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<b>let </b><font color="Maroon">N</font> be  <a href="setfam_1.html#V1">with_non-empty_elements</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">S</font> be  non <a href="struct_0.html#V3">empty</a> non <a href="ami_1.html#V2">void</a> <a href="ami_1.html#V5">IC-Ins-separated</a> <a href="ami_1.html#V8">definite</a>  <a href="ami_1.html#L1">AMI-Struct</a> of <font color="Maroon">N</font>;<br/>



<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:30_1"/>
<font color="Maroon">S</font> is <a href="amistd_1.html#V4" target="_self">standard</a>
 ;<br/><b>then 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 <font color="Maroon">S</font><b> such that </b><br/><a NAME="E3:30_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">f</font> is <a href="funct_2.html#V3">bijective</a>
 <b>and </b><br/><a NAME="E4:30_1"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">m</font>, <font color="Olive">n</font> being   <a href="subset_1.html#M2">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" target="_self">&lt;=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font> )
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E5:30_1"/>
 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 <font color="Maroon">S</font> st <br/>( <font color="Olive">f</font> is <a href="funct_2.html#V3">bijective</a> &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <br/> ( <font color="Olive">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" target="_self">SUCC</a> <span class="p1">(<span class="default"><font color="Olive">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#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">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" target="_self">SUCC</a> <span class="p1">(<span class="default"><font color="Olive">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> ) ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1"><b>proof </b></a><div class="add">

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

<b>thus </b><a NAME="E1:30_1_1"/>
<font color="Maroon">f</font> is <a href="funct_2.html#V3">bijective</a>
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E2:30_1_1"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">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" target="_self">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#M2">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" target="_self">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>, , </i>;<br/>


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

<b>given </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 <font color="Maroon">S</font><b> such that </b><a NAME="E2:30"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">f</font> is <a href="funct_2.html#V3">bijective</a>
 <b>and </b><br/><a NAME="E3:30"/><i><font color="Green">E40</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">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" target="_self">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#M2">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" target="_self">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> ) )
 ;<br/>

<b>take </b>
<font color="Maroon">f</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="amistd_1.html#D10" target="_self">AMISTD_1:def 10</a><br/>

<b>thus </b><a NAME="E4:30"/>
<font color="Maroon">f</font> is <a href="funct_2.html#V3">bijective</a>
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E5:30"/>
for <font color="Olive">m</font>, <font color="Olive">n</font> being   <a href="subset_1.html#M2">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" target="_self">&lt;=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font> )
 <b>by </b><i>, , </i>;<br/>


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