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<b>let </b><font color="Maroon" title="c2">N</font> be  <a href="setfam_1.html#V1" title="SETFAM_1:attr.1">with_non-empty_elements</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">T</font> being non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="ami_1.html#V2" title="AMI_1:attr.2">stored-program</a> <a href="ami_1.html#V5" title="AMI_1:attr.5">IC-Ins-separated</a> <a href="ami_1.html#V7" title="AMI_1:attr.7">definite</a> <a href="amistd_1.html#V4" title="AMISTD_1:attr.4">standard</a>  <a href="ami_1.html#L1" title="AMI_1:struct.1">AMI-Struct</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<font color="Maroon" title="c2">N</font><br/> for <font color="Olive" title="b2">k1</font>, <font color="Olive" title="b3">k2</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st  <a href="amistd_1.html#K5" title="AMISTD_1:func.5">il.</a> <font color="Olive" title="b1">T</font>,<font color="Olive" title="b2">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="amistd_1.html#K5" title="AMISTD_1:func.5">il.</a> <font color="Olive" title="b1">T</font>,<font color="Olive" title="b3">k2</font> holds <br/><font color="Olive" title="b2">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">k2</font></span><br/><b>let </b><font color="Maroon" title="c3">T</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="ami_1.html#V2" title="AMI_1:attr.2">stored-program</a> <a href="ami_1.html#V5" title="AMI_1:attr.5">IC-Ins-separated</a> <a href="ami_1.html#V7" title="AMI_1:attr.7">definite</a> <a href="amistd_1.html#V4" title="AMISTD_1:attr.4">standard</a>  <a href="ami_1.html#L1" title="AMI_1:struct.1">AMI-Struct</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<font color="Maroon" title="c2">N</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">k1</font>, <font color="Olive" title="b2">k2</font> being <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st  <a href="amistd_1.html#K5" title="AMISTD_1:func.5">il.</a> <font color="Maroon" title="c3">T</font>,<font color="Olive" title="b1">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="amistd_1.html#K5" title="AMISTD_1:func.5">il.</a> <font color="Maroon" title="c3">T</font>,<font color="Olive" title="b2">k2</font> holds <br/><font color="Olive" title="b1">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">k2</font></span><br/><b>let </b><font color="Maroon" title="c4">k1</font>, <font color="Maroon" title="c5">k2</font> be  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="amistd_1.html#K5" title="AMISTD_1:func.5">il.</a> <font color="Maroon" title="c3">T</font>,<font color="Maroon" title="c4">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="amistd_1.html#K5" title="AMISTD_1:func.5">il.</a> <font color="Maroon" title="c3">T</font>,<font color="Maroon" title="c5">k2</font> implies <font color="Maroon" title="c4">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k2</font> )</span><br/>





<b>assume </b><a NAME="E1:46"/><i><font color="Green" title="E23">A1</font></i>: 
 <a href="amistd_1.html#K5" title="AMISTD_1:func.5">il.</a> <font color="Maroon" title="c3">T</font>,<font color="Maroon" title="c4">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="amistd_1.html#K5" title="AMISTD_1:func.5">il.</a> <font color="Maroon" title="c3">T</font>,<font color="Maroon" title="c5">k2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k2</font></span><br/>

<a NAME="E2:46"/><i><font color="Green" title="E24">A2</font></i>: 
( <font color="Maroon" title="c4">k1</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  &amp; <font color="Maroon" title="c5">k2</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">f1</font> being    <a href="ami_1.html#M6" title="AMI_1:mode.6">IL-Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<font color="Maroon" title="c3">T</font><b> such that </b><br/><a NAME="E4:46"/><i><font color="Green" title="E25">A3</font></i>: 
( <font color="Maroon" title="c6">f1</font> is <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> &amp; ( for <font color="Olive" title="b1">m</font>, <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS: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="c6">f1</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</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="c6">f1</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Olive" title="b2">n</font> ) ) &amp;  <a href="amistd_1.html#K5" title="AMISTD_1:func.5">il.</a> <font color="Maroon" title="c3">T</font>,<font color="Maroon" title="c4">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">k1</font> )
 <b>by </b><i><a class="ref" href="amistd_1.html#D12" target="_self" title="AMISTD_1:def.12">Def12</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">f2</font> being    <a href="ami_1.html#M6" title="AMI_1:mode.6">IL-Function</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,<font color="Maroon" title="c3">T</font><b> such that </b><br/><a NAME="E6:46"/><i><font color="Green" title="E26">A4</font></i>: 
( <font color="Maroon" title="c7">f2</font> is <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> &amp; ( for <font color="Olive" title="b1">m</font>, <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS: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="c7">f2</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</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="c7">f2</font> <a href="ami_1.html#K26" title="AMI_1:func.26">.</a> <font color="Olive" title="b2">n</font> ) ) &amp;  <a href="amistd_1.html#K5" title="AMISTD_1:func.5">il.</a> <font color="Maroon" title="c3">T</font>,<font color="Maroon" title="c5">k2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">k2</font> )
 <b>by </b><i><a class="ref" href="amistd_1.html#D12" target="_self" title="AMISTD_1:def.12">Def12</a></i>;<br/>
<a NAME="E7:46"/><i><font color="Green" title="E27">A5</font></i>: 
<font color="Maroon" title="c6">f1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c7">f2</font>
 
<b>by </b><i><a class="txt" href="amistd_1.html#E4:46"><i><font color="Green" title="E25">A3</font></i></a>, <a class="txt" href="amistd_1.html#E6:46"><i><font color="Green" title="E26">A4</font></i></a>, <a class="ref" href="amistd_1.html#T17" target="_self" title="AMISTD_1:th.17">Th17</a></i>;<br/>
<a NAME="E8:46"/>
( <font color="Maroon" title="c6">f1</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )
 
<b>by </b><i><a class="txt" href="amistd_1.html#E4:46"><i><font color="Green" title="E25">A3</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="funct_2.html#D4" title="FUNCT_2:def.4">FUNCT_2:def 4</a></i>;<br/>
<b>hence </b><a NAME="E9:46"/>
<font color="Maroon" title="c4">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k2</font>
 <b>by </b><i><a class="txt" href="amistd_1.html#E1:46"><i><font color="Green" title="E23">A1</font></i></a>, <a class="txt" href="amistd_1.html#E2:46"><i><font color="Green" title="E24">A2</font></i></a>, <a class="txt" href="amistd_1.html#E4:46"><i><font color="Green" title="E25">A3</font></i></a>, <a class="txt" href="amistd_1.html#E6:46"><i><font color="Green" title="E26">A4</font></i></a>, <a class="txt" href="amistd_1.html#E7:46"><i><font color="Green" title="E27">A5</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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