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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">T</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="amistd_1.html#V4" target="_self">standard</a>  <a href="ami_1.html#L1">AMI-Struct</a> of <font color="Maroon">N</font>;<br/><b>let </b><font color="Maroon">F</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="ami_3.html#V1">programmed</a> <a href="amistd_1.html#V9" target="_self">lower</a>  <a href="ami_1.html#M1">FinPartState</a> of <font color="Maroon">T</font>;<br/>





<b>consider </b><font color="Maroon">l</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E2:82"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">l</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<a NAME="E3:82"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> the <a href="ami_1.html#U2">Instruction-Locations</a> of <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="ami_3.html#D13">AMI_3:def 13</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">l</font> = <font color="Maroon">l</font> as   <a href="ami_1.html#NM2">Instruction-Location</a> of <font color="Maroon">T</font> <b>by </b><i/>;<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 <font color="Maroon">T</font><b> such that </b><br/><a NAME="E6:82"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">f</font> is <a href="funct_2.html#V3">bijective</a>
 <b>and </b><br/><a NAME="E7:82"/><i><font color="Green">E39</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>and </b><br/><a NAME="E8:82"/><i><font color="Green">E40</font></i>: 
 <a href="amistd_1.html#K5" target="_self">il.</a> <font color="Maroon">T</font>,0 <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> 0
 <b>by </b><i/>;<br/>
<a NAME="E9:82"/>
<font color="Maroon">f</font> is <a href="funct_2.html#V2">onto</a>
 
<b>by </b><i>, <a class="ref" href="funct_2.html#D4">FUNCT_2:def 4</a></i>;<br/>
<a NAME="E10:82"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="ami_1.html#U2">Instruction-Locations</a> of <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E12:82"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E13:82"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/>
<a NAME="E15:82"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">x</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/>
<a NAME="E16:82"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> 0 <a href="amistd_1.html#R1" target="_self">&lt;=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E17:82"/>
 <a href="amistd_1.html#K5" target="_self">il.</a> <font color="Maroon">T</font>,0 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font>
 <b>by </b><i>, , , <a class="txt" href="amistd_1.html#E7:82"><i><font color="Green">E39</font></i></a></i>;<br/>


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