<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c2">z</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"> for <font color="Olive" title="b1">N</font> being <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> <br/> for <font color="Olive" title="b2">l</font> being   <a href="ami_1.html#M3" title="AMI_1:mode.3">Instruction-Location</a> of  <a href="amistd_1.html#K4" title="AMISTD_1:func.4">STC</a> <font color="Olive" title="b1">N</font><br/> for <font color="Olive" title="b3">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="ami_1.html#U2" title="AMI_1:sel.2">Instructions</a> of <span class="p1">(<span class="default"><a href="amistd_1.html#K4" title="AMISTD_1:func.4">STC</a> <font color="Olive" title="b1">N</font></span>)</span>  st <font color="Olive" title="b2">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">z</font> &amp;  <a href="ami_1.html#K21" title="AMI_1:func.21">InsCode</a> <font color="Olive" title="b3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 holds <br/> <a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <font color="Olive" title="b3">i</font>,<font color="Olive" title="b2">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><br/><b>let </b><font color="Maroon" title="c3">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">l</font> being   <a href="ami_1.html#M3" title="AMI_1:mode.3">Instruction-Location</a> of  <a href="amistd_1.html#K4" title="AMISTD_1:func.4">STC</a> <font color="Maroon" title="c3">N</font><br/> for <font color="Olive" title="b2">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="ami_1.html#U2" title="AMI_1:sel.2">Instructions</a> of <span class="p1">(<span class="default"><a href="amistd_1.html#K4" title="AMISTD_1:func.4">STC</a> <font color="Maroon" title="c3">N</font></span>)</span>  st <font color="Olive" title="b1">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">z</font> &amp;  <a href="ami_1.html#K21" title="AMI_1:func.21">InsCode</a> <font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 holds <br/> <a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <font color="Olive" title="b2">i</font>,<font color="Olive" title="b1">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><br/><b>let </b><font color="Maroon" title="c4">l</font> be    <a href="ami_1.html#M3" title="AMI_1:mode.3">Instruction-Location</a> of  <a href="amistd_1.html#K4" title="AMISTD_1:func.4">STC</a> <font color="Maroon" title="c3">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">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="ami_1.html#U2" title="AMI_1:sel.2">Instructions</a> of <span class="p1">(<span class="default"><a href="amistd_1.html#K4" title="AMISTD_1:func.4">STC</a> <font color="Maroon" title="c3">N</font></span>)</span>  st <font color="Maroon" title="c4">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">z</font> &amp;  <a href="ami_1.html#K21" title="AMI_1:func.21">InsCode</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 holds <br/> <a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <font color="Olive" title="b1">i</font>,<font color="Maroon" title="c4">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><br/><b>let </b><font color="Maroon" title="c5">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="ami_1.html#U2" title="AMI_1:sel.2">Instructions</a> of <span class="p1">(<span class="default"><a href="amistd_1.html#K4" title="AMISTD_1:func.4">STC</a> <font color="Maroon" title="c3">N</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">z</font> &amp;  <a href="ami_1.html#K21" title="AMI_1:func.21">InsCode</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies  <a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c4">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )</span><br/>







<b>assume </b><b>that </b><br/><a NAME="E1:39"/><i><font color="Green" title="E19">A1</font></i>: 
<font color="Maroon" title="c4">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">z</font>
 <b>and </b><br/><a NAME="E2:39"/><i><font color="Green" title="E20">A2</font></i>: 
 <a href="ami_1.html#K21" title="AMI_1:func.21">InsCode</a> <font color="Maroon" title="c5">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c4">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><br/>

<b>set </b><font color="Maroon" title="c6">M</font> =  <a href="amistd_1.html#K4" title="AMISTD_1:func.4">STC</a> <font color="Maroon" title="c3">N</font>;<br/>
<b>set </b><font color="Maroon" title="c7">F</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <span class="p3">(<span class="default"><a href="ami_1.html#K9" title="AMI_1:func.9">Following</a> <font color="Olive" title="b1">s</font></span>)</span></span>)</span> where <font color="Olive" title="b1">s</font> is   <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <span class="p1">(<span class="default"><a href="amistd_1.html#K4" title="AMISTD_1:func.4">STC</a> <font color="Maroon" title="c3">N</font></span>)</span> : (  <a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">l</font> &amp; <font color="Olive" title="b1">s</font> <a href="ami_1.html#K13" title="AMI_1:func.13">.</a> <font color="Maroon" title="c4">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">i</font> ) </span>}</span> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c8">y</font> be    <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"> ( ( <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <span class="p3">(<span class="default"><a href="ami_1.html#K9" title="AMI_1:func.9">Following</a> <font color="Olive" title="b1">s</font></span>)</span></span>)</span> where <font color="Olive" title="b1">s</font> is   <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <span class="p1">(<span class="default"><a href="amistd_1.html#K4" title="AMISTD_1:func.4">STC</a> <font color="Maroon" title="c3">N</font></span>)</span> : (  <a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">l</font> &amp; <font color="Olive" title="b1">s</font> <a href="ami_1.html#K13" title="AMI_1:func.13">.</a> <font color="Maroon" title="c4">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">i</font> ) </span>}</span>  implies <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> ) &amp; ( <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> implies <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <span class="p3">(<span class="default"><a href="ami_1.html#K9" title="AMI_1:func.9">Following</a> <font color="Olive" title="b1">s</font></span>)</span></span>)</span> where <font color="Olive" title="b1">s</font> is   <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <span class="p1">(<span class="default"><a href="amistd_1.html#K4" title="AMISTD_1:func.4">STC</a> <font color="Maroon" title="c3">N</font></span>)</span> : (  <a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">l</font> &amp; <font color="Olive" title="b1">s</font> <a href="ami_1.html#K13" title="AMI_1:func.13">.</a> <font color="Maroon" title="c4">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">i</font> ) </span>}</span>  ) )</span><br/><div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> implies <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <span class="p3">(<span class="default"><a href="ami_1.html#K9" title="AMI_1:func.9">Following</a> <font color="Olive" title="b1">s</font></span>)</span></span>)</span> where <font color="Olive" title="b1">s</font> is   <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <span class="p1">(<span class="default"><a href="amistd_1.html#K4" title="AMISTD_1:func.4">STC</a> <font color="Maroon" title="c3">N</font></span>)</span> : (  <a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">l</font> &amp; <font color="Olive" title="b1">s</font> <a href="ami_1.html#K13" title="AMI_1:func.13">.</a> <font color="Maroon" title="c4">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">i</font> ) </span>}</span>  )</span>
<div class="add"><b>assume </b><a NAME="E1:39_1_1"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <span class="p3">(<span class="default"><a href="ami_1.html#K9" title="AMI_1:func.9">Following</a> <font color="Olive" title="b1">s</font></span>)</span></span>)</span> where <font color="Olive" title="b1">s</font> is   <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <span class="p1">(<span class="default"><a href="amistd_1.html#K4" title="AMISTD_1:func.4">STC</a> <font color="Maroon" title="c3">N</font></span>)</span> : (  <a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">l</font> &amp; <font color="Olive" title="b1">s</font> <a href="ami_1.html#K13" title="AMI_1:func.13">.</a> <font color="Maroon" title="c4">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">i</font> ) </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><br/><b>then consider </b><font color="Maroon" title="c9">s</font> being   <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <span class="p1">(<span class="default"><a href="amistd_1.html#K4" title="AMISTD_1:func.4">STC</a> <font color="Maroon" title="c3">N</font></span>)</span><b> such that </b><br/><a NAME="E3:39_1_1"/><i><font color="Green" title="E21">A3</font></i>: 
( <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K9" title="AMI_1:func.9">Following</a> <font color="Maroon" title="c9">s</font></span>)</span> &amp;  <a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <font color="Maroon" title="c9">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">l</font> &amp; <font color="Maroon" title="c9">s</font> <a href="ami_1.html#K13" title="AMI_1:func.13">.</a> <font color="Maroon" title="c4">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">i</font> )
 ;<br/><font color="Maroon" title="c8">y</font> = 
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c2">z</font>
<b>by </b><i><a class="txt" href="#E1:39"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="#E2:39"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="#E3:39_1_1"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="amistd_1.html#E34"><i><font color="Green" title="E15">Lm4</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c2">z</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1

;<br/><b>hence </b><a NAME="E5:39_1_1"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:39_1"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <span class="p3">(<span class="default"><a href="ami_1.html#K9" title="AMI_1:func.9">Following</a> <font color="Olive" title="b1">s</font></span>)</span></span>)</span> where <font color="Olive" title="b1">s</font> is   <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <span class="p1">(<span class="default"><a href="amistd_1.html#K4" title="AMISTD_1:func.4">STC</a> <font color="Maroon" title="c3">N</font></span>)</span> : (  <a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">l</font> &amp; <font color="Olive" title="b1">s</font> <a href="ami_1.html#K13" title="AMI_1:func.13">.</a> <font color="Maroon" title="c4">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">i</font> ) </span>}</span> </span><br/><b>then </b><i><font color="Green" title="E21">A4</font></i>: <font color="Maroon" title="c8">y</font> = 
<font color="Maroon" title="c2">z</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>
<br/>.= 
 <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c2">z</font>

;<br/><b>consider </b><font color="Maroon" title="c9">w</font> being   <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <span class="p1">(<span class="default"><a href="amistd_1.html#K4" title="AMISTD_1:func.4">STC</a> <font color="Maroon" title="c3">N</font></span>)</span>;<br/><b>set </b><font color="Maroon" title="c10">f</font> = <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <font color="Maroon" title="c5">i</font>;<br/><a NAME="E5:39_1"/><i><font color="Green" title="E22">A5</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <font color="Maroon" title="c5">i</font></span>)</span> <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="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>;<br/><a NAME="E6:39_1"/><i><font color="Green" title="E23">A6</font></i>: 
<font color="Maroon" title="c4">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS: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>reconsider </b><font color="Maroon" title="c11">s</font> = <font color="Maroon" title="c9">w</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <font color="Maroon" title="c5">i</font></span>)</span> as   <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <span class="p1">(<span class="default"><a href="amistd_1.html#K4" title="AMISTD_1:func.4">STC</a> <font color="Maroon" title="c3">N</font></span>)</span> <b>by </b><i><a class="ref" href="amistd_1.html#T11" target="_self" title="AMISTD_1:th.11">Th11</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c12">l'</font> = <font color="Maroon" title="c4">l</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_1.html#K3" title="AMI_1:func.3">ObjectKind</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <span class="p2">(<span class="default"><a href="amistd_1.html#K4" title="AMISTD_1:func.4">STC</a> <font color="Maroon" title="c3">N</font></span>)</span></span>)</span> <b>by </b><i><a class="txt" href="#E6:39_1"><i><font color="Green" title="E23">A6</font></i></a>, <a class="ref" href="ami_1.html#D11" title="AMI_1:def.11">AMI_1:def 11</a></i>;<br/><b>set </b><font color="Maroon" title="c13">u</font> = <span class="p1">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <span class="p2">(<span class="default"><a href="amistd_1.html#K4" title="AMISTD_1:func.4">STC</a> <font color="Maroon" title="c3">N</font></span>)</span></span>)</span> <a href="ami_1.html#K16" title="AMI_1:func.16">.--&gt;</a> <font color="Maroon" title="c12">l'</font>;<br/><b>set </b><font color="Maroon" title="c14">t</font> = <font color="Maroon" title="c11">s</font> <a href="ami_1.html#K36" title="AMI_1:func.36">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <span class="p3">(<span class="default"><a href="amistd_1.html#K4" title="AMISTD_1:func.4">STC</a> <font color="Maroon" title="c3">N</font></span>)</span></span>)</span> <a href="ami_1.html#K16" title="AMI_1:func.16">.--&gt;</a> <font color="Maroon" title="c12">l'</font></span>)</span>;<br/><a NAME="E9:39_1"/><i><font color="Green" title="E24">A7</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <span class="p3">(<span class="default"><a href="amistd_1.html#K4" title="AMISTD_1:func.4">STC</a> <font color="Maroon" title="c3">N</font></span>)</span></span>)</span> <a href="ami_1.html#K16" title="AMI_1:func.16">.--&gt;</a> <font color="Maroon" title="c12">l'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <span class="p3">(<span class="default"><a href="amistd_1.html#K4" title="AMISTD_1:func.4">STC</a> <font color="Maroon" title="c3">N</font></span>)</span></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>;<br/><a NAME="E10:39_1"/><b>then </b>
 <a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <span class="p1">(<span class="default"><a href="amistd_1.html#K4" title="AMISTD_1:func.4">STC</a> <font color="Maroon" title="c3">N</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <span class="p3">(<span class="default"><a href="amistd_1.html#K4" title="AMISTD_1:func.4">STC</a> <font color="Maroon" title="c3">N</font></span>)</span></span>)</span> <a href="ami_1.html#K16" title="AMI_1:func.16">.--&gt;</a> <font color="Maroon" title="c12">l'</font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>then </b><i><font color="Green" title="E25">A8</font></i>:  <a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">s</font> <a href="ami_1.html#K36" title="AMI_1:func.36">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <span class="p4">(<span class="default"><a href="amistd_1.html#K4" title="AMISTD_1:func.4">STC</a> <font color="Maroon" title="c3">N</font></span>)</span></span>)</span> <a href="ami_1.html#K16" title="AMI_1:func.16">.--&gt;</a> <font color="Maroon" title="c12">l'</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <span class="p3">(<span class="default"><a href="amistd_1.html#K4" title="AMISTD_1:func.4">STC</a> <font color="Maroon" title="c3">N</font></span>)</span></span>)</span> <a href="ami_1.html#K16" title="AMI_1:func.16">.--&gt;</a> <font color="Maroon" title="c12">l'</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <span class="p2">(<span class="default"><a href="amistd_1.html#K4" title="AMISTD_1:func.4">STC</a> <font color="Maroon" title="c3">N</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon" title="c2">z</font>
<b>by </b><i><a class="txt" href="#E1:39"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="funcop_1.html#T87" title="FUNCOP_1:th.87">FUNCOP_1:87</a></i>
;<br/><a NAME="E12:39_1"/>
<font color="Maroon" title="c4">l</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <span class="p1">(<span class="default"><a href="amistd_1.html#K4" title="AMISTD_1:func.4">STC</a> <font color="Maroon" title="c3">N</font></span>)</span>
 <b>by </b><i><a class="ref" href="ami_1.html#T48" title="AMI_1:th.48">AMI_1:48</a></i>;<br/><a NAME="E13:39_1"/><b>then </b>
not <font color="Maroon" title="c4">l</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <span class="p3">(<span class="default"><a href="amistd_1.html#K4" title="AMISTD_1:func.4">STC</a> <font color="Maroon" title="c3">N</font></span>)</span></span>)</span> <a href="ami_1.html#K16" title="AMI_1:func.16">.--&gt;</a> <font color="Maroon" title="c12">l'</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E9:39_1"><i><font color="Green" title="E24">A7</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>then </b><i><font color="Green" title="E26">A9</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c11">s</font> <a href="ami_1.html#K36" title="AMI_1:func.36">+*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <span class="p4">(<span class="default"><a href="amistd_1.html#K4" title="AMISTD_1:func.4">STC</a> <font color="Maroon" title="c3">N</font></span>)</span></span>)</span> <a href="ami_1.html#K16" title="AMI_1:func.16">.--&gt;</a> <font color="Maroon" title="c12">l'</font></span>)</span></span>)</span> <a href="ami_1.html#K13" title="AMI_1:func.13">.</a> <font color="Maroon" title="c4">l</font> = 
<font color="Maroon" title="c11">s</font> <a href="ami_1.html#K13" title="AMI_1:func.13">.</a> <font color="Maroon" title="c4">l</font>
<b>by </b><i><a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <font color="Maroon" title="c5">i</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">l</font>
<b>by </b><i><a class="txt" href="#E5:39_1"><i><font color="Green" title="E22">A5</font></i></a>, <a class="txt" href="#E6:39_1"><i><font color="Green" title="E23">A6</font></i></a>, <a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></i>
<br/>.= 
<font color="Maroon" title="c5">i</font>
<b>by </b><i><a class="txt" href="#E6:39_1"><i><font color="Green" title="E23">A6</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
;<br/><a NAME="E15:39_1"/><b>then </b>
 <a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K9" title="AMI_1:func.9">Following</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">s</font> <a href="ami_1.html#K36" title="AMI_1:func.36">+*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <span class="p5">(<span class="default"><a href="amistd_1.html#K4" title="AMISTD_1:func.4">STC</a> <font color="Maroon" title="c3">N</font></span>)</span></span>)</span> <a href="ami_1.html#K16" title="AMI_1:func.16">.--&gt;</a> <font color="Maroon" title="c12">l'</font></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c2">z</font>
 <b>by </b><i><a class="txt" href="#E1:39"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="#E2:39"><i><font color="Green" title="E20">A2</font></i></a>, <a class="txt" href="#E11:39_1"><i><font color="Green" title="E25">A8</font></i></a>, <a class="txt" href="amistd_1.html#E34"><i><font color="Green" title="E15">Lm4</font></i></a></i>;<br/><b>hence </b><a NAME="E16:39_1"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <span class="p3">(<span class="default"><a href="ami_1.html#K9" title="AMI_1:func.9">Following</a> <font color="Olive" title="b1">s</font></span>)</span></span>)</span> where <font color="Olive" title="b1">s</font> is   <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <span class="p1">(<span class="default"><a href="amistd_1.html#K4" title="AMISTD_1:func.4">STC</a> <font color="Maroon" title="c3">N</font></span>)</span> : (  <a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">l</font> &amp; <font color="Olive" title="b1">s</font> <a href="ami_1.html#K13" title="AMI_1:func.13">.</a> <font color="Maroon" title="c4">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">i</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="#E1:39"><i><font color="Green" title="E19">A1</font></i></a>, <a class="txt" href="#E3:39_1"><i><font color="Green" title="E21">A4</font></i></a>, <a class="txt" href="#E11:39_1"><i><font color="Green" title="E25">A8</font></i></a>, <a class="txt" href="#E14:39_1"><i><font color="Green" title="E26">A9</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:39"/>
 <a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c4">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">z</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
