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<span class="kw">let </span><font color="Maroon" title="c1">a</font> be   <a href="ami_3.html#NM1" title="AMI_3:NM.1">Data-Location</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">il</font>, <font color="Olive" title="b2">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</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#K9" title="AMI_3:func.9">&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"><font color="Olive" title="b1">il</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</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>, <font color="Maroon" title="c3">k</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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#K9" title="AMI_3:func.9">&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"><font color="Maroon" title="c2">il</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><br/>





<span class="kw">set </span><font color="Maroon" title="c4">t</font> =  the   <a href="memstr_0.html#NM2" title="MEMSTR_0:NM.2">State</a> of <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">Q</font> =  the   <a href="compos_1.html#NM2" title="COMPOS_1:NM.2">Instruction-Sequence</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="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10" onmouseover="rs('xboole_0/D10')" onmouseout="rh()">XBOOLE_0:def 10</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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"><font color="Maroon" title="c2">il</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</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#K9" title="AMI_3:func.9">&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="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">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#K9" title="AMI_3:func.9">&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"><font color="Maroon" title="c2">il</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )</span><br/><span class="kw">assume </span><a NAME="E1:28_1"/>
<font color="Maroon" title="c6">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#K9" title="AMI_3:func.9">&gt;0_goto</a> <font color="Maroon" title="c3">k</font></span>)</span>,<font color="Maroon" title="c2">il</font>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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"><font color="Maroon" title="c2">il</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</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="c7">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> <span class="p1">(<span class="default"><a href="memstr_0.html#K2" title="MEMSTR_0:func.2">the_Values_of</a> <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a></span>)</span><span class="kw"> such that </span><br/><a NAME="E3:28_1"/><span class="lab"><font color="Green" title="E11">A1</font></span>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="memstr_0.html#K5" title="MEMSTR_0:func.5">IC</a> <span class="p1">(<span class="default"><a href="extpro_1.html#K2" title="EXTPRO_1:func.2">Exec</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="ami_3.html#K9" title="AMI_3:func.9">&gt;0_goto</a> <font color="Maroon" title="c3">k</font></span>)</span>,<font color="Maroon" title="c7">s</font>)</span>)</span> &amp;  <a href="memstr_0.html#K5" title="MEMSTR_0:func.5">IC</a> <font color="Maroon" title="c7">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">il</font> )
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:28_1_1"/>
( <font color="Maroon" title="c7">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c7">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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#K5" title="NUMBERS:func.5">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:28_1_1_1"/>
<font color="Maroon" title="c7">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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"><font color="Maroon" title="c2">il</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><br/><div class="add"><a NAME="E2:28_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">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="#E3:28_1"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="ref" href="ami_3.html#T9" onmouseover="rs('ami_3/T9')" onmouseout="rh()">AMI_3:9</a></span>;<br/><span class="kw">hence </span><a NAME="E3:28_1_1_1"/>
<font color="Maroon" title="c6">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"><font color="Maroon" title="c2">il</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</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" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="28_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:28_1_1_2"/>
<font color="Maroon" title="c7">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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#K5" title="NUMBERS:func.5">0</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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"><font color="Maroon" title="c2">il</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><br/><div class="add"><a NAME="E2:28_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">il</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:28_1"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="ref" href="ami_3.html#T9" onmouseover="rs('ami_3/T9')" onmouseout="rh()">AMI_3:9</a></span>;<br/><span class="kw">hence </span><a NAME="E3:28_1_1_2"/>
<font color="Maroon" title="c6">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"><font color="Maroon" title="c2">il</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</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" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c6">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"><font color="Maroon" title="c2">il</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> or <font color="Maroon" title="c6">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#K9" title="AMI_3:func.9">&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">set </span><font color="Maroon" title="c7">I</font> = <font color="Maroon" title="c1">a</font> <a href="ami_3.html#K9" title="AMI_3:func.9">&gt;0_goto</a> <font color="Maroon" title="c3">k</font>;<br/>
<a NAME="E2:28"/><span class="lab"><font color="Green" title="E11">A2</font></span>: 
 <a href="memstr_0.html#NK2" title="MEMSTR_0:NK.2">IC</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#T2" onmouseover="rs('ami_5/T2')" onmouseout="rh()">AMI_5:2</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:28"/><span class="lab"><font color="Green" title="E12">A3</font></span>: 
<font color="Maroon" title="c6">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"><font color="Maroon" title="c2">il</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">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#K9" title="AMI_3:func.9">&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="c8">n</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="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">il1</font> = <font color="Maroon" title="c8">n</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="memstr_0.html#K4" title="MEMSTR_0:func.4">Values</a> <span class="p1">(<span class="default"><a href="memstr_0.html#NK2" title="MEMSTR_0:NK.2">IC</a> </span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="memstr_0.html#D6" onmouseover="rs('memstr_0/D6')" onmouseout="rh()">MEMSTR_0:def 6</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">u</font> =  the   <a href="memstr_0.html#NM2" title="MEMSTR_0:NM.2">State</a> of <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> <a href="funct_7.html#K1" title="FUNCT_7:func.1">+*</a> (<span class="p1">(<span class="default"><a href="memstr_0.html#NK2" title="MEMSTR_0:NK.2">IC</a> </span>)</span>,<font color="Maroon" title="c9">il1</font>) 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> <span class="p1">(<span class="default"><a href="memstr_0.html#K2" title="MEMSTR_0:func.2">the_Values_of</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="card_3.html#T107" onmouseover="rs('card_3/T107')" onmouseout="rh()">CARD_3:107</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">P</font> =  the   <a href="compos_1.html#NM2" title="COMPOS_1:NM.2">Instruction-Sequence</a> of <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> <a href="funct_7.html#K1" title="FUNCT_7:func.1">+*</a> (<font color="Maroon" title="c2">il</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="ami_3.html#K9" title="AMI_3:func.9">&gt;0_goto</a> <font color="Maroon" title="c3">k</font></span>)</span>) as   <a href="compos_1.html#NM2" title="COMPOS_1:NM.2">Instruction-Sequence</a> of <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_2"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:28_2"/>
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font> or <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">il</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:28"><span class="lab"><font color="Green" title="E12">A3</font></span></a>, <a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_2_1"><span class="kw">suppose </span></a><a NAME="E1:28_2_1"/><span class="lab"><font color="Green" title="E13">A4</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">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#K9" title="AMI_3:func.9">&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="c12">v</font> = <font color="Maroon" title="c10">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#K11" title="AMI_3:func.11">.--&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> <span class="p1">(<span class="default"><a href="memstr_0.html#K2" title="MEMSTR_0:func.2">the_Values_of</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="card_3.html#T107" onmouseover="rs('card_3/T107')" onmouseout="rh()">CARD_3:107</a></span>;<br/><a NAME="E3:28_2_1"/><span class="lab"><font color="Green" title="E14">A5</font></span>: 
 <a href="memstr_0.html#NK2" title="MEMSTR_0:NK.2">IC</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a>  the   <a href="memstr_0.html#NM2" title="MEMSTR_0:NM.2">State</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="memstr_0.html#T2" onmouseover="rs('memstr_0/T2')" onmouseout="rh()">MEMSTR_0:2</a></span>;<br/><a NAME="E4:28_2_1"/>
 not  <a href="memstr_0.html#NK2" title="MEMSTR_0:NK.2">IC</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="ami_3.html#K11" title="AMI_3:func.11">.--&gt;</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:28"><span class="lab"><font color="Green" title="E11">A2</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A7</font></span>:  <a href="memstr_0.html#K5" title="MEMSTR_0:func.5">IC</a> <font color="Maroon" title="c12">v</font> = 
 <a href="memstr_0.html#K5" title="MEMSTR_0:func.5">IC</a> <font color="Maroon" title="c10">u</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T11" onmouseover="rs('funct_4/T11')" onmouseout="rh()">FUNCT_4:11</a></span>
<br/>.= 
<font color="Maroon" title="c8">n</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:28_2_1"><span class="lab"><font color="Green" title="E14">A5</font></span></a>, <a class="ref" href="funct_7.html#T31" onmouseover="rs('funct_7/T31')" onmouseout="rh()">FUNCT_7:31</a></span>
;<br/><span class="kw">reconsider </span><font color="Maroon" title="c13">il</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="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>;<br/><a NAME="E7:28_2_1"/><span class="lab"><font color="Green" title="E16">A8</font></span>: 
<font color="Maroon" title="c11">P</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">il</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">P</font> <a href="compos_1.html#K3" title="COMPOS_1:func.3">.</a> <font color="Maroon" title="c13">il</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#T143" onmouseover="rs('pboole/T143')" onmouseout="rh()">PBOOLE:143</a></span>;<br/><a NAME="E8:28_2_1"/>
<font color="Maroon" title="c13">il</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 ;<br/><a NAME="E9:28_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">il</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a>  the   <a href="compos_1.html#NM2" title="COMPOS_1:NM.2">Instruction-Sequence</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="partfun1.html#D2" onmouseover="rs('partfun1/D2')" onmouseout="rh()">PARTFUN1:def 2</a></span>;<br/><a NAME="E10:28_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A9</font></span>: 
<font color="Maroon" title="c11">P</font> <a href="compos_1.html#K3" title="COMPOS_1:func.3">.</a> <font color="Maroon" title="c13">il</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> <a href="ami_3.html#K9" title="AMI_3:func.9">&gt;0_goto</a> <font color="Maroon" title="c3">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T31" onmouseover="rs('funct_7/T31')" onmouseout="rh()">FUNCT_7:31</a></span>;<br/><a NAME="E11:28_2_1"/>
<font color="Maroon" title="c1">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="ami_3.html#K11" title="AMI_3:func.11">.--&gt;</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><span class="kw">then </span><font color="Maroon" title="c12">v</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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#K11" title="AMI_3:func.11">.--&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#T13" onmouseover="rs('funct_4/T13')" onmouseout="rh()">FUNCT_4:13</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T72" onmouseover="rs('funcop_1/T72')" onmouseout="rh()">FUNCOP_1:72</a></span>
;<br/><a NAME="E13:28_2_1"/><span class="kw">then </span>
 <a href="memstr_0.html#K5" title="MEMSTR_0:func.5">IC</a> <span class="p1">(<span class="default"><a href="extpro_1.html#K4" title="EXTPRO_1:func.4">Following</a> (<font color="Maroon" title="c11">P</font>,<font color="Maroon" title="c12">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="#E5:28_2_1"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="txt" href="#E10:28_2_1"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="txt" href="#E7:28_2_1"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="ref" href="ami_3.html#T9" onmouseover="rs('ami_3/T9')" onmouseout="rh()">AMI_3:9</a></span>;<br/><span class="kw">hence </span><a NAME="E14:28_2_1"/>
<font color="Maroon" title="c6">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#K9" title="AMI_3:func.9">&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="#E1:28_2_1"><span class="lab"><font color="Green" title="E13">A4</font></span></a>, <a class="txt" href="#E5:28_2_1"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="txt" href="#E10:28_2_1"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="txt" href="#E7:28_2_1"><span class="lab"><font color="Green" title="E16">A8</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="28_2_2"><span class="kw">suppose </span></a><a NAME="E1:28_2_2"/><span class="lab"><font color="Green" title="E13">A10</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">il</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">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#K9" title="AMI_3:func.9">&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="c12">v</font> = <font color="Maroon" title="c10">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#K11" title="AMI_3:func.11">.--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">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> <span class="p1">(<span class="default"><a href="memstr_0.html#K2" title="MEMSTR_0:func.2">the_Values_of</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="card_3.html#T107" onmouseover="rs('card_3/T107')" onmouseout="rh()">CARD_3:107</a></span>;<br/><a NAME="E3:28_2_2"/><span class="lab"><font color="Green" title="E14">A11</font></span>: 
 <a href="memstr_0.html#NK2" title="MEMSTR_0:NK.2">IC</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a>  the   <a href="memstr_0.html#NM2" title="MEMSTR_0:NM.2">State</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="memstr_0.html#T2" onmouseover="rs('memstr_0/T2')" onmouseout="rh()">MEMSTR_0:2</a></span>;<br/><a NAME="E4:28_2_2"/>
 not  <a href="memstr_0.html#NK2" title="MEMSTR_0:NK.2">IC</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="ami_3.html#K11" title="AMI_3:func.11">.--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:28"><span class="lab"><font color="Green" title="E11">A2</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A13</font></span>:  <a href="memstr_0.html#K5" title="MEMSTR_0:func.5">IC</a> <font color="Maroon" title="c12">v</font> = 
 <a href="memstr_0.html#K5" title="MEMSTR_0:func.5">IC</a> <font color="Maroon" title="c10">u</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T11" onmouseover="rs('funct_4/T11')" onmouseout="rh()">FUNCT_4:11</a></span>
<br/>.= 
<font color="Maroon" title="c8">n</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:28_2_2"><span class="lab"><font color="Green" title="E14">A11</font></span></a>, <a class="ref" href="funct_7.html#T31" onmouseover="rs('funct_7/T31')" onmouseout="rh()">FUNCT_7:31</a></span>
;<br/><span class="kw">reconsider </span><font color="Maroon" title="c13">il</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="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>;<br/><a NAME="E7:28_2_2"/><span class="lab"><font color="Green" title="E16">A14</font></span>: 
<font color="Maroon" title="c11">P</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">il</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">P</font> <a href="compos_1.html#K3" title="COMPOS_1:func.3">.</a> <font color="Maroon" title="c13">il</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#T143" onmouseover="rs('pboole/T143')" onmouseout="rh()">PBOOLE:143</a></span>;<br/><a NAME="E8:28_2_2"/>
<font color="Maroon" title="c13">il</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> 
 ;<br/><a NAME="E9:28_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c13">il</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a>  the   <a href="compos_1.html#NM2" title="COMPOS_1:NM.2">Instruction-Sequence</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="partfun1.html#D2" onmouseover="rs('partfun1/D2')" onmouseout="rh()">PARTFUN1:def 2</a></span>;<br/><a NAME="E10:28_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A15</font></span>: 
<font color="Maroon" title="c11">P</font> <a href="compos_1.html#K3" title="COMPOS_1:func.3">.</a> <font color="Maroon" title="c13">il</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> <a href="ami_3.html#K9" title="AMI_3:func.9">&gt;0_goto</a> <font color="Maroon" title="c3">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_7.html#T31" onmouseover="rs('funct_7/T31')" onmouseout="rh()">FUNCT_7:31</a></span>;<br/><a NAME="E11:28_2_2"/>
<font color="Maroon" title="c1">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font> <a href="ami_3.html#K11" title="AMI_3:func.11">.--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><span class="kw">then </span><font color="Maroon" title="c12">v</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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#K11" title="AMI_3:func.11">.--&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">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#T13" onmouseover="rs('funct_4/T13')" onmouseout="rh()">FUNCT_4:13</a></span>
<br/>.= 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T72" onmouseover="rs('funcop_1/T72')" onmouseout="rh()">FUNCOP_1:72</a></span>
;<br/><a NAME="E13:28_2_2"/><span class="kw">then </span>
 <a href="memstr_0.html#K5" title="MEMSTR_0:func.5">IC</a> <span class="p1">(<span class="default"><a href="extpro_1.html#K4" title="EXTPRO_1:func.4">Following</a> (<font color="Maroon" title="c11">P</font>,<font color="Maroon" title="c12">v</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">il</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:28_2_2"><span class="lab"><font color="Green" title="E15">A13</font></span></a>, <a class="txt" href="#E10:28_2_2"><span class="lab"><font color="Green" title="E17">A15</font></span></a>, <a class="txt" href="#E7:28_2_2"><span class="lab"><font color="Green" title="E16">A14</font></span></a>, <a class="ref" href="ami_3.html#T9" onmouseover="rs('ami_3/T9')" onmouseout="rh()">AMI_3:9</a></span>;<br/><span class="kw">hence </span><a NAME="E14:28_2_2"/>
<font color="Maroon" title="c6">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#K9" title="AMI_3:func.9">&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="#E1:28_2_2"><span class="lab"><font color="Green" title="E13">A10</font></span></a>, <a class="txt" href="#E5:28_2_2"><span class="lab"><font color="Green" title="E15">A13</font></span></a>, <a class="txt" href="#E10:28_2_2"><span class="lab"><font color="Green" title="E17">A15</font></span></a>, <a class="txt" href="#E7:28_2_2"><span class="lab"><font color="Green" title="E16">A14</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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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