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

<span class="kw">let </span><font color="Maroon" title="c2">N</font> be   <a href="measure6.html#NV2" title="MEASURE6:NV.2">with_zero</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</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">S</font> being   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="memstr_0.html#V2" title="MEMSTR_0:attr.2">with_non-empty_values</a>   <a href="memstr_0.html#V3" title="MEMSTR_0:attr.3">IC-Ins-separated</a>   <a href="extpro_1.html#L1" title="EXTPRO_1:struct.1">AMI-Struct</a> over <font color="Maroon" title="c2">N</font><br/>  for <font color="Olive" title="b2">il</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b3">i</font> being   <a href="compos_1.html#NM1" title="COMPOS_1:NM.1">Instruction</a> of <font color="Olive" title="b1">S</font>  st <font color="Olive" title="b3">i</font> is  <a href="amistd_1.html#V4" title="AMISTD_1:attr.4">sequential</a>  holds <br/> <a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> (<font color="Olive" title="b3">i</font>,<font color="Olive" title="b2">il</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">il</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">S</font> be   non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>   <a href="memstr_0.html#V2" title="MEMSTR_0:attr.2">with_non-empty_values</a>   <a href="memstr_0.html#V3" title="MEMSTR_0:attr.3">IC-Ins-separated</a>   <a href="extpro_1.html#L1" title="EXTPRO_1:struct.1">AMI-Struct</a> over <font color="Maroon" title="c2">N</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">  for <font color="Olive" title="b1">il</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b2">i</font> being   <a href="compos_1.html#NM1" title="COMPOS_1:NM.1">Instruction</a> of <font color="Maroon" title="c3">S</font>  st <font color="Olive" title="b2">i</font> is  <a href="amistd_1.html#V4" title="AMISTD_1:attr.4">sequential</a>  holds <br/> <a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> (<font color="Olive" title="b2">i</font>,<font color="Olive" title="b1">il</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">il</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c4">il</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">  for <font color="Olive" title="b1">i</font> being   <a href="compos_1.html#NM1" title="COMPOS_1:NM.1">Instruction</a> of <font color="Maroon" title="c3">S</font>  st <font color="Olive" title="b1">i</font> is  <a href="amistd_1.html#V4" title="AMISTD_1:attr.4">sequential</a>  holds <br/> <a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> (<font color="Olive" title="b1">i</font>,<font color="Maroon" title="c4">il</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">il</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c5">i</font> be   <a href="compos_1.html#NM1" title="COMPOS_1:NM.1">Instruction</a> of <font color="Maroon" title="c3">S</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="c5">i</font> is  <a href="amistd_1.html#V4" title="AMISTD_1:attr.4">sequential</a>  implies  <a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> (<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c4">il</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">il</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )</span><br/>







<span class="kw">assume </span><a NAME="E1:32"/><span class="lab"><font color="Green" title="E14">A1</font></span>: 
 for <font color="Olive" title="b1">s</font> being   <a href="memstr_0.html#NM2" title="MEMSTR_0:NM.2">State</a> of <font color="Maroon" title="c3">S</font> holds  <span class="p1">(<span class="default"><a href="extpro_1.html#K2" title="EXTPRO_1:func.2">Exec</a> (<font color="Maroon" title="c5">i</font>,<font color="Olive" title="b1">s</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="memstr_0.html#NK2" title="MEMSTR_0:NK.2">IC</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="memstr_0.html#K5" title="MEMSTR_0:func.5">IC</a> <font color="Olive" title="b1">s</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="amistd_1.html#D8" target="_self" onmouseover="rs('amistd_1/D8')" onmouseout="rh()">AMISTD_1:def 8</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> (<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c4">il</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">il</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><br/>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="32_1"><span class="kw">now </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">  for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">il</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> iff <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="memstr_0.html#K5" title="MEMSTR_0:func.5">IC</a> <span class="p3">(<span class="default"><a href="extpro_1.html#K2" title="EXTPRO_1:func.2">Exec</a> (<font color="Maroon" title="c5">i</font>,<font color="Olive" title="b2">ss</font>)</span>)</span></span>)</span> where <font color="Olive" title="b2">ss</font> is    <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> <font color="Maroon" title="c3">S</font></span>)</span> :  <a href="memstr_0.html#K5" title="MEMSTR_0:func.5">IC</a> <font color="Olive" title="b2">ss</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">il</font> </span> } </span>  )</span></a><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> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">il</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> iff <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="memstr_0.html#K5" title="MEMSTR_0:func.5">IC</a> <span class="p3">(<span class="default"><a href="extpro_1.html#K2" title="EXTPRO_1:func.2">Exec</a> (<font color="Maroon" title="c5">i</font>,<font color="Olive" title="b1">ss</font>)</span>)</span></span>)</span> where <font color="Olive" title="b1">ss</font> is    <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> <font color="Maroon" title="c3">S</font></span>)</span> :  <a href="memstr_0.html#K5" title="MEMSTR_0:func.5">IC</a> <font color="Olive" title="b1">ss</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">il</font> </span> } </span>  )</span><br/><div><span class="lab"><font color="Green" title="E15">A2</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="32_1_1"><span class="kw">now </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#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">il</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 implies <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="memstr_0.html#K5" title="MEMSTR_0:func.5">IC</a> <span class="p3">(<span class="default"><a href="extpro_1.html#K2" title="EXTPRO_1:func.2">Exec</a> (<font color="Maroon" title="c5">i</font>,<font color="Olive" title="b1">ss</font>)</span>)</span></span>)</span> where <font color="Olive" title="b1">ss</font> is    <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> <font color="Maroon" title="c3">S</font></span>)</span> :  <a href="memstr_0.html#K5" title="MEMSTR_0:func.5">IC</a> <font color="Olive" title="b1">ss</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">il</font> </span> } </span>  )</span></a><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c7">ll</font> = <font color="Maroon" title="c4">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="c8">il1</font> = <font color="Maroon" title="c7">ll</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">set </span><font color="Maroon" title="c9">I</font> = <font color="Maroon" title="c5">i</font>;<br/><span class="kw">set </span><font color="Maroon" title="c10">t</font> =  the   <a href="memstr_0.html#NM2" title="MEMSTR_0:NM.2">State</a> of <font color="Maroon" title="c3">S</font>;<br/><span class="kw">set </span><font color="Maroon" title="c11">P</font> =  the   <a href="compos_1.html#NM2" title="COMPOS_1:NM.2">Instruction-Sequence</a> of <font color="Maroon" title="c3">S</font>;<br/><span class="kw">assume </span><a NAME="E3:32_1_1"/><span class="lab"><font color="Green" title="E16">A3</font></span>: 
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">il</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="memstr_0.html#K5" title="MEMSTR_0:func.5">IC</a> <span class="p3">(<span class="default"><a href="extpro_1.html#K2" title="EXTPRO_1:func.2">Exec</a> (<font color="Maroon" title="c5">i</font>,<font color="Olive" title="b1">ss</font>)</span>)</span></span>)</span> where <font color="Olive" title="b1">ss</font> is    <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> <font color="Maroon" title="c3">S</font></span>)</span> :  <a href="memstr_0.html#K5" title="MEMSTR_0:func.5">IC</a> <font color="Olive" title="b1">ss</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">il</font> </span> } </span> </span><br/><span class="kw">reconsider </span><font color="Maroon" title="c12">u</font> =  the   <a href="memstr_0.html#NM2" title="MEMSTR_0:NM.2">State</a> of <font color="Maroon" title="c3">S</font> <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="c8">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> <font color="Maroon" title="c3">S</font></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="E5:32_1_1"/>
<font color="Maroon" title="c7">ll</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="E6:32_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A4</font></span>: 
<font color="Maroon" title="c4">il</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a>  the   <a href="compos_1.html#NM2" title="COMPOS_1:NM.2">Instruction-Sequence</a> of <font color="Maroon" title="c3">S</font>
 <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/><span class="lab"><font color="Green" title="E18">A5</font></span>: <span class="p1">(<span class="default"> the   <a href="compos_1.html#NM2" title="COMPOS_1:NM.2">Instruction-Sequence</a> of <font color="Maroon" title="c3">S</font> <a href="funct_7.html#K1" title="FUNCT_7:func.1">+*</a> (<font color="Maroon" title="c4">il</font>,<font color="Maroon" title="c5">i</font>)</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">ll</font> = 
<span class="p1">(<span class="default"> the   <a href="compos_1.html#NM2" title="COMPOS_1:NM.2">Instruction-Sequence</a> of <font color="Maroon" title="c3">S</font> <a href="funct_7.html#K1" title="FUNCT_7:func.1">+*</a> (<font color="Maroon" title="c4">il</font>,<font color="Maroon" title="c5">i</font>)</span>)</span> <a href="compos_1.html#K3" title="COMPOS_1:func.3">.</a> <font color="Maroon" title="c7">ll</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/>.= 
<font color="Maroon" title="c5">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:32_1_1"><span class="lab"><font color="Green" title="E17">A4</font></span></a>, <a class="ref" href="funct_7.html#T31" onmouseover="rs('funct_7/T31')" onmouseout="rh()">FUNCT_7:31</a></span>
;<br/><a NAME="E8:32_1_1"/>
 <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="relset_1.html#K1" title="RELSET_1:func.1">dom</a>  the   <a href="memstr_0.html#NM2" title="MEMSTR_0:NM.2">State</a> of <font color="Maroon" title="c3">S</font>
 <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="E9:32_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A6</font></span>: 
 <a href="memstr_0.html#K5" title="MEMSTR_0:func.5">IC</a> <font color="Maroon" title="c12">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">il</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="E10:32_1_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> (<span class="p2">(<span class="default"> the   <a href="compos_1.html#NM2" title="COMPOS_1:NM.2">Instruction-Sequence</a> of <font color="Maroon" title="c3">S</font> <a href="funct_7.html#K1" title="FUNCT_7:func.1">+*</a> (<font color="Maroon" title="c4">il</font>,<font color="Maroon" title="c5">i</font>)</span>)</span>,<font color="Maroon" title="c12">u</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">il</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32"><span class="lab"><font color="Green" title="E14">A1</font></span></a>, <a class="txt" href="#E7:32_1_1"><span class="lab"><font color="Green" title="E18">A5</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E11:32_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="memstr_0.html#K5" title="MEMSTR_0:func.5">IC</a> <span class="p3">(<span class="default"><a href="extpro_1.html#K2" title="EXTPRO_1:func.2">Exec</a> (<font color="Maroon" title="c5">i</font>,<font color="Olive" title="b1">ss</font>)</span>)</span></span>)</span> where <font color="Olive" title="b1">ss</font> is    <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> <font color="Maroon" title="c3">S</font></span>)</span> :  <a href="memstr_0.html#K5" title="MEMSTR_0:func.5">IC</a> <font color="Olive" title="b1">ss</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">il</font> </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:32_1_1"><span class="lab"><font color="Green" title="E16">A3</font></span></a>, <a class="txt" href="#E9:32_1_1"><span class="lab"><font color="Green" title="E19">A6</font></span></a>, <a class="txt" href="#E7:32_1_1"><span class="lab"><font color="Green" title="E18">A5</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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="32_1_2"><span class="kw">now </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> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="memstr_0.html#K5" title="MEMSTR_0:func.5">IC</a> <span class="p3">(<span class="default"><a href="extpro_1.html#K2" title="EXTPRO_1:func.2">Exec</a> (<font color="Maroon" title="c5">i</font>,<font color="Olive" title="b1">ss</font>)</span>)</span></span>)</span> where <font color="Olive" title="b1">ss</font> is    <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> <font color="Maroon" title="c3">S</font></span>)</span> :  <a href="memstr_0.html#K5" title="MEMSTR_0:func.5">IC</a> <font color="Olive" title="b1">ss</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">il</font> </span> } </span>  implies <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">il</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:32_1_2"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="memstr_0.html#K5" title="MEMSTR_0:func.5">IC</a> <span class="p3">(<span class="default"><a href="extpro_1.html#K2" title="EXTPRO_1:func.2">Exec</a> (<font color="Maroon" title="c5">i</font>,<font color="Olive" title="b1">ss</font>)</span>)</span></span>)</span> where <font color="Olive" title="b1">ss</font> is    <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> <font color="Maroon" title="c3">S</font></span>)</span> :  <a href="memstr_0.html#K5" title="MEMSTR_0:func.5">IC</a> <font color="Olive" title="b1">ss</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">il</font> </span> } </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#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">il</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span><br/><a NAME="E2:32_1_2"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">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> <font color="Maroon" title="c3">S</font></span>)</span> st <br/>( <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> (<font color="Maroon" title="c5">i</font>,<font color="Olive" title="b1">s</font>)</span>)</span> &amp;  <a href="memstr_0.html#K5" title="MEMSTR_0:func.5">IC</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">il</font> )
 ;<br/><span class="kw">hence </span><a NAME="E3:32_1_2"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">il</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32"><span class="lab"><font color="Green" title="E14">A1</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><span class="kw">hence </span><a NAME="E3:32_1"/>
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">il</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> iff <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="memstr_0.html#K5" title="MEMSTR_0:func.5">IC</a> <span class="p3">(<span class="default"><a href="extpro_1.html#K2" title="EXTPRO_1:func.2">Exec</a> (<font color="Maroon" title="c5">i</font>,<font color="Olive" title="b1">ss</font>)</span>)</span></span>)</span> where <font color="Olive" title="b1">ss</font> is    <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> <font color="Maroon" title="c3">S</font></span>)</span> :  <a href="memstr_0.html#K5" title="MEMSTR_0:func.5">IC</a> <font color="Olive" title="b1">ss</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">il</font> </span> } </span>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32_1"><span class="lab"><font color="Green" title="E15">A2</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</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>
<span class="kw">hence </span><a NAME="E3:32"/>
 <a href="amistd_1.html#K1" title="AMISTD_1:func.1">NIC</a> (<font color="Maroon" title="c5">i</font>,<font color="Maroon" title="c4">il</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">il</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#T2" onmouseover="rs('tarski/T2')" onmouseout="rh()">TARSKI: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>
