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<span class="kw">let </span><font color="Maroon" title="c1">I1</font>, <font color="Maroon" title="c2">I2</font> be   <a href="afinsq_1.html#NM2" title="AFINSQ_1:NM.2">XFinSequence</a> of the <a href="ami_1.html#U2" title="AMI_1:sel.2">Instructions</a> of <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</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">D</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> <br/> for <font color="Olive" title="b2">il</font>, <font color="Olive" title="b3">ps</font>, <font color="Olive" title="b4">ds</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a> <br/> for <font color="Olive" title="b5">s</font> being   <a href="scm_1.html#M1" title="SCM_1:mode.1">State-consisting</a> of <font color="Olive" title="b2">il</font>,<font color="Olive" title="b3">ps</font>,<font color="Olive" title="b4">ds</font>,<font color="Maroon" title="c1">I1</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c2">I2</font>,<font color="Olive" title="b1">D</font> holds <br/> ( <font color="Olive" title="b5">s</font> is    <a href="scm_1.html#M1" title="SCM_1:mode.1">State-consisting</a> of <font color="Olive" title="b2">il</font>,<font color="Olive" title="b3">ps</font>,<font color="Olive" title="b4">ds</font>,<font color="Maroon" title="c1">I1</font>,<font color="Olive" title="b1">D</font> &amp; <font color="Olive" title="b5">s</font> is    <a href="scm_1.html#M1" title="SCM_1:mode.1">State-consisting</a> of <font color="Olive" title="b2">il</font>,<font color="Olive" title="b3">ps</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c1">I1</font></span>)</span>,<font color="Olive" title="b4">ds</font>,<font color="Maroon" title="c2">I2</font>,<font color="Olive" title="b1">D</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">D</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</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>, <font color="Olive" title="b2">ps</font>, <font color="Olive" title="b3">ds</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a> <br/> for <font color="Olive" title="b4">s</font> being   <a href="scm_1.html#M1" title="SCM_1:mode.1">State-consisting</a> of <font color="Olive" title="b1">il</font>,<font color="Olive" title="b2">ps</font>,<font color="Olive" title="b3">ds</font>,<font color="Maroon" title="c1">I1</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c2">I2</font>,<font color="Maroon" title="c3">D</font> holds <br/> ( <font color="Olive" title="b4">s</font> is    <a href="scm_1.html#M1" title="SCM_1:mode.1">State-consisting</a> of <font color="Olive" title="b1">il</font>,<font color="Olive" title="b2">ps</font>,<font color="Olive" title="b3">ds</font>,<font color="Maroon" title="c1">I1</font>,<font color="Maroon" title="c3">D</font> &amp; <font color="Olive" title="b4">s</font> is    <a href="scm_1.html#M1" title="SCM_1:mode.1">State-consisting</a> of <font color="Olive" title="b1">il</font>,<font color="Olive" title="b2">ps</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c1">I1</font></span>)</span>,<font color="Olive" title="b3">ds</font>,<font color="Maroon" title="c2">I2</font>,<font color="Maroon" title="c3">D</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">il</font>, <font color="Maroon" title="c5">ps</font>, <font color="Maroon" title="c6">ds</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">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">s</font> being   <a href="scm_1.html#M1" title="SCM_1:mode.1">State-consisting</a> of <font color="Maroon" title="c4">il</font>,<font color="Maroon" title="c5">ps</font>,<font color="Maroon" title="c6">ds</font>,<font color="Maroon" title="c1">I1</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c2">I2</font>,<font color="Maroon" title="c3">D</font> holds <br/> ( <font color="Olive" title="b1">s</font> is    <a href="scm_1.html#M1" title="SCM_1:mode.1">State-consisting</a> of <font color="Maroon" title="c4">il</font>,<font color="Maroon" title="c5">ps</font>,<font color="Maroon" title="c6">ds</font>,<font color="Maroon" title="c1">I1</font>,<font color="Maroon" title="c3">D</font> &amp; <font color="Olive" title="b1">s</font> is    <a href="scm_1.html#M1" title="SCM_1:mode.1">State-consisting</a> of <font color="Maroon" title="c4">il</font>,<font color="Maroon" title="c5">ps</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c1">I1</font></span>)</span>,<font color="Maroon" title="c6">ds</font>,<font color="Maroon" title="c2">I2</font>,<font color="Maroon" title="c3">D</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c7">s</font> be    <a href="scm_1.html#M1" title="SCM_1:mode.1">State-consisting</a> of <font color="Maroon" title="c4">il</font>,<font color="Maroon" title="c5">ps</font>,<font color="Maroon" title="c6">ds</font>,<font color="Maroon" title="c1">I1</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c2">I2</font>,<font color="Maroon" title="c3">D</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="c7">s</font> is    <a href="scm_1.html#M1" title="SCM_1:mode.1">State-consisting</a> of <font color="Maroon" title="c4">il</font>,<font color="Maroon" title="c5">ps</font>,<font color="Maroon" title="c6">ds</font>,<font color="Maroon" title="c1">I1</font>,<font color="Maroon" title="c3">D</font> &amp; <font color="Maroon" title="c7">s</font> is    <a href="scm_1.html#M1" title="SCM_1:mode.1">State-consisting</a> of <font color="Maroon" title="c4">il</font>,<font color="Maroon" title="c5">ps</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c1">I1</font></span>)</span>,<font color="Maroon" title="c6">ds</font>,<font color="Maroon" title="c2">I2</font>,<font color="Maroon" title="c3">D</font> )</span><br/>







<a NAME="E1:1"/><span class="lab"><font color="Green" title="E2">A1</font></span>: 
 <a href="ami_1.html#K6" title="AMI_1:func.6">IC</a> <font color="Maroon" title="c7">s</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="scm_1.html#D1" title="SCM_1:def.1">SCM_1:def 1</a></span>;<br/>
<a NAME="E2:1"/><span class="lab"><font color="Green" title="E3">A2</font></span>: 
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">I1</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c2">I2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c1">I1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c2">I2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="afinsq_1.html#T20" title="AFINSQ_1:th.20">AFINSQ_1:20</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E4">A3</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">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"> ( <font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c2">I2</font> implies <font color="Maroon" title="c7">s</font> <a href="ami_1.html#K12" title="AMI_1:func.12">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">ps</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c1">I1</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">I2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k</font> )</span><br/><span class="kw">assume </span><a NAME="E1:1_1"/>
<font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c2">I2</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="c7">s</font> <a href="ami_1.html#K12" title="AMI_1:func.12">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">ps</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c1">I1</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">I2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k</font></span><br/><a NAME="E2:1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E5">A5</font></span>: 
<span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c1">I1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">I1</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c2">I2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1"><span class="lab"><font color="Green" title="E3">A2</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><span class="lab"><font color="Green" title="E6">A6</font></span>: <font color="Maroon" title="c7">s</font> <a href="ami_1.html#K12" title="AMI_1:func.12">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">ps</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c1">I1</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c8">k</font></span>)</span> = 
<font color="Maroon" title="c7">s</font> <a href="ami_1.html#K12" title="AMI_1:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">ps</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c1">I1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">I1</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c2">I2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c1">I1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c8">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1"><span class="lab"><font color="Green" title="E5">A5</font></span></a>, <a class="ref" href="scm_1.html#D1" title="SCM_1:def.1">SCM_1:def 1</a></span>
;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c1">I1</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c2">I2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c1">I1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c8">k</font></span>)</span> = 
<font color="Maroon" title="c2">I2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c1">I1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c1">I1</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1"><span class="lab"><font color="Green" title="E3">A2</font></span></a>, <a class="txt" href="#E2:1_1"><span class="lab"><font color="Green" title="E5">A5</font></span></a>, <a class="ref" href="afinsq_1.html#T21" title="AFINSQ_1:th.21">AFINSQ_1:21</a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>
<br/>.= 
<font color="Maroon" title="c2">I2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k</font>

;<br/><span class="kw">hence </span><a NAME="E5:1_1"/>
<font color="Maroon" title="c7">s</font> <a href="ami_1.html#K12" title="AMI_1:func.12">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">ps</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c1">I1</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">I2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1"><span class="lab"><font color="Green" title="E6">A6</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="lab"><font color="Green" title="E5">A7</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">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"> ( <font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c1">I1</font> implies <font color="Maroon" title="c7">s</font> <a href="ami_1.html#K12" title="AMI_1:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">ps</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k</font> )</span><br/><span class="kw">assume </span><a NAME="E1:1_2"/><span class="lab"><font color="Green" title="E6">A8</font></span>: 
<font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c1">I1</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="c7">s</font> <a href="ami_1.html#K12" title="AMI_1:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">ps</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k</font></span><br/><a NAME="E2:1_2"/>
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c1">I1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c1">I1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c2">I2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/><a NAME="E3:1_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">I1</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c2">I2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1"><span class="lab"><font color="Green" title="E3">A2</font></span></a>, <a class="txt" href="#E1:1_2"><span class="lab"><font color="Green" title="E6">A8</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E4:1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E7">A9</font></span>: 
<font color="Maroon" title="c7">s</font> <a href="ami_1.html#K12" title="AMI_1:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">ps</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">I1</font> <a href="ordinal4.html#K1" title="ORDINAL4:func.1">^</a> <font color="Maroon" title="c2">I2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="scm_1.html#D1" title="SCM_1:def.1">SCM_1:def 1</a></span>;<br/><a NAME="E5:1_2"/>
<font color="Maroon" title="c8">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="afinsq_1.html#K2" title="AFINSQ_1:func.2">dom</a> <font color="Maroon" title="c1">I1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_2"><span class="lab"><font color="Green" title="E6">A8</font></span></a>, <a class="ref" href="nat_1.html#T45" title="NAT_1:th.45">NAT_1:45</a></span>;<br/><span class="kw">hence </span><a NAME="E6:1_2"/>
<font color="Maroon" title="c7">s</font> <a href="ami_1.html#K12" title="AMI_1:func.12">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">ps</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">I1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_2"><span class="lab"><font color="Green" title="E7">A9</font></span></a>, <a class="ref" href="afinsq_1.html#D4" title="AFINSQ_1:def.4">AFINSQ_1:def 4</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>
<a NAME="E5:1"/>
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D</font> holds <br/><font color="Maroon" title="c7">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">ds</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="scm_1.html#D1" title="SCM_1:def.1">SCM_1:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:1"/>
<font color="Maroon" title="c7">s</font> is    <a href="scm_1.html#M1" title="SCM_1:mode.1">State-consisting</a> of <font color="Maroon" title="c4">il</font>,<font color="Maroon" title="c5">ps</font>,<font color="Maroon" title="c6">ds</font>,<font color="Maroon" title="c1">I1</font>,<font color="Maroon" title="c3">D</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1"><span class="lab"><font color="Green" title="E2">A1</font></span></a>, <a class="txt" href="#E4:1"><span class="lab"><font color="Green" title="E5">A7</font></span></a>, <a class="ref" href="scm_1.html#D1" title="SCM_1:def.1">SCM_1:def 1</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="c7">s</font> is    <a href="scm_1.html#M1" title="SCM_1:mode.1">State-consisting</a> of <font color="Maroon" title="c4">il</font>,<font color="Maroon" title="c5">ps</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c1">I1</font></span>)</span>,<font color="Maroon" title="c6">ds</font>,<font color="Maroon" title="c2">I2</font>,<font color="Maroon" title="c3">D</font></span><br/>

<a NAME="E7:1"/>
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ami_2.html#K5" title="AMI_2:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">D</font> holds <br/><font color="Maroon" title="c7">s</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">ds</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Olive" title="b1">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">D</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="scm_1.html#D1" title="SCM_1:def.1">SCM_1:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:1"/>
<font color="Maroon" title="c7">s</font> is    <a href="scm_1.html#M1" title="SCM_1:mode.1">State-consisting</a> of <font color="Maroon" title="c4">il</font>,<font color="Maroon" title="c5">ps</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <font color="Maroon" title="c1">I1</font></span>)</span>,<font color="Maroon" title="c6">ds</font>,<font color="Maroon" title="c2">I2</font>,<font color="Maroon" title="c3">D</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1"><span class="lab"><font color="Green" title="E2">A1</font></span></a>, <a class="txt" href="#E3:1"><span class="lab"><font color="Green" title="E4">A3</font></span></a>, <a class="ref" href="scm_1.html#D1" title="SCM_1:def.1">SCM_1:def 1</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>
