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

<b>let </b><font color="Maroon">z</font> be  <a href="ordinal1.html#V7">natural</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><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">l</font> be   <a href="ami_1.html#NM2">Instruction-Location</a> of <span class="p1">(<span class="default"><a href="amistd_1.html#K4" target="_self">STC</a> <font color="Maroon">N</font></span>)</span>;<br/>





<b>assume </b><a NAME="E1:43"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font>
 ;<br/>

<b>set </b><font color="Maroon">K</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="amistd_1.html#K1" target="_self">NIC</a> <font color="Olive">i</font>,<font color="Maroon">l</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p3">(<span class="default"><a href="amistd_1.html#K2" target="_self">JUMP</a> <font color="Olive">i</font></span>)</span></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of the <a href="ami_1.html#U4">Instructions</a> of <span class="p1">(<span class="default"><a href="amistd_1.html#K4" target="_self">STC</a> <font color="Maroon">N</font></span>)</span> : verum </span>}</span> ;<br/>
<b>set </b><font color="Maroon">M</font> =  <a href="amistd_1.html#K4" target="_self">STC</a> <font color="Maroon">N</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:43_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="amistd_1.html#K1" target="_self">NIC</a> <font color="Olive">i</font>,<font color="Maroon">l</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p3">(<span class="default"><a href="amistd_1.html#K2" target="_self">JUMP</a> <font color="Olive">i</font></span>)</span></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of the <a href="ami_1.html#U4">Instructions</a> of <span class="p1">(<span class="default"><a href="amistd_1.html#K4" target="_self">STC</a> <font color="Maroon">N</font></span>)</span> : verum </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">ii</font> being    <a href="subset_1.html#M2">Element</a> of the <a href="ami_1.html#U4">Instructions</a> of <span class="p1">(<span class="default"><a href="amistd_1.html#K4" target="_self">STC</a> <font color="Maroon">N</font></span>)</span><b> such that </b><br/><a NAME="E3:43_1_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="amistd_1.html#K1" target="_self">NIC</a> <font color="Maroon">ii</font>,<font color="Maroon">l</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="amistd_1.html#K2" target="_self">JUMP</a> <font color="Maroon">ii</font></span>)</span>
 ;<br/><b>reconsider </b><font color="Maroon">ii</font> = <font color="Maroon">ii</font> as   <a href="ami_1.html#NM3">Instruction</a> of <span class="p1">(<span class="default"><a href="amistd_1.html#K4" target="_self">STC</a> <font color="Maroon">N</font></span>)</span> ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:43_1_1_1_1"/>
(  <a href="ami_5.html#K1">InsCode</a> <font color="Maroon">ii</font> <a href="hidden.html#R1">=</a> 1 or  <a href="ami_5.html#K1">InsCode</a> <font color="Maroon">ii</font> <a href="hidden.html#R1">=</a> 0 )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:43_1_1_1_1_1"/><i><font color="Green">E39</font></i>: 
 <a href="ami_5.html#K1">InsCode</a> <font color="Maroon">ii</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:43_1_1_1_1_1"/>
 <a href="amistd_1.html#K2" target="_self">JUMP</a> <font color="Maroon">ii</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i/>;<br/><a NAME="E3:43_1_1_1_1_1"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, , , </i>;<br/><b>hence </b><a NAME="E4:43_1_1_1_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">z</font></span><a href="tarski.html#K1">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:43_1_1_1_1_2"/>
 <a href="ami_5.html#K1">InsCode</a> <font color="Maroon">ii</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:43_1_1_1_1_2"/><b>then </b><i><font color="Green">E40</font></i>: 
<font color="Maroon">ii</font> is <a href="ami_1.html#V3">halting</a>
 <b>by </b><i/>;<br/><a NAME="E3:43_1_1_1_1_2"/>
 <a href="amistd_1.html#K2" target="_self">JUMP</a> <font color="Maroon">ii</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i/>;<br/><a NAME="E4:43_1_1_1_1_2"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">z</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, , , </i>;<br/><b>hence </b><a NAME="E5:43_1_1_1_1_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">z</font></span><a href="tarski.html#K1">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:43_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">z</font></span><a href="tarski.html#K1">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K2">}</a></span>
 ;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:43_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">z</font></span><a href="tarski.html#K1">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K2">}</a></span>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:43_1_2"/>
( <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">z</font></span><a href="tarski.html#K1">}</a></span> or <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span><a href="tarski.html#K1">}</a></span> )
 </span><b>by </b><i>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_2_1"><b>suppose </b></a><a NAME="E1:43_1_2_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">z</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/><div class="add"><b>set </b><font color="Maroon">i</font> = <span class="p1"><a href="domain_1.html#K1">[</a><span class="default">0,0</span><a href="domain_1.html#K1">]</a></span>;<br/><a NAME="E2:43_1_2_1"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default">0,0</span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default">1,0</span><a href="domain_1.html#K1">]</a></span>,<span class="p2"><a href="domain_1.html#K1">[</a><span class="default">0,0</span><a href="domain_1.html#K1">]</a></span></span><a href="domain_1.html#K7">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>then reconsider </b><font color="Maroon">i</font> = <span class="p1"><a href="domain_1.html#K1">[</a><span class="default">0,0</span><a href="domain_1.html#K1">]</a></span> as   <a href="ami_1.html#NM3">Instruction</a> of <span class="p1">(<span class="default"><a href="amistd_1.html#K4" target="_self">STC</a> <font color="Maroon">N</font></span>)</span> <b>by </b><i/>;<br/><a NAME="E4:43_1_2_1"/><i><font color="Green">E66</font></i>: 
 <a href="amistd_1.html#K2" target="_self">JUMP</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i/>;<br/> <a href="ami_5.html#K1">InsCode</a> <font color="Maroon">i</font> = 
<font color="Maroon">i</font> <a href="domain_1.html#K2">`1</a> 
<b>by </b><i><a class="ref" href="ami_5.html#D1">AMI_5:def 1</a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
;<br/><a NAME="E6:43_1_2_1"/><b>then </b>
<font color="Maroon">i</font> is <a href="ami_1.html#V3">halting</a>
 <b>by </b><i/>;<br/><a NAME="E7:43_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="amistd_1.html#K1" target="_self">NIC</a> <font color="Maroon">i</font>,<font color="Maroon">l</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="amistd_1.html#K2" target="_self">JUMP</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i>, , , </i>;<br/><b>hence </b><a NAME="E8:43_1_2_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="amistd_1.html#K1" target="_self">NIC</a> <font color="Olive">i</font>,<font color="Maroon">l</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p3">(<span class="default"><a href="amistd_1.html#K2" target="_self">JUMP</a> <font color="Olive">i</font></span>)</span></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of the <a href="ami_1.html#U4">Instructions</a> of <span class="p1">(<span class="default"><a href="amistd_1.html#K4" target="_self">STC</a> <font color="Maroon">N</font></span>)</span> : verum </span>}</span> 
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_2_2"><b>suppose </b></a><a NAME="E1:43_1_2_2"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span><a href="tarski.html#K1">}</a></span>
 ;<br/><div class="add"><b>set </b><font color="Maroon">i</font> = <span class="p1"><a href="domain_1.html#K1">[</a><span class="default">1,0</span><a href="domain_1.html#K1">]</a></span>;<br/><a NAME="E2:43_1_2_2"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default">1,0</span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><span class="p2"><a href="domain_1.html#K1">[</a><span class="default">1,0</span><a href="domain_1.html#K1">]</a></span>,<span class="p2"><a href="domain_1.html#K1">[</a><span class="default">0,0</span><a href="domain_1.html#K1">]</a></span></span><a href="domain_1.html#K7">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>then reconsider </b><font color="Maroon">i</font> = <span class="p1"><a href="domain_1.html#K1">[</a><span class="default">1,0</span><a href="domain_1.html#K1">]</a></span> as   <a href="ami_1.html#NM3">Instruction</a> of <span class="p1">(<span class="default"><a href="amistd_1.html#K4" target="_self">STC</a> <font color="Maroon">N</font></span>)</span> <b>by </b><i/>;<br/><a NAME="E4:43_1_2_2"/><i><font color="Green">E68</font></i>: 
 <a href="amistd_1.html#K2" target="_self">JUMP</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i/>;<br/> <a href="ami_5.html#K1">InsCode</a> <font color="Maroon">i</font> = 
<font color="Maroon">i</font> <a href="domain_1.html#K2">`1</a> 
<b>by </b><i><a class="ref" href="ami_5.html#D1">AMI_5:def 1</a></i>
<br/>.= 
1
<b>by </b><i><a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
;<br/><a NAME="E6:43_1_2_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="amistd_1.html#K1" target="_self">NIC</a> <font color="Maroon">i</font>,<font color="Maroon">l</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="amistd_1.html#K2" target="_self">JUMP</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font>
 <b>by </b><i>, , , </i>;<br/><b>hence </b><a NAME="E7:43_1_2_2"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="amistd_1.html#K1" target="_self">NIC</a> <font color="Olive">i</font>,<font color="Maroon">l</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p3">(<span class="default"><a href="amistd_1.html#K2" target="_self">JUMP</a> <font color="Olive">i</font></span>)</span></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of the <a href="ami_1.html#U4">Instructions</a> of <span class="p1">(<span class="default"><a href="amistd_1.html#K4" target="_self">STC</a> <font color="Maroon">N</font></span>)</span> : verum </span>}</span> 
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E3:43"/><b>then </b><i><font color="Green">E69</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="amistd_1.html#K1" target="_self">NIC</a> <font color="Olive">i</font>,<font color="Maroon">l</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p3">(<span class="default"><a href="amistd_1.html#K2" target="_self">JUMP</a> <font color="Olive">i</font></span>)</span></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of the <a href="ami_1.html#U4">Instructions</a> of <span class="p1">(<span class="default"><a href="amistd_1.html#K4" target="_self">STC</a> <font color="Maroon">N</font></span>)</span> : verum </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">z</font></span><a href="tarski.html#K1">}</a></span>,<span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">z</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span><a href="tarski.html#K1">}</a></span></span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>
<b>thus </b><a NAME="E4:43"/>
 <a href="amistd_1.html#K3" target="_self">SUCC</a> <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">z</font>,<span class="p2">(<span class="default"><font color="Maroon">z</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span></span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T32">ZFMISC_1:32</a></i>;<br/>


</div>
