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<b>let </b><font color="Maroon">R</font> be  <a href="scmring1.html#V1">good</a> <a href="funcsdom.html#NM1">Ring</a>;<br/><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>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M2">Element</a> of the <a href="ami_1.html#U2">Instruction-Locations</a> of <a href="ami_3.html#K1">SCM</a>  =  <a href="ami_3.html#K16">il.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>consider </b><font color="Maroon">f</font> being   <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 <a href="ami_3.html#K1">SCM</a> <b> such that </b><br/><a NAME="E2:78"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">k</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4">FUNCT_2:sch 4</a>();<br/></i>
<a NAME="E3:78"/>
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">R</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>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as   <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">R</font></span>)</span> ;<br/>
<a NAME="E5:78"/><i><font color="Green">E51</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">k</font> <a href="hidden.html#R1">=</a>  <a href="ami_3.html#K16">il.</a> <font color="Olive">k</font>
 
<b>by </b><i><a class="ref" href="scmring3.html#T10" target="_self">Th10</a></i>;<br/>
<a NAME="E6:78"/><b>then </b><i><font color="Green">E64</font></i>: 
<font color="Maroon">f</font> is <a href="funct_2.html#V3">bijective</a>
 
<b>by </b><i><a class="ref" href="scmring3.html#T8" target="_self">Th8</a></i>;<br/>
<a NAME="E7:78"/><i><font color="Green">E67</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <br/> ( <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">k</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">k</font></span>)</span> &amp; ( for <font color="Olive">j</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">j</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">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">k</font></span>)</span> holds <br/><font color="Olive">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> ) )
 
<b>by </b><i><a class="ref" href="scmring3.html#T11" target="_self">Th11</a>, <a class="ref" href="scmring3.html#T8" target="_self">Th8</a></i>;<br/>
<b>reconsider </b><font color="Maroon">k</font> = <font color="Maroon">k</font> as    <a href="subset_1.html#M1">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="E9:78"/>
 ex <font color="Olive">f</font> being  <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">R</font></span>)</span> st <br/>( <font color="Olive">f</font> is <a href="funct_2.html#V3">bijective</a> &amp; ( for <font color="Olive">m</font>, <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <br/> ( <font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> iff <font color="Olive">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">m</font> <a href="amistd_1.html#R1">&lt;=</a> <font color="Olive">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font> ) ) &amp;  <a href="ami_3.html#K16">il.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_1"><b>proof </b></a><div class="add">

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

<b>thus </b><a NAME="E1:78_1"/>
<font color="Maroon">f</font> is <a href="funct_2.html#V3">bijective</a>
 <b>by </b><i><a class="ref" href="scmring3.html#T11" target="_self">Th11</a>, <a class="ref" href="scmring3.html#T8" target="_self">Th8</a></i>;<br/>

<b>thus </b><a NAME="E2:78_1"/>
for <font color="Olive">m</font>, <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <br/> ( <font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> iff <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">m</font> <a href="amistd_1.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font> )
 <b>by </b><i><a class="ref" href="scmring3.html#T12" target="_self">Th12</a>, <a class="ref" href="scmring3.html#T13" target="_self">Th13</a>, <a class="ref" href="amistd_1.html#T18">AMISTD_1:18</a></i>;<br/>

<b>thus </b><a NAME="E3:78_1"/>
 <a href="ami_3.html#K16">il.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="scmring3.html#T10" target="_self">Th10</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:78"/>
 <a href="amistd_1.html#K5">il.</a> <span class="p1">(<span class="default"><a href="scmring2.html#K1">SCM</a> <font color="Maroon">R</font></span>)</span>,<font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="ami_3.html#K16">il.</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="amistd_1.html#D12">AMISTD_1:def 12</a></i>;<br/>


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