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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="scmring1.html#V1">good</a> <a href="funcsdom.html#NM1">Ring</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,the <a href="ami_1.html#U2">Instruction-Locations</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1">SCM</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>



<b>assume </b><a NAME="E1:76"/><i><font color="Green">E71</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="ami_3.html#K16">il.</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/>

<a NAME="E2:76"/><i><font color="Green">E72</font></i>: 
the <a href="ami_1.html#U2">Instruction-Locations</a> of <a href="ami_3.html#K1">SCM</a>  <a href="hidden.html#R1">=</a> the <a href="ami_1.html#U2">Instruction-Locations</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1">SCM</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="ami_3.html#D1">AMI_3:def 1</a>, <a class="ref" href="scmring2.html#D1">SCMRING2:def 1</a></i>;<br/>
<b>thus </b><a NAME="E3:76"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="funct_2.html#V3">bijective</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:76_1"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
  <i><font color="Red">:: according to </font></i><a class="ref" href="funct_2.html#D4">FUNCT_2:def 4</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:76_1_1"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>and </b><br/><a NAME="E2:76_1_1"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="scmring3.html#E1:76_1_1"><i><font color="Green">E74</font></i></a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E4:76_1_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="ami_3.html#K16">il.</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="ami_3.html#K16">il.</a> <font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="txt" href="scmring3.html#E1:76"><i><font color="Green">E71</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:76_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="scmring3.html#E2:76_1_1"><i><font color="Green">E75</font></i></a>, <a class="ref" href="ami_3.html#T53">AMI_3:53</a></i>;<br/>


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

<b>thus </b><a NAME="E2:76_1"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="funct_2.html#V2">onto</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:76_1_2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a> the <a href="ami_1.html#U2">Instruction-Locations</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1">SCM</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a>,<a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>thus </b><a NAME="E2:76_1_2"/>
the <a href="ami_1.html#U2">Instruction-Locations</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1">SCM</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></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="E1:76_1_2_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> the <a href="ami_1.html#U2">Instruction-Locations</a> of <span class="p1">(<span class="default"><a href="scmring2.html#K1">SCM</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:76_1_2_1"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="ami_3.html#K16">il.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="scmring3.html#E2:76"><i><font color="Green">E72</font></i></a>, <a class="ref" href="ami_5.html#T19">AMI_5:19</a></i>;<br/>
<a NAME="E4:76_1_2_1"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:76_1_2_1"/><b>then </b>
(  <a href="ami_3.html#K16">il.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="scmring3.html#E1:76"><i><font color="Green">E71</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:76_1_2_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="scmring3.html#E3:76_1_2_1"><i><font color="Green">E74</font></i></a>, <a class="ref" href="funct_1.html#D5">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">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E4:76"/><i><font color="Green">E74</font></i>: 
 <a href="amistd_1.html#K3">SUCC</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="scmring2.html#K11">Next</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i><a class="ref" href="scmring3.html#T64" target="_self">Th64</a></i>;<br/>
<a NAME="E5:76"/><i><font color="Green">E75</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="ami_3.html#K16">il.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="ami_3.html#K16">il.</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="txt" href="scmring3.html#E1:76"><i><font color="Green">E71</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="subset_1.html#M2">Element</a> of  <a href="ami_2.html#K3">SCM-Instr-Loc</a> <b> such that </b><br/><a NAME="E7:76"/><i><font color="Green">E76</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="ami_2.html#K15">Next</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="scmring2.html#K11">Next</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="scmring2.html#D10">SCMRING2:def 10</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="subset_1.html#M2">Element</a> of  <a href="ami_2.html#K3">SCM-Instr-Loc</a> <b> such that </b><br/><a NAME="E9:76"/><i><font color="Green">E77</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="ami_3.html#K16">il.</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="ami_3.html#K11">Next</a> <span class="p1">(<span class="default"><a href="ami_3.html#K16">il.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ami_2.html#K15">Next</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="ami_3.html#D11">AMI_3:def 11</a></i>;<br/>
<i><font color="Green">E78</font></i>: <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
 <a href="ami_3.html#K16">il.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="scmring3.html#E1:76"><i><font color="Green">E71</font></i></a></i>
<br/>.= 
 <a href="ami_3.html#K11">Next</a> <span class="p1">(<span class="default"><a href="ami_3.html#K16">il.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="ami_3.html#T54">AMI_3:54</a></i>
;<br/>
<b>hence </b><a NAME="E11:76"/>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="amistd_1.html#K3">SUCC</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="scmring3.html#E4:76"><i><font color="Green">E74</font></i></a>, <a class="txt" href="scmring3.html#E5:76"><i><font color="Green">E75</font></i></a>, <a class="txt" href="scmring3.html#E7:76"><i><font color="Green">E76</font></i></a>, <a class="txt" href="scmring3.html#E9:76"><i><font color="Green">E77</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E12:76"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="amistd_1.html#K3">SUCC</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 ;<br/>

<a NAME="E13:76"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="scmring3.html#E3:76"><i><font color="Green">E73</font></i></a>, <a class="ref" href="funct_2.html#D4">FUNCT_2:def 4</a></i>;<br/>
<a NAME="E14:76"/><i><font color="Green">E81</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:76_3"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font> or <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="scmring2.html#K11">Next</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> )
 </span><b>by </b><i><a class="txt" href="scmring3.html#E4:76"><i><font color="Green">E74</font></i></a>, <a class="txt" href="scmring3.html#E12:76"><i><font color="Green">E79</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_3_1"><b>suppose </b></a><a NAME="E1:76_3_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:76_3_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="scmring3.html#E13:76"><i><font color="Green">E80</font></i></a>, <a class="txt" href="scmring3.html#E14:76"><i><font color="Green">E81</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="76_3_2"><b>suppose </b></a><a NAME="E1:76_3_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="scmring2.html#K11">Next</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:76_3_2"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="scmring3.html#E5:76"><i><font color="Green">E75</font></i></a>, <a class="txt" href="scmring3.html#E7:76"><i><font color="Green">E76</font></i></a>, <a class="txt" href="scmring3.html#E9:76"><i><font color="Green">E77</font></i></a>, <a class="txt" href="scmring3.html#E10:76"><i><font color="Green">E78</font></i></a>, <a class="txt" href="scmring3.html#E13:76"><i><font color="Green">E80</font></i></a>, <a class="txt" href="scmring3.html#E14:76"><i><font color="Green">E81</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><b>hence </b><a NAME="E3:76_3_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
