Green" title="E66">A11</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="67_2_1"><b>suppose </b></a><a NAME="E1:67_2_1"/>
<font color="Maroon" title="c2">f</font> <a href="ami_1.html#K25" title="AMI_1:func.25">.</a> <font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f</font> <a href="ami_1.html#K25" title="AMI_1:func.25">.</a> <font color="Maroon" title="c3">k</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:67_2_1"/>
<font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">j</font>
 <b>by </b><i><a class="txt" href="scmring3.html#E7:67"><i><font color="Green" title="E67">A12</font></i></a>, <a class="txt" href="scmring3.html#E8:67"><i><font color="Green" title="E68">A13</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="67_2_2"><b>suppose </b></a><a NAME="E1:67_2_2"/>
<font color="Maroon" title="c2">f</font> <a href="ami_1.html#K25" title="AMI_1:func.25">.</a> <font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="scmnorm.html#K1" title="SCMNORM:func.1">Next</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="ami_1.html#K25" title="AMI_1:func.25">.</a> <font color="Maroon" title="c3">k</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:67_2_2"/><b>then </b>
<font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="scmring3.html#E4:67"><i><font color="Green" title="E65">A7</font></i></a>, <a class="txt" href="scmring3.html#E7:67"><i><font color="Green" title="E67">A12</font></i></a>, <a class="txt" href="scmring3.html#E8:67"><i><font color="Green" title="E68">A13</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="E3:67_2_2"/>
<font color="Maroon" title="c3">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">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>

</div>
