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<div class="add">

<b>let </b><font color="Maroon" title="c1">k</font> be  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <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>-&gt; </b>   <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>  =  <a href="ami_3.html#K12" title="AMI_3:func.12">il.</a> $1;<br/>
<a NAME="E1:58"/><i><font color="Green" title="E50">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">H<sub>1</sub></font>(<font color="Olive" title="b1">k</font>) is    <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>consider </b><font color="Maroon" title="c2">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a> , <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:58"/><i><font color="Green" title="E51">A2</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="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">k</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S9" title="FUNCT_2:sch.9">FUNCT_2:sch 9</a>(<a class="txt" href="ami_6.html#E1:58"><i><font color="Green" title="E50">A1</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon" title="c3">f</font> = <font color="Maroon" title="c2">f</font> as    <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="ami_3.html#K1" title="AMI_3:func.1">SCM</a>  <b>by </b><i><a class="ref" href="ami_1.html#D36" title="AMI_1:def.36">AMI_1:def 36</a></i>;<br/>
<a NAME="E5:58"/><i><font color="Green" title="E52">A3</font></i>: 
<font color="Maroon" title="c3">f</font> is <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>
 
<b>by </b><i><a class="txt" href="ami_6.html#E3:58"><i><font color="Green" title="E51">A2</font></i></a>, <a class="ref" href="ami_6.html#T53" target="_self" title="AMI_6:th.53">Th53</a></i>;<br/>
<a NAME="E6:58"/><i><font color="Green" title="E53">A4</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 <br/> ( <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0: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="c3">f</font> <a href="ami_1.html#K25" title="AMI_1:func.25">.</a> <font color="Olive" title="b1">k</font></span>)</span> &amp; ( for <font color="Olive" title="b2">j</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>   st <font color="Maroon" title="c3">f</font> <a href="ami_1.html#K25" title="AMI_1:func.25">.</a> <font color="Olive" title="b2">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="c3">f</font> <a href="ami_1.html#K25" title="AMI_1:func.25">.</a> <font color="Olive" title="b1">k</font></span>)</span> holds <br/><font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">j</font> ) )
 
<b>by </b><i><a class="txt" href="ami_6.html#E3:58"><i><font color="Green" title="E51">A2</font></i></a>, <a class="ref" href="ami_6.html#T53" target="_self" title="AMI_6:th.53">Th53</a></i>;<br/>
<a NAME="E7:58"/>
 ex <font color="Olive" title="b1">f</font> being   <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="ami_3.html#K1" title="AMI_3:func.1">SCM</a>  st <br/>( <font color="Olive" title="b1">f</font> is <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> &amp; ( for <font color="Olive" title="b2">m</font>, <font color="Olive" title="b3">n</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 <br/> ( <font color="Olive" title="b2">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">n</font> iff <font color="Olive" title="b1">f</font> <a href="ami_1.html#K25" title="AMI_1:func.25">.</a> <font color="Olive" title="b2">m</font> <a href="amistd_1.html#R1" title="AMISTD_1:pred.1">&lt;=</a> <font color="Olive" title="b1">f</font> <a href="ami_1.html#K25" title="AMI_1:func.25">.</a> <font color="Olive" title="b3">n</font> ) ) &amp;  <a href="ami_3.html#K12" title="AMI_3:func.12">il.</a> <font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">k</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon" title="c3">f</font>
;<br/>

<b>thus </b><a NAME="E1:58_1"/>
<font color="Maroon" title="c3">f</font> is <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>
 <b>by </b><i><a class="txt" href="ami_6.html#E3:58"><i><font color="Green" title="E51">A2</font></i></a>, <a class="ref" href="ami_6.html#T53" target="_self" title="AMI_6:th.53">Th53</a></i>;<br/>

<b>thus </b><a NAME="E2:58_1"/>
for <font color="Olive" title="b1">m</font>, <font color="Olive" title="b2">n</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 <br/> ( <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">n</font> iff <font color="Maroon" title="c3">f</font> <a href="ami_1.html#K25" title="AMI_1:func.25">.</a> <font color="Olive" title="b1">m</font> <a href="amistd_1.html#R1" title="AMISTD_1:pred.1">&lt;=</a> <font color="Maroon" title="c3">f</font> <a href="ami_1.html#K25" title="AMI_1:func.25">.</a> <font color="Olive" title="b2">n</font> )
 <b>by </b><i><a class="txt" href="ami_6.html#E5:58"><i><font color="Green" title="E52">A3</font></i></a>, <a class="txt" href="ami_6.html#E6:58"><i><font color="Green" title="E53">A4</font></i></a>, <a class="ref" href="amistd_1.html#T18" title="AMISTD_1:th.18">AMISTD_1:18</a></i>;<br/>

<a NAME="E3:58_1"/>
 <a href="ami_3.html#K12" title="AMI_3:func.12">il.</a> <font color="Maroon" title="c1">k</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> 
 
<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>hence </b><a NAME="E4:58_1"/>
 <a href="ami_3.html#K12" title="AMI_3:func.12">il.</a> <font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">k</font>
 <b>by </b><i><a class="txt" href="ami_6.html#E3:58"><i><font color="Green" title="E51">A2</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:58"/>
 <a href="amistd_1.html#K5" title="AMISTD_1:func.5">il.</a> <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> ,<font color="Maroon" title="c1">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="c1">k</font>
 <b>by </b><i><a class="ref" href="amistd_1.html#D12" title="AMISTD_1:def.12">AMISTD_1:def 12</a></i>;<br/>


</div>
