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<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">a</font> be    <a href="ami_3.html#M1" title="AMI_3:mode.1">Data-Location</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">il</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a> <br/> for <font color="Olive" title="b2">k</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  holds   <a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="ami_3.html#K10" title="AMI_3:func.10">&gt;0_goto</a> <font color="Olive" title="b2">k</font></span>)</span>,<font color="Olive" title="b1">il</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Olive" title="b2">k</font>,<span class="p2">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Olive" title="b1">il</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">il</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">k</font> being  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  holds   <a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="ami_3.html#K10" title="AMI_3:func.10">&gt;0_goto</a> <font color="Olive" title="b1">k</font></span>)</span>,<font color="Maroon" title="c2">il</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Olive" title="b1">k</font>,<span class="p2">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c2">il</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">k</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="ami_3.html#K10" title="AMI_3:func.10">&gt;0_goto</a> <font color="Maroon" title="c3">k</font></span>)</span>,<font color="Maroon" title="c2">il</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">k</font>,<span class="p2">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c2">il</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><br/>





<span class="kw">consider </span><font color="Maroon" title="c4">t</font> being   <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> ;<br/>
<div><span class="kw">hereby </span> <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">k</font>,<span class="p2">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c2">il</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="ami_3.html#K10" title="AMI_3:func.10">&gt;0_goto</a> <font color="Maroon" title="c3">k</font></span>)</span>,<font color="Maroon" title="c2">il</font></span>
<div class="add"><span class="kw">let </span><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="ami_3.html#K10" title="AMI_3:func.10">&gt;0_goto</a> <font color="Maroon" title="c3">k</font></span>)</span>,<font color="Maroon" title="c2">il</font> implies <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">k</font>,<span class="p2">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c2">il</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )</span><br/><span class="kw">assume </span><a NAME="E1:62_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="ami_3.html#K10" title="AMI_3:func.10">&gt;0_goto</a> <font color="Maroon" title="c3">k</font></span>)</span>,<font color="Maroon" title="c2">il</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">k</font>,<span class="p2">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c2">il</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">s</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> the <a href="ami_1.html#U3" title="AMI_1:sel.3">Object-Kind</a> of <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> <span class="kw"> such that </span><br/><a NAME="E3:62_1"/><span class="lab"><font color="Green" title="E46">A1</font></span>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K6" title="AMI_1:func.6">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"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c6">s</font></span>)</span>,<font color="Maroon" title="c6">s</font></span>)</span> &amp;  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c6">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">il</font> &amp; <span class="p1">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c6">s</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c2">il</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> <a href="ami_3.html#K10" title="AMI_3:func.10">&gt;0_goto</a> <font color="Maroon" title="c3">k</font> )
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:62_1_1"/>
( <font color="Maroon" title="c6">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c6">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:62_1_1_1"/>
<font color="Maroon" title="c6">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">k</font>,<span class="p2">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c2">il</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><br/><div class="add"><a NAME="E2:62_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:62_1"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="ref" href="ami_3.html#T15" title="AMI_3:th.15">AMI_3:15</a></span>;<br/><span class="kw">hence </span><a NAME="E3:62_1_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">k</font>,<span class="p2">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c2">il</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:62_1_1_2"/>
<font color="Maroon" title="c6">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">k</font>,<span class="p2">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c2">il</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><br/><div class="add"><a NAME="E2:62_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">x</font> <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">il</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:62_1"><span class="lab"><font color="Green" title="E46">A1</font></span></a>, <a class="ref" href="ami_3.html#T15" title="AMI_3:th.15">AMI_3:15</a></span>;<br/><span class="kw">hence </span><a NAME="E3:62_1_1_2"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">k</font>,<span class="p2">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c2">il</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div>
<span class="kw">end;</span></div>

<span class="kw">let </span><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">k</font>,<span class="p2">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c2">il</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> or <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="ami_3.html#K10" title="AMI_3:func.10">&gt;0_goto</a> <font color="Maroon" title="c3">k</font></span>)</span>,<font color="Maroon" title="c2">il</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c6">I</font> = <font color="Maroon" title="c1">a</font> <a href="ami_3.html#K10" title="AMI_3:func.10">&gt;0_goto</a> <font color="Maroon" title="c3">k</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="ami_1.html#U3" title="AMI_1:sel.3">Object-Kind</a> of <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">il</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="ami_1.html#D14" title="AMI_1:def.14">AMI_1:def 14</a></span>;<br/>
<a NAME="E4:62"/><span class="lab"><font color="Green" title="E46">A2</font></span>: 
 <a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ami_5.html#T20" title="AMI_5:th.20">AMI_5:20</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">il1</font> = <font color="Maroon" title="c2">il</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> <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="ami_1.html#D11" title="AMI_1:def.11">AMI_1:def 11</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c8">p</font> = <span class="p1">(<span class="default"><a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> </span>)</span>,<font color="Maroon" title="c2">il</font> <a href="funct_4.html#K4" title="FUNCT_4:func.4">--&gt;</a> <font color="Maroon" title="c7">il1</font>,<font color="Maroon" title="c6">I</font> as   <a href="ami_1.html#NM3" title="AMI_1:NM.3">PartState</a> of <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ami_1.html#T149" title="AMI_1:th.149">AMI_1:149</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">u</font> = <font color="Maroon" title="c4">t</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <font color="Maroon" title="c8">p</font> as   <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a>  ;<br/>
<a NAME="E8:62"/><span class="lab"><font color="Green" title="E47">A3</font></span>: 
<font color="Maroon" title="c1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">il</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ami_6.html#T1" target="_self" title="AMI_6:th.1">Th1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">n</font> = <font color="Maroon" title="c7">il1</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>  ;<br/>
<span class="kw">assume </span><a NAME="E10:62"/><span class="lab"><font color="Green" title="E48">A4</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">k</font>,<span class="p2">(<span class="default"><a href="ordinal1.html#K1" title="ORDINAL1:func.1">succ</a> <font color="Maroon" title="c2">il</font></span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="ami_3.html#K10" title="AMI_3:func.10">&gt;0_goto</a> <font color="Maroon" title="c3">k</font></span>)</span>,<font color="Maroon" title="c2">il</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_2"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:62_2"/>
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> or <font color="Maroon" title="c5">x</font> <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">il</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:62"><span class="lab"><font color="Green" title="E48">A4</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_2_1"><span class="kw">suppose </span></a><a NAME="E1:62_2_1"/><span class="lab"><font color="Green" title="E49">A5</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="ami_3.html#K10" title="AMI_3:func.10">&gt;0_goto</a> <font color="Maroon" title="c3">k</font></span>)</span>,<font color="Maroon" title="c2">il</font></span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c11">v</font> = <font color="Maroon" title="c9">u</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="ami_3.html#K12" title="AMI_3:func.12">.--&gt;</a> 1</span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> the <a href="ami_1.html#U3" title="AMI_1:sel.3">Object-Kind</a> of <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#T155" title="PBOOLE:th.155">PBOOLE:155</a></span>;<br/><a NAME="E3:62_2_1"/><span class="lab"><font color="Green" title="E50">A6</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="ami_3.html#K12" title="AMI_3:func.12">.--&gt;</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></span>;<br/><a NAME="E4:62_2_1"/><span class="kw">then </span>
not  <a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a>  <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"><font color="Maroon" title="c1">a</font> <a href="ami_3.html#K12" title="AMI_3:func.12">.--&gt;</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:62"><span class="lab"><font color="Green" title="E46">A2</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A7</font></span>:  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c11">v</font> = 
 <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c9">u</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></span>
<br/>.= 
<font color="Maroon" title="c10">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ami_1.html#T129" title="AMI_1:th.129">AMI_1:129</a></span>
;<br/><a NAME="E6:62_2_1"/><span class="lab"><font color="Green" title="E52">U</font></span>: 
not <font color="Maroon" title="c2">il</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"><font color="Maroon" title="c1">a</font> <a href="ami_3.html#K12" title="AMI_3:func.12">.--&gt;</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:62"><span class="lab"><font color="Green" title="E47">A3</font></span></a>, <a class="txt" href="#E3:62_2_1"><span class="lab"><font color="Green" title="E50">A6</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><span class="lab"><font color="Green" title="E53">A8</font></span>: <span class="p1">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c11">v</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c2">il</font> = 
<font color="Maroon" title="c11">v</font> <a href="ami_1.html#K12" title="AMI_1:func.12">.</a> <font color="Maroon" title="c2">il</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ami_1.html#T150" title="AMI_1:th.150">AMI_1:150</a></span>
<br/>.= 
<font color="Maroon" title="c9">u</font> <a href="ami_1.html#K12" title="AMI_1:func.12">.</a> <font color="Maroon" title="c10">n</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:62_2_1"><span class="lab"><font color="Green" title="E52">U</font></span></a>, <a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></span>
<br/>.= 
<font color="Maroon" title="c6">I</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ami_1.html#T129" title="AMI_1:th.129">AMI_1:129</a></span>
;<br/><a NAME="E8:62_2_1"/>
<font color="Maroon" title="c1">a</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"><font color="Maroon" title="c1">a</font> <a href="ami_3.html#K12" title="AMI_3:func.12">.--&gt;</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:62_2_1"><span class="lab"><font color="Green" title="E50">A6</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><span class="kw">then </span><font color="Maroon" title="c11">v</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <font color="Maroon" title="c1">a</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="ami_3.html#K12" title="AMI_3:func.12">.--&gt;</a> 1</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">a</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T87" title="FUNCOP_1:th.87">FUNCOP_1:87</a></span>
;<br/><a NAME="E10:62_2_1"/><span class="kw">then </span>
 <a href="ami_1.html#K6" title="AMI_1:func.6">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"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c11">v</font></span>)</span>,<font color="Maroon" title="c11">v</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:62_2_1"><span class="lab"><font color="Green" title="E51">A7</font></span></a>, <a class="txt" href="#E7:62_2_1"><span class="lab"><font color="Green" title="E53">A8</font></span></a>, <a class="ref" href="ami_3.html#T15" title="AMI_3:th.15">AMI_3:15</a></span>;<br/><span class="kw">hence </span><a NAME="E11:62_2_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="ami_3.html#K10" title="AMI_3:func.10">&gt;0_goto</a> <font color="Maroon" title="c3">k</font></span>)</span>,<font color="Maroon" title="c2">il</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62_2_1"><span class="lab"><font color="Green" title="E49">A5</font></span></a>, <a class="txt" href="#E5:62_2_1"><span class="lab"><font color="Green" title="E51">A7</font></span></a>, <a class="txt" href="#E7:62_2_1"><span class="lab"><font color="Green" title="E53">A8</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="62_2_2"><span class="kw">suppose </span></a><a NAME="E1:62_2_2"/><span class="lab"><font color="Green" title="E49">A9</font></span>: 
<font color="Maroon" title="c5">x</font> <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">il</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="ami_3.html#K10" title="AMI_3:func.10">&gt;0_goto</a> <font color="Maroon" title="c3">k</font></span>)</span>,<font color="Maroon" title="c2">il</font></span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c11">v</font> = <font color="Maroon" title="c9">u</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="ami_3.html#K12" title="AMI_3:func.12">.--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> the <a href="ami_1.html#U3" title="AMI_1:sel.3">Object-Kind</a> of <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#T155" title="PBOOLE:th.155">PBOOLE:155</a></span>;<br/><a NAME="E3:62_2_2"/><span class="lab"><font color="Green" title="E50">A10</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="ami_3.html#K12" title="AMI_3:func.12">.--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></span>;<br/><a NAME="E4:62_2_2"/><span class="kw">then </span>
not  <a href="ami_1.html#K2" title="AMI_1:func.2">IC</a> <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a>  <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"><font color="Maroon" title="c1">a</font> <a href="ami_3.html#K12" title="AMI_3:func.12">.--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:62"><span class="lab"><font color="Green" title="E46">A2</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A11</font></span>:  <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c11">v</font> = 
 <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c9">u</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></span>
<br/>.= 
<font color="Maroon" title="c10">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ami_1.html#T129" title="AMI_1:th.129">AMI_1:129</a></span>
;<br/><a NAME="E6:62_2_2"/><span class="lab"><font color="Green" title="E52">U</font></span>: 
not <font color="Maroon" title="c2">il</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"><font color="Maroon" title="c1">a</font> <a href="ami_3.html#K12" title="AMI_3:func.12">.--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:62"><span class="lab"><font color="Green" title="E47">A3</font></span></a>, <a class="txt" href="#E3:62_2_2"><span class="lab"><font color="Green" title="E50">A10</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><span class="lab"><font color="Green" title="E53">A12</font></span>: <span class="p1">(<span class="default"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c11">v</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c2">il</font> = 
<font color="Maroon" title="c11">v</font> <a href="ami_1.html#K12" title="AMI_1:func.12">.</a> <font color="Maroon" title="c2">il</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ami_1.html#T150" title="AMI_1:th.150">AMI_1:150</a></span>
<br/>.= 
<font color="Maroon" title="c9">u</font> <a href="ami_1.html#K12" title="AMI_1:func.12">.</a> <font color="Maroon" title="c10">n</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:62_2_2"><span class="lab"><font color="Green" title="E52">U</font></span></a>, <a class="ref" href="funct_4.html#T12" title="FUNCT_4:th.12">FUNCT_4:12</a></span>
<br/>.= 
<font color="Maroon" title="c6">I</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ami_1.html#T129" title="AMI_1:th.129">AMI_1:129</a></span>
;<br/><a NAME="E8:62_2_2"/>
<font color="Maroon" title="c1">a</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"><font color="Maroon" title="c1">a</font> <a href="ami_3.html#K12" title="AMI_3:func.12">.--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:62_2_2"><span class="lab"><font color="Green" title="E50">A10</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><span class="kw">then </span><font color="Maroon" title="c11">v</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <font color="Maroon" title="c1">a</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="ami_3.html#K12" title="AMI_3:func.12">.--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">a</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T14" title="FUNCT_4:th.14">FUNCT_4:14</a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T87" title="FUNCOP_1:th.87">FUNCOP_1:87</a></span>
;<br/><a NAME="E10:62_2_2"/><span class="kw">then </span>
 <a href="ami_1.html#K6" title="AMI_1:func.6">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"><a href="ami_1.html#K8" title="AMI_1:func.8">ProgramPart</a> <font color="Maroon" title="c11">v</font></span>)</span>,<font color="Maroon" title="c11">v</font></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">il</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:62_2_2"><span class="lab"><font color="Green" title="E51">A11</font></span></a>, <a class="txt" href="#E7:62_2_2"><span class="lab"><font color="Green" title="E53">A12</font></span></a>, <a class="ref" href="ami_3.html#T15" title="AMI_3:th.15">AMI_3:15</a></span>;<br/><span class="kw">hence </span><a NAME="E11:62_2_2"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="ami_3.html#K10" title="AMI_3:func.10">&gt;0_goto</a> <font color="Maroon" title="c3">k</font></span>)</span>,<font color="Maroon" title="c2">il</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:62_2_2"><span class="lab"><font color="Green" title="E49">A9</font></span></a>, <a class="txt" href="#E5:62_2_2"><span class="lab"><font color="Green" title="E51">A11</font></span></a>, <a class="txt" href="#E7:62_2_2"><span class="lab"><font color="Green" title="E53">A12</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

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