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

<a NAME="E1:30"/><i><font color="Green" title="E27">A1</font></i>: 
for <font color="Olive" title="b1">t</font> being  <a href="dtconstr.html#NM2" title="DTCONSTR:NM.2">Terminal</a> of <a href="scm_comp.html#K1" title="SCM_COMP:func.1">SCM-AE</a>  holds  <font color="Maroon">S<sub>2</sub></font>[ <a href="scm_comp.html#K3" title="SCM_COMP:func.3">root-tree</a> <font color="Olive" title="b1">t</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">t</font> be   <a href="dtconstr.html#NM2" title="DTCONSTR:NM.2">Terminal</a> of <a href="scm_comp.html#K1" title="SCM_COMP:func.1">SCM-AE</a> ;<br/><b>let </b><font color="Maroon" title="c14">aux</font>, <font color="Maroon" title="c15">n</font>, <font color="Maroon" title="c16">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> ;<br/><b>let </b><font color="Maroon" title="c17">s</font> be    <a href="scm_1.html#M1" title="SCM_1:mode.1">State-consisting</a> of <font color="Maroon" title="c15">n</font>,<font color="Maroon" title="c15">n</font>,<font color="Maroon" title="c16">k</font>, <a href="scm_comp.html#K15" title="SCM_COMP:func.15">SCM-Compile</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K3" title="SCM_COMP:func.3">root-tree</a> <font color="Maroon" title="c13">t</font></span>)</span>,<font color="Maroon" title="c14">aux</font>, <a href="lang1.html#K2" title="LANG1:func.2">&lt;*&gt;</a> <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> ;<br/>





<b>assume </b><a NAME="E1:30_1"/>
<font color="Maroon" title="c14">aux</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="scm_comp.html#K17" title="SCM_COMP:func.17">max_Data-Loc_in</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K3" title="SCM_COMP:func.3">root-tree</a> <font color="Maroon" title="c13">t</font></span>)</span>
 ;<br/>

<b>take </b><font color="Maroon" title="c18">i</font> =  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ;<br/>

<b>take </b><font color="Maroon" title="c19">u</font> =  <a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c17">s</font>,1;<br/>

<b>thus </b><a NAME="E2:30_1"/>
<font color="Maroon" title="c19">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c17">s</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ;<br/>

<a NAME="E3:30_1"/><i><font color="Green" title="E28">A2</font></i>: 
 <a href="scm_comp.html#K15" title="SCM_COMP:func.15">SCM-Compile</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K3" title="SCM_COMP:func.3">root-tree</a> <font color="Maroon" title="c13">t</font></span>)</span>,<font color="Maroon" title="c14">aux</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c14">aux</font></span>)</span> <a href="ami_3.html#K3" title="AMI_3:func.3">:=</a> <span class="p3">(<span class="default"><a href="scm_comp.html#K4" title="SCM_COMP:func.4">@</a> <font color="Maroon" title="c13">t</font></span>)</span></span>)</span></span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="scm_comp.html#T9" target="_self" title="SCM_COMP:th.9">Th9</a></i>;<br/>
<b>hence </b><a NAME="E4:30_1"/>
<font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K15" title="SCM_COMP:func.15">SCM-Compile</a> <span class="p2">(<span class="default"><a href="scm_comp.html#K3" title="SCM_COMP:func.3">root-tree</a> <font color="Maroon" title="c13">t</font></span>)</span>,<font color="Maroon" title="c14">aux</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>;<br/>

<a NAME="E5:30_1"/><i><font color="Green" title="E29">A3</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c14">aux</font></span>)</span> <a href="ami_3.html#K3" title="AMI_3:func.3">:=</a> <span class="p3">(<span class="default"><a href="scm_comp.html#K4" title="SCM_COMP:func.4">@</a> <font color="Maroon" title="c13">t</font></span>)</span></span>)</span></span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 &amp; <font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">n</font> &amp; <span class="p1"><a href="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c14">aux</font></span>)</span> <a href="ami_3.html#K3" title="AMI_3:func.3">:=</a> <span class="p3">(<span class="default"><a href="scm_comp.html#K4" title="SCM_COMP:func.4">@</a> <font color="Maroon" title="c13">t</font></span>)</span></span>)</span></span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c14">aux</font></span>)</span> <a href="ami_3.html#K3" title="AMI_3:func.3">:=</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K4" title="SCM_COMP:func.4">@</a> <font color="Maroon" title="c13">t</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>;<br/>
<a NAME="E6:30_1"/><b>then </b><i><font color="Green" title="E30">A4</font></i>: 
( <font color="Maroon" title="c17">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c17">s</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp;  <a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <font color="Maroon" title="c17">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_3.html#K12" title="AMI_3:func.12">il.</a> <font color="Maroon" title="c15">n</font> &amp; <font color="Maroon" title="c17">s</font> <a href="ami_1.html#K13" title="AMI_1:func.13">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">il.</a> <font color="Maroon" title="c15">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c14">aux</font></span>)</span> <a href="ami_3.html#K3" title="AMI_3:func.3">:=</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K4" title="SCM_COMP:func.4">@</a> <font color="Maroon" title="c13">t</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="scm_comp.html#E3:30_1"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="ami_1.html#T13" title="AMI_1:th.13">AMI_1:13</a>, <a class="ref" href="scm_1.html#D1" title="SCM_1:def.1">SCM_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E7:30_1"/>
 <a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c17">s</font>,<font color="Maroon" title="c18">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_3.html#K12" title="AMI_3:func.12">il.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c18">i</font></span>)</span>
 ;<br/>

<b>thus </b><a NAME="E8:30_1"/>
 <a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <font color="Maroon" title="c19">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_3.html#K12" title="AMI_3:func.12">il.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="scm_comp.html#E6:30_1"><i><font color="Green" title="E30">A4</font></i></a>, <a class="ref" href="scm_1.html#T18" title="SCM_1:th.18">SCM_1:18</a></i>;<br/>

<b>thus </b><font color="Maroon" title="c19">u</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> <font color="Maroon" title="c14">aux</font></span>)</span> = 
<font color="Maroon" title="c17">s</font> <a href="scm_comp.html#K11" title="SCM_COMP:func.11">.</a> <font color="Maroon" title="c13">t</font>
<b>by </b><i><a class="txt" href="scm_comp.html#E5:30_1"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="scm_comp.html#E6:30_1"><i><font color="Green" title="E30">A4</font></i></a>, <a class="ref" href="scm_1.html#T18" title="SCM_1:th.18">SCM_1:18</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="scm_comp.html#K3" title="SCM_COMP:func.3">root-tree</a> <font color="Maroon" title="c13">t</font></span>)</span> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c17">s</font>
<b>by </b><i><a class="ref" href="scm_comp.html#T6" target="_self" title="SCM_COMP:th.6">Th6</a></i>
;<br/>

<b>let </b><font color="Maroon" title="c20">dn</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> ;<br/>

<b>assume </b><a NAME="E10:30_1"/>
<font color="Maroon" title="c20">dn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">aux</font>
 ;<br/>

<a NAME="E11:30_1"/><b>then </b>
 <a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c20">dn</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c14">aux</font>
 
<b>by </b><i><a class="ref" href="ami_3.html#T52" title="AMI_3:th.52">AMI_3:52</a></i>;<br/>
<b>hence </b><a NAME="E12:30_1"/>
<font color="Maroon" title="c17">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> <font color="Maroon" title="c20">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">u</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> <font color="Maroon" title="c20">dn</font></span>)</span>
 <b>by </b><i><a class="txt" href="scm_comp.html#E5:30_1"><i><font color="Green" title="E29">A3</font></i></a>, <a class="txt" href="scm_comp.html#E6:30_1"><i><font color="Green" title="E30">A4</font></i></a>, <a class="ref" href="scm_1.html#T18" title="SCM_1:th.18">SCM_1:18</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:30"/><i><font color="Green" title="E28">A5</font></i>: 
for <font color="Olive" title="b1">nt</font> being  <a href="dtconstr.html#NM3" title="DTCONSTR:NM.3">NonTerminal</a> of <a href="scm_comp.html#K1" title="SCM_COMP:func.1">SCM-AE</a> <br/> for <font color="Olive" title="b2">tl</font>, <font color="Olive" title="b3">tr</font> being  <a href="scm_comp.html#NM1" title="SCM_COMP:NM.1">bin-term</a>  st <font color="Olive" title="b1">nt</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a> <span class="p1"><a href="lang1.html#K4" title="LANG1:func.4">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="bintree1.html#K1" title="BINTREE1:func.1">root-label</a> <font color="Olive" title="b2">tl</font></span>)</span>,<span class="p2">(<span class="default"><a href="bintree1.html#K1" title="BINTREE1:func.1">root-label</a> <font color="Olive" title="b3">tr</font></span>)</span></span><a href="lang1.html#K4" title="LANG1:func.4">*&gt;</a></span> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b2">tl</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b3">tr</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Olive" title="b2">tl</font>,<font color="Olive" title="b3">tr</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">nt</font> be   <a href="dtconstr.html#NM3" title="DTCONSTR:NM.3">NonTerminal</a> of <a href="scm_comp.html#K1" title="SCM_COMP:func.1">SCM-AE</a> ;<br/><b>let </b><font color="Maroon" title="c14">tl</font>, <font color="Maroon" title="c15">tr</font> be   <a href="scm_comp.html#NM1" title="SCM_COMP:NM.1">bin-term</a>;<br/>



<b>assume </b><a NAME="E1:30_2"/><i><font color="Green" title="E29">A6</font></i>: 
( <font color="Maroon" title="c13">nt</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a> <span class="p1"><a href="lang1.html#K4" title="LANG1:func.4">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="bintree1.html#K1" title="BINTREE1:func.1">root-label</a> <font color="Maroon" title="c14">tl</font></span>)</span>,<span class="p2">(<span class="default"><a href="bintree1.html#K1" title="BINTREE1:func.1">root-label</a> <font color="Maroon" title="c15">tr</font></span>)</span></span><a href="lang1.html#K4" title="LANG1:func.4">*&gt;</a></span> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c14">tl</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c15">tr</font>] )
 ;<br/>

<b>let </b><font color="Maroon" title="c16">aux</font>, <font color="Maroon" title="c17">n</font>, <font color="Maroon" title="c18">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> ;<br/><b>let </b><font color="Maroon" title="c19">s</font> be    <a href="scm_1.html#M1" title="SCM_1:mode.1">State-consisting</a> of <font color="Maroon" title="c17">n</font>,<font color="Maroon" title="c17">n</font>,<font color="Maroon" title="c18">k</font>, <a href="scm_comp.html#K15" title="SCM_COMP:func.15">SCM-Compile</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c14">tl</font>,<font color="Maroon" title="c15">tr</font></span>)</span>,<font color="Maroon" title="c16">aux</font>, <a href="lang1.html#K2" title="LANG1:func.2">&lt;*&gt;</a> <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> ;<br/>



<b>assume </b><a NAME="E2:30_2"/><i><font color="Green" title="E30">A7</font></i>: 
<font color="Maroon" title="c16">aux</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="scm_comp.html#K17" title="SCM_COMP:func.17">max_Data-Loc_in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c14">tl</font>,<font color="Maroon" title="c15">tr</font></span>)</span>
 ;<br/>

<a NAME="E3:30_2"/><i><font color="Green" title="E31">A8</font></i>: 
 <a href="scm_comp.html#K15" title="SCM_COMP:func.15">SCM-Compile</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c14">tl</font>,<font color="Maroon" title="c15">tr</font></span>)</span>,<font color="Maroon" title="c16">aux</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scm_comp.html#K15" title="SCM_COMP:func.15">SCM-Compile</a> <font color="Maroon" title="c14">tl</font>,<font color="Maroon" title="c16">aux</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="scm_comp.html#K15" title="SCM_COMP:func.15">SCM-Compile</a> <font color="Maroon" title="c15">tr</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c16">aux</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K14" title="SCM_COMP:func.14">Selfwork</a> <font color="Maroon" title="c13">nt</font>,<font color="Maroon" title="c16">aux</font></span>)</span>
 
<b>by </b><i><a class="txt" href="scm_comp.html#E1:30_2"><i><font color="Green" title="E29">A6</font></i></a>, <a class="ref" href="scm_comp.html#T10" target="_self" title="SCM_COMP:th.10">Th10</a></i>;<br/>
<a NAME="E4:30_2"/><b>then </b><i><font color="Green" title="E32">A9</font></i>: 
 <a href="scm_comp.html#K15" title="SCM_COMP:func.15">SCM-Compile</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c14">tl</font>,<font color="Maroon" title="c15">tr</font></span>)</span>,<font color="Maroon" title="c16">aux</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K15" title="SCM_COMP:func.15">SCM-Compile</a> <font color="Maroon" title="c14">tl</font>,<font color="Maroon" title="c16">aux</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scm_comp.html#K15" title="SCM_COMP:func.15">SCM-Compile</a> <font color="Maroon" title="c15">tr</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c16">aux</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="scm_comp.html#K14" title="SCM_COMP:func.14">Selfwork</a> <font color="Maroon" title="c13">nt</font>,<font color="Maroon" title="c16">aux</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T45" title="FINSEQ_1:th.45">FINSEQ_1:45</a></i>;<br/>
<a NAME="E5:30_2"/>
 <a href="scm_comp.html#K17" title="SCM_COMP:func.17">max_Data-Loc_in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c14">tl</font>,<font color="Maroon" title="c15">tr</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="limfunc1.html#K1" title="LIMFUNC1:func.1">max</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K17" title="SCM_COMP:func.17">max_Data-Loc_in</a> <font color="Maroon" title="c14">tl</font></span>)</span>,<span class="p1">(<span class="default"><a href="scm_comp.html#K17" title="SCM_COMP:func.17">max_Data-Loc_in</a> <font color="Maroon" title="c15">tr</font></span>)</span>
 
<b>by </b><i><a class="ref" href="scm_comp.html#T12" target="_self" title="SCM_COMP:th.12">Th12</a></i>;<br/>
<a NAME="E6:30_2"/><b>then </b>
(  <a href="scm_comp.html#K17" title="SCM_COMP:func.17">max_Data-Loc_in</a> <font color="Maroon" title="c14">tl</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="scm_comp.html#K17" title="SCM_COMP:func.17">max_Data-Loc_in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c14">tl</font>,<font color="Maroon" title="c15">tr</font></span>)</span> &amp;  <a href="scm_comp.html#K17" title="SCM_COMP:func.17">max_Data-Loc_in</a> <font color="Maroon" title="c15">tr</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="scm_comp.html#K17" title="SCM_COMP:func.17">max_Data-Loc_in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c14">tl</font>,<font color="Maroon" title="c15">tr</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T25" title="XXREAL_0:th.25">XXREAL_0:25</a></i>;<br/>
<a NAME="E7:30_2"/><b>then </b><i><font color="Green" title="E33">A10</font></i>: 
(  <a href="scm_comp.html#K17" title="SCM_COMP:func.17">max_Data-Loc_in</a> <font color="Maroon" title="c14">tl</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">aux</font> &amp;  <a href="scm_comp.html#K17" title="SCM_COMP:func.17">max_Data-Loc_in</a> <font color="Maroon" title="c15">tr</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">aux</font> )
 
<b>by </b><i><a class="txt" href="scm_comp.html#E2:30_2"><i><font color="Green" title="E30">A7</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E8:30_2"/><b>then </b><i><font color="Green" title="E34">A11</font></i>: 
 <a href="scm_comp.html#K17" title="SCM_COMP:func.17">max_Data-Loc_in</a> <font color="Maroon" title="c15">tr</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">aux</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E9:30_2"/>
<font color="Maroon" title="c19">s</font> is    <a href="scm_1.html#M1" title="SCM_1:mode.1">State-consisting</a> of <font color="Maroon" title="c17">n</font>,<font color="Maroon" title="c17">n</font>,<font color="Maroon" title="c18">k</font>, <a href="scm_comp.html#K15" title="SCM_COMP:func.15">SCM-Compile</a> <font color="Maroon" title="c14">tl</font>,<font color="Maroon" title="c16">aux</font>, <a href="lang1.html#K2" title="LANG1:func.2">&lt;*&gt;</a> <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> 
 
<b>by </b><i><a class="txt" href="scm_comp.html#E4:30_2"><i><font color="Green" title="E32">A9</font></i></a>, <a class="ref" href="scm_comp.html#T1" target="_self" title="SCM_COMP:th.1">Th1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c20">i1</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> , <font color="Maroon" title="c21">u1</font> being   <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> <b> such that </b><br/><a NAME="E11:30_2"/><i><font color="Green" title="E35">A12</font></i>: 
( <font color="Maroon" title="c21">u1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c19">s</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c20">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c20">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K15" title="SCM_COMP:func.15">SCM-Compile</a> <font color="Maroon" title="c14">tl</font>,<font color="Maroon" title="c16">aux</font></span>)</span> &amp;  <a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c19">s</font>,<font color="Maroon" title="c20">i1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_3.html#K12" title="AMI_3:func.12">il.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c20">i1</font></span>)</span> &amp;  <a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <font color="Maroon" title="c21">u1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_3.html#K12" title="AMI_3:func.12">il.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon" title="c21">u1</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> <font color="Maroon" title="c16">aux</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">tl</font> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c19">s</font> &amp; ( for <font color="Olive" title="b1">dn</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">dn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">aux</font> holds <br/><font color="Maroon" title="c19">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> <font color="Olive" title="b1">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">u1</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> <font color="Olive" title="b1">dn</font></span>)</span> ) )
 <b>by </b><i><a class="txt" href="scm_comp.html#E1:30_2"><i><font color="Green" title="E29">A6</font></i></a>, <a class="txt" href="scm_comp.html#E7:30_2"><i><font color="Green" title="E33">A10</font></i></a></i>;<br/>
<a NAME="E12:30_2"/>
<font color="Maroon" title="c21">u1</font> is    <a href="scm_1.html#M1" title="SCM_1:mode.1">State-consisting</a> of <font color="Maroon" title="c17">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c17">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c18">k</font>,<span class="p1">(<span class="default"><a href="scm_comp.html#K15" title="SCM_COMP:func.15">SCM-Compile</a> <font color="Maroon" title="c15">tr</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">aux</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K14" title="SCM_COMP:func.14">Selfwork</a> <font color="Maroon" title="c13">nt</font>,<font color="Maroon" title="c16">aux</font></span>)</span>, <a href="lang1.html#K2" title="LANG1:func.2">&lt;*&gt;</a> <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> 
 
<b>by </b><i><a class="txt" href="scm_comp.html#E4:30_2"><i><font color="Green" title="E32">A9</font></i></a>, <a class="txt" href="scm_comp.html#E11:30_2"><i><font color="Green" title="E35">A12</font></i></a>, <a class="ref" href="scm_comp.html#T2" target="_self" title="SCM_COMP:th.2">Th2</a></i>;<br/>
<a NAME="E13:30_2"/><b>then </b>
<font color="Maroon" title="c21">u1</font> is    <a href="scm_1.html#M1" title="SCM_1:mode.1">State-consisting</a> of <font color="Maroon" title="c17">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c17">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c18">k</font>, <a href="scm_comp.html#K15" title="SCM_COMP:func.15">SCM-Compile</a> <font color="Maroon" title="c15">tr</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c16">aux</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>, <a href="lang1.html#K2" title="LANG1:func.2">&lt;*&gt;</a> <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> 
 
<b>by </b><i><a class="ref" href="scm_comp.html#T1" target="_self" title="SCM_COMP:th.1">Th1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c22">i2</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> , <font color="Maroon" title="c23">u2</font> being   <a href="ami_1.html#NM5" title="AMI_1:NM.5">State</a> of <a href="ami_3.html#K1" title="AMI_3:func.1">SCM</a> <b> such that </b><br/><a NAME="E15:30_2"/><i><font color="Green" title="E36">A13</font></i>: 
( <font color="Maroon" title="c23">u2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c21">u1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <font color="Maroon" title="c22">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K15" title="SCM_COMP:func.15">SCM-Compile</a> <font color="Maroon" title="c15">tr</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c16">aux</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> &amp;  <a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c21">u1</font>,<font color="Maroon" title="c22">i2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_3.html#K12" title="AMI_3:func.12">il.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c20">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c22">i2</font></span>)</span> &amp;  <a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <font color="Maroon" title="c23">u2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_3.html#K12" title="AMI_3:func.12">il.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c20">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon" title="c23">u2</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="c16">aux</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">tr</font> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c21">u1</font> &amp; ( for <font color="Olive" title="b1">dn</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">dn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">aux</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 holds <br/><font color="Maroon" title="c21">u1</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> <font color="Olive" title="b1">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">u2</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> <font color="Olive" title="b1">dn</font></span>)</span> ) )
 <b>by </b><i><a class="txt" href="scm_comp.html#E1:30_2"><i><font color="Green" title="E29">A6</font></i></a>, <a class="txt" href="scm_comp.html#E8:30_2"><i><font color="Green" title="E34">A11</font></i></a></i>;<br/>
<a NAME="E16:30_2"/><i><font color="Green" title="E37">A14</font></i>: 
<font color="Maroon" title="c23">u2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c19">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="scm_comp.html#E11:30_2"><i><font color="Green" title="E35">A12</font></i></a>, <a class="txt" href="scm_comp.html#E15:30_2"><i><font color="Green" title="E36">A13</font></i></a>, <a class="ref" href="ami_1.html#T51" title="AMI_1:th.51">AMI_1:51</a></i>;<br/>
<a NAME="E17:30_2"/><i><font color="Green" title="E38">A15</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c14">tl</font>,<font color="Maroon" title="c15">tr</font></span>)</span> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c19">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">nt</font> <a href="scm_comp.html#K10" title="SCM_COMP:func.10">-Meaning_on</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">tl</font> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c19">s</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c15">tr</font> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c19">s</font></span>)</span>
 
<b>by </b><i><a class="ref" href="scm_comp.html#T7" target="_self" title="SCM_COMP:th.7">Th7</a></i>;<br/>
<div><i><font color="Green" title="E39">A16</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c24">n</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> ;<br/><b>assume </b><a NAME="E1:30_2_1"/>
<font color="Maroon" title="c24">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="scm_comp.html#K17" title="SCM_COMP:func.17">max_Data-Loc_in</a> <font color="Maroon" title="c15">tr</font>
 ;<br/><a NAME="E2:30_2_1"/><b>then </b>
<font color="Maroon" title="c24">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">aux</font>
 <b>by </b><i><a class="txt" href="scm_comp.html#E7:30_2"><i><font color="Green" title="E33">A10</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E3:30_2_1"/>
<font color="Maroon" title="c19">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> <font color="Maroon" title="c24">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">u1</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> <font color="Maroon" title="c24">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="scm_comp.html#E11:30_2"><i><font color="Green" title="E35">A12</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E19:30_2"/><i><font color="Green" title="E40">A17</font></i>: 
<font color="Maroon" title="c16">aux</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">aux</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E20:30_2"/><i><font color="Green" title="E41">A18</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scm_comp.html#K15" title="SCM_COMP:func.15">SCM-Compile</a> <font color="Maroon" title="c14">tl</font>,<font color="Maroon" title="c16">aux</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><a href="scm_comp.html#K15" title="SCM_COMP:func.15">SCM-Compile</a> <font color="Maroon" title="c15">tr</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c16">aux</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="scm_comp.html#E11:30_2"><i><font color="Green" title="E35">A12</font></i></a>, <a class="txt" href="scm_comp.html#E15:30_2"><i><font color="Green" title="E36">A13</font></i></a>, <a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_2_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:30_2_2"/>
( <font color="Maroon" title="c13">nt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> or <font color="Maroon" title="c13">nt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> or <font color="Maroon" title="c13">nt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> or <font color="Maroon" title="c13">nt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,3</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> or <font color="Maroon" title="c13">nt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,4</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 </span><b>by </b><i><a class="ref" href="scm_comp.html#T3" target="_self" title="SCM_COMP:th.3">Th3</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_2_2_1"><b>suppose </b></a><a NAME="E1:30_2_2_1"/>
<font color="Maroon" title="c13">nt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 ;<br/><div class="add"><a NAME="E2:30_2_2_1"/><b>then </b><i><font color="Green" title="E42">A19</font></i>: 
(  <a href="scm_comp.html#K14" title="SCM_COMP:func.14">Selfwork</a> <font color="Maroon" title="c13">nt</font>,<font color="Maroon" title="c16">aux</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="ami_3.html#K4" title="AMI_3:func.4">AddTo</a> <span class="p3">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c16">aux</font></span>)</span>,<span class="p3">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c16">aux</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span> &amp; <font color="Maroon" title="c13">nt</font> <a href="scm_comp.html#K10" title="SCM_COMP:func.10">-Meaning_on</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">tl</font> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c19">s</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c15">tr</font> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c19">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">tl</font> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c19">s</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">tr</font> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c19">s</font></span>)</span> )
 <b>by </b><i><a class="ref" href="scm_comp.html#D8" target="_self" title="SCM_COMP:def.8">Def8</a>, <a class="ref" href="scm_comp.html#D10" target="_self" title="SCM_COMP:def.10">Def10</a></i>;<br/><b>take </b><font color="Maroon" title="c24">i</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c20">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>;<br/><b>take </b><font color="Maroon" title="c25">u</font> =  <a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c19">s</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c24">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>;<br/><b>thus </b><a NAME="E3:30_2_2_1"/>
<font color="Maroon" title="c25">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c19">s</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c24">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ;<br/><a NAME="E4:30_2_2_1"/><i><font color="Green" title="E43">A20</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K14" title="SCM_COMP:func.14">Selfwork</a> <font color="Maroon" title="c13">nt</font>,<font color="Maroon" title="c16">aux</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="scm_comp.html#E2:30_2_2_1"><i><font color="Green" title="E42">A19</font></i></a>, <a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>;<br/><a NAME="E5:30_2_2_1"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K15" title="SCM_COMP:func.15">SCM-Compile</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c14">tl</font>,<font color="Maroon" title="c15">tr</font></span>)</span>,<font color="Maroon" title="c16">aux</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="scm_comp.html#E3:30_2"><i><font color="Green" title="E31">A8</font></i></a>, <a class="txt" href="scm_comp.html#E20:30_2"><i><font color="Green" title="E41">A18</font></i></a>, <a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>;<br/><a NAME="E6:30_2_2_1"/><b>then </b>
<font color="Maroon" title="c24">i</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> <span class="p1">(<span class="default"><a href="scm_comp.html#K15" title="SCM_COMP:func.15">SCM-Compile</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c14">tl</font>,<font color="Maroon" title="c15">tr</font></span>)</span>,<font color="Maroon" title="c16">aux</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>then </b><i><font color="Green" title="E44">A21</font></i>: <font color="Maroon" title="c19">s</font> <a href="ami_1.html#K13" title="AMI_1:func.13">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">il.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">i</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="scm_comp.html#K15" title="SCM_COMP:func.15">SCM-Compile</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c14">tl</font>,<font color="Maroon" title="c15">tr</font></span>)</span>,<font color="Maroon" title="c16">aux</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="scm_1.html#D1" title="SCM_1:def.1">SCM_1:def 1</a></i>
<br/>.= 
 <a href="ami_3.html#K4" title="AMI_3:func.4">AddTo</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c16">aux</font></span>)</span>,<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="c16">aux</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="scm_comp.html#E3:30_2"><i><font color="Green" title="E31">A8</font></i></a>, <a class="txt" href="scm_comp.html#E20:30_2"><i><font color="Green" title="E41">A18</font></i></a>, <a class="txt" href="scm_comp.html#E2:30_2_2_1"><i><font color="Green" title="E42">A19</font></i></a>, <a class="ref" href="finseq_1.html#T59" title="FINSEQ_1:th.59">FINSEQ_1:59</a></i>
;<br/><b>thus </b><a NAME="E8:30_2_2_1"/>
<font color="Maroon" title="c24">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K15" title="SCM_COMP:func.15">SCM-Compile</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c14">tl</font>,<font color="Maroon" title="c15">tr</font></span>)</span>,<font color="Maroon" title="c16">aux</font></span>)</span>
 <b>by </b><i><a class="txt" href="scm_comp.html#E3:30_2"><i><font color="Green" title="E31">A8</font></i></a>, <a class="txt" href="scm_comp.html#E20:30_2"><i><font color="Green" title="E41">A18</font></i></a>, <a class="txt" href="scm_comp.html#E4:30_2_2_1"><i><font color="Green" title="E43">A20</font></i></a>, <a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>;<br/><b>thus </b><a NAME="E9:30_2_2_1"/>
 <a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c19">s</font>,<font color="Maroon" title="c24">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_3.html#K12" title="AMI_3:func.12">il.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="scm_comp.html#E11:30_2"><i><font color="Green" title="E35">A12</font></i></a>, <a class="txt" href="scm_comp.html#E15:30_2"><i><font color="Green" title="E36">A13</font></i></a>, <a class="ref" href="ami_1.html#T51" title="AMI_1:th.51">AMI_1:51</a></i>;<br/><b>thus </b> <a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <font color="Maroon" title="c25">u</font> = 
 <a href="ami_3.html#K12" title="AMI_3:func.12">il.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="scm_comp.html#E15:30_2"><i><font color="Green" title="E36">A13</font></i></a>, <a class="txt" href="scm_comp.html#E16:30_2"><i><font color="Green" title="E37">A14</font></i></a>, <a class="txt" href="scm_comp.html#E7:30_2_2_1"><i><font color="Green" title="E44">A21</font></i></a>, <a class="ref" href="scm_1.html#T19" title="SCM_1:th.19">SCM_1:19</a></i>
<br/>.= 
 <a href="ami_3.html#K12" title="AMI_3:func.12">il.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c24">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>

;<br/><b>thus </b><font color="Maroon" title="c25">u</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> <font color="Maroon" title="c16">aux</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c23">u2</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p2">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c16">aux</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">u2</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p2">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">aux</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="scm_comp.html#E15:30_2"><i><font color="Green" title="E36">A13</font></i></a>, <a class="txt" href="scm_comp.html#E16:30_2"><i><font color="Green" title="E37">A14</font></i></a>, <a class="txt" href="scm_comp.html#E7:30_2_2_1"><i><font color="Green" title="E44">A21</font></i></a>, <a class="ref" href="scm_1.html#T19" title="SCM_1:th.19">SCM_1:19</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c21">u1</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p2">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c16">aux</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">tr</font> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c21">u1</font></span>)</span>
<b>by </b><i><a class="txt" href="scm_comp.html#E15:30_2"><i><font color="Green" title="E36">A13</font></i></a>, <a class="txt" href="scm_comp.html#E19:30_2"><i><font color="Green" title="E40">A17</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c14">tl</font>,<font color="Maroon" title="c15">tr</font></span>)</span> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c19">s</font>
<b>by </b><i><a class="txt" href="scm_comp.html#E11:30_2"><i><font color="Green" title="E35">A12</font></i></a>, <a class="txt" href="scm_comp.html#E17:30_2"><i><font color="Green" title="E38">A15</font></i></a>, <a class="txt" href="scm_comp.html#E18:30_2"><i><font color="Green" title="E39">A16</font></i></a>, <a class="txt" href="scm_comp.html#E2:30_2_2_1"><i><font color="Green" title="E42">A19</font></i></a>, <a class="ref" href="scm_comp.html#T13" target="_self" title="SCM_COMP:th.13">Th13</a></i>
;<br/><b>let </b><font color="Maroon" title="c26">dn</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> ;<br/><b>assume </b><a NAME="E12:30_2_2_1"/><i><font color="Green" title="E45">A22</font></i>: 
<font color="Maroon" title="c26">dn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">aux</font>
 ;<br/><a NAME="E13:30_2_2_1"/><b>then </b>
 <a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c26">dn</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c16">aux</font>
 <b>by </b><i><a class="ref" href="ami_3.html#T52" title="AMI_3:th.52">AMI_3:52</a></i>;<br/><a NAME="E14:30_2_2_1"/><b>then </b><i><font color="Green" title="E46">A23</font></i>: 
<font color="Maroon" title="c25">u</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> <font color="Maroon" title="c26">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">u2</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> <font color="Maroon" title="c26">dn</font></span>)</span>
 <b>by </b><i><a class="txt" href="scm_comp.html#E15:30_2"><i><font color="Green" title="E36">A13</font></i></a>, <a class="txt" href="scm_comp.html#E16:30_2"><i><font color="Green" title="E37">A14</font></i></a>, <a class="txt" href="scm_comp.html#E7:30_2_2_1"><i><font color="Green" title="E44">A21</font></i></a>, <a class="ref" href="scm_1.html#T19" title="SCM_1:th.19">SCM_1:19</a></i>;<br/><a NAME="E15:30_2_2_1"/>
<font color="Maroon" title="c26">dn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">aux</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="scm_comp.html#E12:30_2_2_1"><i><font color="Green" title="E45">A22</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E16:30_2_2_1"/><b>then </b>
<font color="Maroon" title="c21">u1</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> <font color="Maroon" title="c26">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">u2</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> <font color="Maroon" title="c26">dn</font></span>)</span>
 <b>by </b><i><a class="txt" href="scm_comp.html#E15:30_2"><i><font color="Green" title="E36">A13</font></i></a></i>;<br/><b>hence </b><a NAME="E17:30_2_2_1"/>
<font color="Maroon" title="c19">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> <font color="Maroon" title="c26">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">u</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> <font color="Maroon" title="c26">dn</font></span>)</span>
 <b>by </b><i><a class="txt" href="scm_comp.html#E11:30_2"><i><font color="Green" title="E35">A12</font></i></a>, <a class="txt" href="scm_comp.html#E12:30_2_2_1"><i><font color="Green" title="E45">A22</font></i></a>, <a class="txt" href="scm_comp.html#E14:30_2_2_1"><i><font color="Green" title="E46">A23</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_2_2_2"><b>suppose </b></a><a NAME="E1:30_2_2_2"/>
<font color="Maroon" title="c13">nt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 ;<br/><div class="add"><a NAME="E2:30_2_2_2"/><b>then </b><i><font color="Green" title="E42">A24</font></i>: 
(  <a href="scm_comp.html#K14" title="SCM_COMP:func.14">Selfwork</a> <font color="Maroon" title="c13">nt</font>,<font color="Maroon" title="c16">aux</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="ami_3.html#K5" title="AMI_3:func.5">SubFrom</a> <span class="p3">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c16">aux</font></span>)</span>,<span class="p3">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c16">aux</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span> &amp; <font color="Maroon" title="c13">nt</font> <a href="scm_comp.html#K10" title="SCM_COMP:func.10">-Meaning_on</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">tl</font> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c19">s</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c15">tr</font> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c19">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">tl</font> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c19">s</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">tr</font> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c19">s</font></span>)</span> )
 <b>by </b><i><a class="ref" href="scm_comp.html#D8" target="_self" title="SCM_COMP:def.8">Def8</a>, <a class="ref" href="scm_comp.html#D10" target="_self" title="SCM_COMP:def.10">Def10</a></i>;<br/><b>take </b><font color="Maroon" title="c24">i</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c20">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>;<br/><b>take </b><font color="Maroon" title="c25">u</font> =  <a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c19">s</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c24">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>;<br/><b>thus </b><a NAME="E3:30_2_2_2"/>
<font color="Maroon" title="c25">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c19">s</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c24">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ;<br/><a NAME="E4:30_2_2_2"/><i><font color="Green" title="E43">A25</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K14" title="SCM_COMP:func.14">Selfwork</a> <font color="Maroon" title="c13">nt</font>,<font color="Maroon" title="c16">aux</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="scm_comp.html#E2:30_2_2_2"><i><font color="Green" title="E42">A24</font></i></a>, <a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>;<br/><a NAME="E5:30_2_2_2"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K15" title="SCM_COMP:func.15">SCM-Compile</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c14">tl</font>,<font color="Maroon" title="c15">tr</font></span>)</span>,<font color="Maroon" title="c16">aux</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="scm_comp.html#E3:30_2"><i><font color="Green" title="E31">A8</font></i></a>, <a class="txt" href="scm_comp.html#E20:30_2"><i><font color="Green" title="E41">A18</font></i></a>, <a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>;<br/><a NAME="E6:30_2_2_2"/><b>then </b>
<font color="Maroon" title="c24">i</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> <span class="p1">(<span class="default"><a href="scm_comp.html#K15" title="SCM_COMP:func.15">SCM-Compile</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c14">tl</font>,<font color="Maroon" title="c15">tr</font></span>)</span>,<font color="Maroon" title="c16">aux</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>then </b><i><font color="Green" title="E44">A26</font></i>: <font color="Maroon" title="c19">s</font> <a href="ami_1.html#K13" title="AMI_1:func.13">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">il.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">i</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="scm_comp.html#K15" title="SCM_COMP:func.15">SCM-Compile</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c14">tl</font>,<font color="Maroon" title="c15">tr</font></span>)</span>,<font color="Maroon" title="c16">aux</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="scm_1.html#D1" title="SCM_1:def.1">SCM_1:def 1</a></i>
<br/>.= 
 <a href="ami_3.html#K5" title="AMI_3:func.5">SubFrom</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c16">aux</font></span>)</span>,<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="c16">aux</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="scm_comp.html#E3:30_2"><i><font color="Green" title="E31">A8</font></i></a>, <a class="txt" href="scm_comp.html#E20:30_2"><i><font color="Green" title="E41">A18</font></i></a>, <a class="txt" href="scm_comp.html#E2:30_2_2_2"><i><font color="Green" title="E42">A24</font></i></a>, <a class="ref" href="finseq_1.html#T59" title="FINSEQ_1:th.59">FINSEQ_1:59</a></i>
;<br/><b>thus </b><a NAME="E8:30_2_2_2"/>
<font color="Maroon" title="c24">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K15" title="SCM_COMP:func.15">SCM-Compile</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c14">tl</font>,<font color="Maroon" title="c15">tr</font></span>)</span>,<font color="Maroon" title="c16">aux</font></span>)</span>
 <b>by </b><i><a class="txt" href="scm_comp.html#E3:30_2"><i><font color="Green" title="E31">A8</font></i></a>, <a class="txt" href="scm_comp.html#E20:30_2"><i><font color="Green" title="E41">A18</font></i></a>, <a class="txt" href="scm_comp.html#E4:30_2_2_2"><i><font color="Green" title="E43">A25</font></i></a>, <a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>;<br/><b>thus </b><a NAME="E9:30_2_2_2"/>
 <a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c19">s</font>,<font color="Maroon" title="c24">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_3.html#K12" title="AMI_3:func.12">il.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="scm_comp.html#E11:30_2"><i><font color="Green" title="E35">A12</font></i></a>, <a class="txt" href="scm_comp.html#E15:30_2"><i><font color="Green" title="E36">A13</font></i></a>, <a class="ref" href="ami_1.html#T51" title="AMI_1:th.51">AMI_1:51</a></i>;<br/><b>thus </b> <a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <font color="Maroon" title="c25">u</font> = 
 <a href="ami_3.html#K12" title="AMI_3:func.12">il.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="scm_comp.html#E15:30_2"><i><font color="Green" title="E36">A13</font></i></a>, <a class="txt" href="scm_comp.html#E16:30_2"><i><font color="Green" title="E37">A14</font></i></a>, <a class="txt" href="scm_comp.html#E7:30_2_2_2"><i><font color="Green" title="E44">A26</font></i></a>, <a class="ref" href="scm_1.html#T20" title="SCM_1:th.20">SCM_1:20</a></i>
<br/>.= 
 <a href="ami_3.html#K12" title="AMI_3:func.12">il.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c24">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>

;<br/><b>thus </b><font color="Maroon" title="c25">u</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> <font color="Maroon" title="c16">aux</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c23">u2</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p2">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c16">aux</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">u2</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p2">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">aux</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="scm_comp.html#E15:30_2"><i><font color="Green" title="E36">A13</font></i></a>, <a class="txt" href="scm_comp.html#E16:30_2"><i><font color="Green" title="E37">A14</font></i></a>, <a class="txt" href="scm_comp.html#E7:30_2_2_2"><i><font color="Green" title="E44">A26</font></i></a>, <a class="ref" href="scm_1.html#T20" title="SCM_1:th.20">SCM_1:20</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c21">u1</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p2">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c16">aux</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">tr</font> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c21">u1</font></span>)</span>
<b>by </b><i><a class="txt" href="scm_comp.html#E15:30_2"><i><font color="Green" title="E36">A13</font></i></a>, <a class="txt" href="scm_comp.html#E19:30_2"><i><font color="Green" title="E40">A17</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c14">tl</font>,<font color="Maroon" title="c15">tr</font></span>)</span> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c19">s</font>
<b>by </b><i><a class="txt" href="scm_comp.html#E11:30_2"><i><font color="Green" title="E35">A12</font></i></a>, <a class="txt" href="scm_comp.html#E17:30_2"><i><font color="Green" title="E38">A15</font></i></a>, <a class="txt" href="scm_comp.html#E18:30_2"><i><font color="Green" title="E39">A16</font></i></a>, <a class="txt" href="scm_comp.html#E2:30_2_2_2"><i><font color="Green" title="E42">A24</font></i></a>, <a class="ref" href="scm_comp.html#T13" target="_self" title="SCM_COMP:th.13">Th13</a></i>
;<br/><b>let </b><font color="Maroon" title="c26">dn</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> ;<br/><b>assume </b><a NAME="E12:30_2_2_2"/><i><font color="Green" title="E45">A27</font></i>: 
<font color="Maroon" title="c26">dn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">aux</font>
 ;<br/><a NAME="E13:30_2_2_2"/><b>then </b>
 <a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c26">dn</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c16">aux</font>
 <b>by </b><i><a class="ref" href="ami_3.html#T52" title="AMI_3:th.52">AMI_3:52</a></i>;<br/><a NAME="E14:30_2_2_2"/><b>then </b><i><font color="Green" title="E46">A28</font></i>: 
<font color="Maroon" title="c25">u</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> <font color="Maroon" title="c26">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">u2</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> <font color="Maroon" title="c26">dn</font></span>)</span>
 <b>by </b><i><a class="txt" href="scm_comp.html#E15:30_2"><i><font color="Green" title="E36">A13</font></i></a>, <a class="txt" href="scm_comp.html#E16:30_2"><i><font color="Green" title="E37">A14</font></i></a>, <a class="txt" href="scm_comp.html#E7:30_2_2_2"><i><font color="Green" title="E44">A26</font></i></a>, <a class="ref" href="scm_1.html#T20" title="SCM_1:th.20">SCM_1:20</a></i>;<br/><a NAME="E15:30_2_2_2"/>
<font color="Maroon" title="c26">dn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">aux</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="scm_comp.html#E12:30_2_2_2"><i><font color="Green" title="E45">A27</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E16:30_2_2_2"/><b>then </b>
<font color="Maroon" title="c21">u1</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> <font color="Maroon" title="c26">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">u2</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> <font color="Maroon" title="c26">dn</font></span>)</span>
 <b>by </b><i><a class="txt" href="scm_comp.html#E15:30_2"><i><font color="Green" title="E36">A13</font></i></a></i>;<br/><b>hence </b><a NAME="E17:30_2_2_2"/>
<font color="Maroon" title="c19">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> <font color="Maroon" title="c26">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">u</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> <font color="Maroon" title="c26">dn</font></span>)</span>
 <b>by </b><i><a class="txt" href="scm_comp.html#E11:30_2"><i><font color="Green" title="E35">A12</font></i></a>, <a class="txt" href="scm_comp.html#E12:30_2_2_2"><i><font color="Green" title="E45">A27</font></i></a>, <a class="txt" href="scm_comp.html#E14:30_2_2_2"><i><font color="Green" title="E46">A28</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_2_2_3"><b>suppose </b></a><a NAME="E1:30_2_2_3"/>
<font color="Maroon" title="c13">nt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,2</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 ;<br/><div class="add"><a NAME="E2:30_2_2_3"/><b>then </b><i><font color="Green" title="E42">A29</font></i>: 
(  <a href="scm_comp.html#K14" title="SCM_COMP:func.14">Selfwork</a> <font color="Maroon" title="c13">nt</font>,<font color="Maroon" title="c16">aux</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="ami_3.html#K6" title="AMI_3:func.6">MultBy</a> <span class="p3">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c16">aux</font></span>)</span>,<span class="p3">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c16">aux</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span> &amp; <font color="Maroon" title="c13">nt</font> <a href="scm_comp.html#K10" title="SCM_COMP:func.10">-Meaning_on</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">tl</font> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c19">s</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c15">tr</font> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c19">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">tl</font> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c19">s</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">tr</font> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c19">s</font></span>)</span> )
 <b>by </b><i><a class="ref" href="scm_comp.html#D8" target="_self" title="SCM_COMP:def.8">Def8</a>, <a class="ref" href="scm_comp.html#D10" target="_self" title="SCM_COMP:def.10">Def10</a></i>;<br/><b>take </b><font color="Maroon" title="c24">i</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c20">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>;<br/><b>take </b><font color="Maroon" title="c25">u</font> =  <a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c19">s</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c24">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>;<br/><b>thus </b><a NAME="E3:30_2_2_3"/>
<font color="Maroon" title="c25">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c19">s</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c24">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ;<br/><a NAME="E4:30_2_2_3"/><i><font color="Green" title="E43">A30</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K14" title="SCM_COMP:func.14">Selfwork</a> <font color="Maroon" title="c13">nt</font>,<font color="Maroon" title="c16">aux</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="scm_comp.html#E2:30_2_2_3"><i><font color="Green" title="E42">A29</font></i></a>, <a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>;<br/><a NAME="E5:30_2_2_3"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K15" title="SCM_COMP:func.15">SCM-Compile</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c14">tl</font>,<font color="Maroon" title="c15">tr</font></span>)</span>,<font color="Maroon" title="c16">aux</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="scm_comp.html#E3:30_2"><i><font color="Green" title="E31">A8</font></i></a>, <a class="txt" href="scm_comp.html#E20:30_2"><i><font color="Green" title="E41">A18</font></i></a>, <a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>;<br/><a NAME="E6:30_2_2_3"/><b>then </b>
<font color="Maroon" title="c24">i</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> <span class="p1">(<span class="default"><a href="scm_comp.html#K15" title="SCM_COMP:func.15">SCM-Compile</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c14">tl</font>,<font color="Maroon" title="c15">tr</font></span>)</span>,<font color="Maroon" title="c16">aux</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>then </b><i><font color="Green" title="E44">A31</font></i>: <font color="Maroon" title="c19">s</font> <a href="ami_1.html#K13" title="AMI_1:func.13">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">il.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">i</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="scm_comp.html#K15" title="SCM_COMP:func.15">SCM-Compile</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c14">tl</font>,<font color="Maroon" title="c15">tr</font></span>)</span>,<font color="Maroon" title="c16">aux</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="scm_1.html#D1" title="SCM_1:def.1">SCM_1:def 1</a></i>
<br/>.= 
 <a href="ami_3.html#K6" title="AMI_3:func.6">MultBy</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c16">aux</font></span>)</span>,<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="c16">aux</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="scm_comp.html#E3:30_2"><i><font color="Green" title="E31">A8</font></i></a>, <a class="txt" href="scm_comp.html#E20:30_2"><i><font color="Green" title="E41">A18</font></i></a>, <a class="txt" href="scm_comp.html#E2:30_2_2_3"><i><font color="Green" title="E42">A29</font></i></a>, <a class="ref" href="finseq_1.html#T59" title="FINSEQ_1:th.59">FINSEQ_1:59</a></i>
;<br/><b>thus </b><a NAME="E8:30_2_2_3"/>
<font color="Maroon" title="c24">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K15" title="SCM_COMP:func.15">SCM-Compile</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c14">tl</font>,<font color="Maroon" title="c15">tr</font></span>)</span>,<font color="Maroon" title="c16">aux</font></span>)</span>
 <b>by </b><i><a class="txt" href="scm_comp.html#E3:30_2"><i><font color="Green" title="E31">A8</font></i></a>, <a class="txt" href="scm_comp.html#E20:30_2"><i><font color="Green" title="E41">A18</font></i></a>, <a class="txt" href="scm_comp.html#E4:30_2_2_3"><i><font color="Green" title="E43">A30</font></i></a>, <a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>;<br/><b>thus </b><a NAME="E9:30_2_2_3"/>
 <a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c19">s</font>,<font color="Maroon" title="c24">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_3.html#K12" title="AMI_3:func.12">il.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="scm_comp.html#E11:30_2"><i><font color="Green" title="E35">A12</font></i></a>, <a class="txt" href="scm_comp.html#E15:30_2"><i><font color="Green" title="E36">A13</font></i></a>, <a class="ref" href="ami_1.html#T51" title="AMI_1:th.51">AMI_1:51</a></i>;<br/><b>thus </b> <a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <font color="Maroon" title="c25">u</font> = 
 <a href="ami_3.html#K12" title="AMI_3:func.12">il.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="scm_comp.html#E15:30_2"><i><font color="Green" title="E36">A13</font></i></a>, <a class="txt" href="scm_comp.html#E16:30_2"><i><font color="Green" title="E37">A14</font></i></a>, <a class="txt" href="scm_comp.html#E7:30_2_2_3"><i><font color="Green" title="E44">A31</font></i></a>, <a class="ref" href="scm_1.html#T21" title="SCM_1:th.21">SCM_1:21</a></i>
<br/>.= 
 <a href="ami_3.html#K12" title="AMI_3:func.12">il.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c24">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>

;<br/><b>thus </b><font color="Maroon" title="c25">u</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> <font color="Maroon" title="c16">aux</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c23">u2</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p2">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c16">aux</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">u2</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p2">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">aux</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="scm_comp.html#E15:30_2"><i><font color="Green" title="E36">A13</font></i></a>, <a class="txt" href="scm_comp.html#E16:30_2"><i><font color="Green" title="E37">A14</font></i></a>, <a class="txt" href="scm_comp.html#E7:30_2_2_3"><i><font color="Green" title="E44">A31</font></i></a>, <a class="ref" href="scm_1.html#T21" title="SCM_1:th.21">SCM_1:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c21">u1</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p2">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c16">aux</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">tr</font> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c21">u1</font></span>)</span>
<b>by </b><i><a class="txt" href="scm_comp.html#E15:30_2"><i><font color="Green" title="E36">A13</font></i></a>, <a class="txt" href="scm_comp.html#E19:30_2"><i><font color="Green" title="E40">A17</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c14">tl</font>,<font color="Maroon" title="c15">tr</font></span>)</span> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c19">s</font>
<b>by </b><i><a class="txt" href="scm_comp.html#E11:30_2"><i><font color="Green" title="E35">A12</font></i></a>, <a class="txt" href="scm_comp.html#E17:30_2"><i><font color="Green" title="E38">A15</font></i></a>, <a class="txt" href="scm_comp.html#E18:30_2"><i><font color="Green" title="E39">A16</font></i></a>, <a class="txt" href="scm_comp.html#E2:30_2_2_3"><i><font color="Green" title="E42">A29</font></i></a>, <a class="ref" href="scm_comp.html#T13" target="_self" title="SCM_COMP:th.13">Th13</a></i>
;<br/><b>let </b><font color="Maroon" title="c26">dn</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> ;<br/><b>assume </b><a NAME="E12:30_2_2_3"/><i><font color="Green" title="E45">A32</font></i>: 
<font color="Maroon" title="c26">dn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">aux</font>
 ;<br/><a NAME="E13:30_2_2_3"/><b>then </b>
 <a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c26">dn</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c16">aux</font>
 <b>by </b><i><a class="ref" href="ami_3.html#T52" title="AMI_3:th.52">AMI_3:52</a></i>;<br/><a NAME="E14:30_2_2_3"/><b>then </b><i><font color="Green" title="E46">A33</font></i>: 
<font color="Maroon" title="c25">u</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> <font color="Maroon" title="c26">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">u2</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> <font color="Maroon" title="c26">dn</font></span>)</span>
 <b>by </b><i><a class="txt" href="scm_comp.html#E15:30_2"><i><font color="Green" title="E36">A13</font></i></a>, <a class="txt" href="scm_comp.html#E16:30_2"><i><font color="Green" title="E37">A14</font></i></a>, <a class="txt" href="scm_comp.html#E7:30_2_2_3"><i><font color="Green" title="E44">A31</font></i></a>, <a class="ref" href="scm_1.html#T21" title="SCM_1:th.21">SCM_1:21</a></i>;<br/><a NAME="E15:30_2_2_3"/>
<font color="Maroon" title="c26">dn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">aux</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="scm_comp.html#E12:30_2_2_3"><i><font color="Green" title="E45">A32</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E16:30_2_2_3"/><b>then </b>
<font color="Maroon" title="c21">u1</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> <font color="Maroon" title="c26">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">u2</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> <font color="Maroon" title="c26">dn</font></span>)</span>
 <b>by </b><i><a class="txt" href="scm_comp.html#E15:30_2"><i><font color="Green" title="E36">A13</font></i></a></i>;<br/><b>hence </b><a NAME="E17:30_2_2_3"/>
<font color="Maroon" title="c19">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> <font color="Maroon" title="c26">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">u</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> <font color="Maroon" title="c26">dn</font></span>)</span>
 <b>by </b><i><a class="txt" href="scm_comp.html#E11:30_2"><i><font color="Green" title="E35">A12</font></i></a>, <a class="txt" href="scm_comp.html#E12:30_2_2_3"><i><font color="Green" title="E45">A32</font></i></a>, <a class="txt" href="scm_comp.html#E14:30_2_2_3"><i><font color="Green" title="E46">A33</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_2_2_4"><b>suppose </b></a><a NAME="E1:30_2_2_4"/>
<font color="Maroon" title="c13">nt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,3</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 ;<br/><div class="add"><a NAME="E2:30_2_2_4"/><b>then </b><i><font color="Green" title="E42">A34</font></i>: 
(  <a href="scm_comp.html#K14" title="SCM_COMP:func.14">Selfwork</a> <font color="Maroon" title="c13">nt</font>,<font color="Maroon" title="c16">aux</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="ami_3.html#K7" title="AMI_3:func.7">Divide</a> <span class="p3">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c16">aux</font></span>)</span>,<span class="p3">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c16">aux</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span> &amp; <font color="Maroon" title="c13">nt</font> <a href="scm_comp.html#K10" title="SCM_COMP:func.10">-Meaning_on</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">tl</font> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c19">s</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c15">tr</font> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c19">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">tl</font> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c19">s</font></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">tr</font> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c19">s</font></span>)</span> )
 <b>by </b><i><a class="ref" href="scm_comp.html#D8" target="_self" title="SCM_COMP:def.8">Def8</a>, <a class="ref" href="scm_comp.html#D10" target="_self" title="SCM_COMP:def.10">Def10</a></i>;<br/><b>take </b><font color="Maroon" title="c24">i</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c20">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>;<br/><b>take </b><font color="Maroon" title="c25">u</font> =  <a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c19">s</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c24">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>;<br/><b>thus </b><a NAME="E3:30_2_2_4"/>
<font color="Maroon" title="c25">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c19">s</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c24">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ;<br/><a NAME="E4:30_2_2_4"/><i><font color="Green" title="E43">A35</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K14" title="SCM_COMP:func.14">Selfwork</a> <font color="Maroon" title="c13">nt</font>,<font color="Maroon" title="c16">aux</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="scm_comp.html#E2:30_2_2_4"><i><font color="Green" title="E42">A34</font></i></a>, <a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>;<br/><a NAME="E5:30_2_2_4"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K15" title="SCM_COMP:func.15">SCM-Compile</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c14">tl</font>,<font color="Maroon" title="c15">tr</font></span>)</span>,<font color="Maroon" title="c16">aux</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="scm_comp.html#E3:30_2"><i><font color="Green" title="E31">A8</font></i></a>, <a class="txt" href="scm_comp.html#E20:30_2"><i><font color="Green" title="E41">A18</font></i></a>, <a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>;<br/><a NAME="E6:30_2_2_4"/><b>then </b>
<font color="Maroon" title="c24">i</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> <span class="p1">(<span class="default"><a href="scm_comp.html#K15" title="SCM_COMP:func.15">SCM-Compile</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c14">tl</font>,<font color="Maroon" title="c15">tr</font></span>)</span>,<font color="Maroon" title="c16">aux</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>then </b><i><font color="Green" title="E44">A36</font></i>: <font color="Maroon" title="c19">s</font> <a href="ami_1.html#K13" title="AMI_1:func.13">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">il.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">i</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="scm_comp.html#K15" title="SCM_COMP:func.15">SCM-Compile</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c14">tl</font>,<font color="Maroon" title="c15">tr</font></span>)</span>,<font color="Maroon" title="c16">aux</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="scm_1.html#D1" title="SCM_1:def.1">SCM_1:def 1</a></i>
<br/>.= 
 <a href="ami_3.html#K7" title="AMI_3:func.7">Divide</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c16">aux</font></span>)</span>,<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="c16">aux</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="scm_comp.html#E3:30_2"><i><font color="Green" title="E31">A8</font></i></a>, <a class="txt" href="scm_comp.html#E20:30_2"><i><font color="Green" title="E41">A18</font></i></a>, <a class="txt" href="scm_comp.html#E2:30_2_2_4"><i><font color="Green" title="E42">A34</font></i></a>, <a class="ref" href="finseq_1.html#T59" title="FINSEQ_1:th.59">FINSEQ_1:59</a></i>
;<br/><a NAME="E8:30_2_2_4"/>
<font color="Maroon" title="c16">aux</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">aux</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/><a NAME="E9:30_2_2_4"/><b>then </b><i><font color="Green" title="E45">A37</font></i>: 
 <a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c16">aux</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">aux</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="ami_3.html#T52" title="AMI_3:th.52">AMI_3:52</a></i>;<br/><b>thus </b><a NAME="E10:30_2_2_4"/>
<font color="Maroon" title="c24">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K15" title="SCM_COMP:func.15">SCM-Compile</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c14">tl</font>,<font color="Maroon" title="c15">tr</font></span>)</span>,<font color="Maroon" title="c16">aux</font></span>)</span>
 <b>by </b><i><a class="txt" href="scm_comp.html#E3:30_2"><i><font color="Green" title="E31">A8</font></i></a>, <a class="txt" href="scm_comp.html#E20:30_2"><i><font color="Green" title="E41">A18</font></i></a>, <a class="txt" href="scm_comp.html#E4:30_2_2_4"><i><font color="Green" title="E43">A35</font></i></a>, <a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>;<br/><b>thus </b><a NAME="E11:30_2_2_4"/>
 <a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c19">s</font>,<font color="Maroon" title="c24">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_3.html#K12" title="AMI_3:func.12">il.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="scm_comp.html#E11:30_2"><i><font color="Green" title="E35">A12</font></i></a>, <a class="txt" href="scm_comp.html#E15:30_2"><i><font color="Green" title="E36">A13</font></i></a>, <a class="ref" href="ami_1.html#T51" title="AMI_1:th.51">AMI_1:51</a></i>;<br/><b>thus </b> <a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <font color="Maroon" title="c25">u</font> = 
 <a href="ami_3.html#K12" title="AMI_3:func.12">il.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c24">i</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="scm_comp.html#E15:30_2"><i><font color="Green" title="E36">A13</font></i></a>, <a class="txt" href="scm_comp.html#E16:30_2"><i><font color="Green" title="E37">A14</font></i></a>, <a class="txt" href="scm_comp.html#E7:30_2_2_4"><i><font color="Green" title="E44">A36</font></i></a>, <a class="txt" href="scm_comp.html#E9:30_2_2_4"><i><font color="Green" title="E45">A37</font></i></a>, <a class="ref" href="scm_1.html#T22" title="SCM_1:th.22">SCM_1:22</a></i>
<br/>.= 
 <a href="ami_3.html#K12" title="AMI_3:func.12">il.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c24">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>

;<br/><b>thus </b><font color="Maroon" title="c25">u</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> <font color="Maroon" title="c16">aux</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c23">u2</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p2">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c16">aux</font></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">u2</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p2">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">aux</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="scm_comp.html#E15:30_2"><i><font color="Green" title="E36">A13</font></i></a>, <a class="txt" href="scm_comp.html#E16:30_2"><i><font color="Green" title="E37">A14</font></i></a>, <a class="txt" href="scm_comp.html#E7:30_2_2_4"><i><font color="Green" title="E44">A36</font></i></a>, <a class="txt" href="scm_comp.html#E9:30_2_2_4"><i><font color="Green" title="E45">A37</font></i></a>, <a class="ref" href="scm_1.html#T22" title="SCM_1:th.22">SCM_1:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c21">u1</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p2">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c16">aux</font></span>)</span></span>)</span> <a href="int_1.html#K5" title="INT_1:func.5">div</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">tr</font> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c21">u1</font></span>)</span>
<b>by </b><i><a class="txt" href="scm_comp.html#E15:30_2"><i><font color="Green" title="E36">A13</font></i></a>, <a class="txt" href="scm_comp.html#E19:30_2"><i><font color="Green" title="E40">A17</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c14">tl</font>,<font color="Maroon" title="c15">tr</font></span>)</span> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c19">s</font>
<b>by </b><i><a class="txt" href="scm_comp.html#E11:30_2"><i><font color="Green" title="E35">A12</font></i></a>, <a class="txt" href="scm_comp.html#E17:30_2"><i><font color="Green" title="E38">A15</font></i></a>, <a class="txt" href="scm_comp.html#E18:30_2"><i><font color="Green" title="E39">A16</font></i></a>, <a class="txt" href="scm_comp.html#E2:30_2_2_4"><i><font color="Green" title="E42">A34</font></i></a>, <a class="ref" href="scm_comp.html#T13" target="_self" title="SCM_COMP:th.13">Th13</a></i>
;<br/><b>let </b><font color="Maroon" title="c26">dn</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> ;<br/><b>assume </b><a NAME="E14:30_2_2_4"/><i><font color="Green" title="E46">A38</font></i>: 
<font color="Maroon" title="c26">dn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">aux</font>
 ;<br/><a NAME="E15:30_2_2_4"/><b>then </b><i><font color="Green" title="E47">A39</font></i>: 
 <a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c26">dn</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c16">aux</font>
 <b>by </b><i><a class="ref" href="ami_3.html#T52" title="AMI_3:th.52">AMI_3:52</a></i>;<br/><a NAME="E16:30_2_2_4"/><i><font color="Green" title="E48">A40</font></i>: 
<font color="Maroon" title="c26">dn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">aux</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="scm_comp.html#E14:30_2_2_4"><i><font color="Green" title="E46">A38</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E17:30_2_2_4"/><b>then </b>
 <a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c26">dn</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">aux</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="ami_3.html#T52" title="AMI_3:th.52">AMI_3:52</a></i>;<br/><a NAME="E18:30_2_2_4"/><b>then </b><i><font color="Green" title="E49">A41</font></i>: 
<font color="Maroon" title="c25">u</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> <font color="Maroon" title="c26">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">u2</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> <font color="Maroon" title="c26">dn</font></span>)</span>
 <b>by </b><i><a class="txt" href="scm_comp.html#E15:30_2"><i><font color="Green" title="E36">A13</font></i></a>, <a class="txt" href="scm_comp.html#E16:30_2"><i><font color="Green" title="E37">A14</font></i></a>, <a class="txt" href="scm_comp.html#E7:30_2_2_4"><i><font color="Green" title="E44">A36</font></i></a>, <a class="txt" href="scm_comp.html#E9:30_2_2_4"><i><font color="Green" title="E45">A37</font></i></a>, <a class="txt" href="scm_comp.html#E15:30_2_2_4"><i><font color="Green" title="E47">A39</font></i></a>, <a class="ref" href="scm_1.html#T22" title="SCM_1:th.22">SCM_1:22</a></i>;<br/><a NAME="E19:30_2_2_4"/>
<font color="Maroon" title="c21">u1</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> <font color="Maroon" title="c26">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">u2</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> <font color="Maroon" title="c26">dn</font></span>)</span>
 <b>by </b><i><a class="txt" href="scm_comp.html#E15:30_2"><i><font color="Green" title="E36">A13</font></i></a>, <a class="txt" href="scm_comp.html#E16:30_2_2_4"><i><font color="Green" title="E48">A40</font></i></a></i>;<br/><b>hence </b><a NAME="E20:30_2_2_4"/>
<font color="Maroon" title="c19">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> <font color="Maroon" title="c26">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">u</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> <font color="Maroon" title="c26">dn</font></span>)</span>
 <b>by </b><i><a class="txt" href="scm_comp.html#E11:30_2"><i><font color="Green" title="E35">A12</font></i></a>, <a class="txt" href="scm_comp.html#E14:30_2_2_4"><i><font color="Green" title="E46">A38</font></i></a>, <a class="txt" href="scm_comp.html#E18:30_2_2_4"><i><font color="Green" title="E49">A41</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_2_2_5"><b>suppose </b></a><a NAME="E1:30_2_2_5"/>
<font color="Maroon" title="c13">nt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,4</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 ;<br/><div class="add"><a NAME="E2:30_2_2_5"/><b>then </b><i><font color="Green" title="E42">A42</font></i>: 
(  <a href="scm_comp.html#K14" title="SCM_COMP:func.14">Selfwork</a> <font color="Maroon" title="c13">nt</font>,<font color="Maroon" title="c16">aux</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="lang1.html#K4" title="LANG1:func.4">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="ami_3.html#K7" title="AMI_3:func.7">Divide</a> <span class="p3">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c16">aux</font></span>)</span>,<span class="p3">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c16">aux</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c16">aux</font></span>)</span> <a href="ami_3.html#K3" title="AMI_3:func.3">:=</a> <span class="p3">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c16">aux</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="lang1.html#K4" title="LANG1:func.4">*&gt;</a></span> &amp; <font color="Maroon" title="c13">nt</font> <a href="scm_comp.html#K10" title="SCM_COMP:func.10">-Meaning_on</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">tl</font> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c19">s</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c15">tr</font> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c19">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">tl</font> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c19">s</font></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">tr</font> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c19">s</font></span>)</span> )
 <b>by </b><i><a class="ref" href="scm_comp.html#D8" target="_self" title="SCM_COMP:def.8">Def8</a>, <a class="ref" href="scm_comp.html#D10" target="_self" title="SCM_COMP:def.10">Def10</a></i>;<br/><b>set </b><font color="Maroon" title="c24">i</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c20">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>;<br/><b>set </b><font color="Maroon" title="c25">u</font> =  <a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c19">s</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c20">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c22">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>;<br/><b>take </b><font color="Maroon" title="c26">k</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1;<br/><b>take </b><font color="Maroon" title="c27">v</font> =  <a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c19">s</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c26">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>;<br/><b>thus </b><a NAME="E3:30_2_2_5"/>
<font color="Maroon" title="c27">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c19">s</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c26">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ;<br/><a NAME="E4:30_2_2_5"/><i><font color="Green" title="E43">A43</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K14" title="SCM_COMP:func.14">Selfwork</a> <font color="Maroon" title="c13">nt</font>,<font color="Maroon" title="c16">aux</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 <b>by </b><i><a class="txt" href="scm_comp.html#E2:30_2_2_5"><i><font color="Green" title="E42">A42</font></i></a>, <a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>;<br/><a NAME="E5:30_2_2_5"/><b>then </b><i><font color="Green" title="E44">A44</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K15" title="SCM_COMP:func.15">SCM-Compile</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c14">tl</font>,<font color="Maroon" title="c15">tr</font></span>)</span>,<font color="Maroon" title="c16">aux</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="scm_comp.html#E3:30_2"><i><font color="Green" title="E31">A8</font></i></a>, <a class="txt" href="scm_comp.html#E20:30_2"><i><font color="Green" title="E41">A18</font></i></a>, <a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>;<br/><b>hence </b><a NAME="E6:30_2_2_5"/>
<font color="Maroon" title="c26">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K15" title="SCM_COMP:func.15">SCM-Compile</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c14">tl</font>,<font color="Maroon" title="c15">tr</font></span>)</span>,<font color="Maroon" title="c16">aux</font></span>)</span>
 ;<br/><a NAME="E7:30_2_2_5"/><i><font color="Green" title="E45">A45</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K14" title="SCM_COMP:func.14">Selfwork</a> <font color="Maroon" title="c13">nt</font>,<font color="Maroon" title="c16">aux</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 2
 <b>by </b><i><a class="txt" href="scm_comp.html#E4:30_2_2_5"><i><font color="Green" title="E43">A43</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E8:30_2_2_5"/><i><font color="Green" title="E46">A46</font></i>: 
( 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 2 &amp; 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 2 )
 <b>by </b><i><a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><a NAME="E9:30_2_2_5"/>
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c20">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c22">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 ;<br/><a NAME="E10:30_2_2_5"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c20">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <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> <span class="p1">(<span class="default"><a href="scm_comp.html#K15" title="SCM_COMP:func.15">SCM-Compile</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c14">tl</font>,<font color="Maroon" title="c15">tr</font></span>)</span>,<font color="Maroon" title="c16">aux</font></span>)</span>
 <b>by </b><i><a class="txt" href="scm_comp.html#E5:30_2_2_5"><i><font color="Green" title="E44">A44</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><b>then </b><i><font color="Green" title="E47">A47</font></i>: <font color="Maroon" title="c19">s</font> <a href="ami_1.html#K13" title="AMI_1:func.13">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">il.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c20">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c22">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="scm_comp.html#K15" title="SCM_COMP:func.15">SCM-Compile</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c14">tl</font>,<font color="Maroon" title="c15">tr</font></span>)</span>,<font color="Maroon" title="c16">aux</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"><span class="p3">(<span class="default"><font color="Maroon" title="c20">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c22">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="scm_1.html#D1" title="SCM_1:def.1">SCM_1:def 1</a></i>
<br/>.= 
<span class="p1"><a href="lang1.html#K4" title="LANG1:func.4">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="ami_3.html#K7" title="AMI_3:func.7">Divide</a> <span class="p3">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c16">aux</font></span>)</span>,<span class="p3">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c16">aux</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c16">aux</font></span>)</span> <a href="ami_3.html#K3" title="AMI_3:func.3">:=</a> <span class="p3">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c16">aux</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="lang1.html#K4" title="LANG1:func.4">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<b>by </b><i><a class="txt" href="scm_comp.html#E3:30_2"><i><font color="Green" title="E31">A8</font></i></a>, <a class="txt" href="scm_comp.html#E20:30_2"><i><font color="Green" title="E41">A18</font></i></a>, <a class="txt" href="scm_comp.html#E2:30_2_2_5"><i><font color="Green" title="E42">A42</font></i></a>, <a class="txt" href="scm_comp.html#E7:30_2_2_5"><i><font color="Green" title="E45">A45</font></i></a>, <a class="txt" href="scm_comp.html#E8:30_2_2_5"><i><font color="Green" title="E46">A46</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>
<br/>.= 
 <a href="ami_3.html#K7" title="AMI_3:func.7">Divide</a> <span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c16">aux</font></span>)</span>,<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="c16">aux</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>
;<br/><a NAME="E12:30_2_2_5"/>
<font color="Maroon" title="c26">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> <span class="p1">(<span class="default"><a href="scm_comp.html#K15" title="SCM_COMP:func.15">SCM-Compile</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c14">tl</font>,<font color="Maroon" title="c15">tr</font></span>)</span>,<font color="Maroon" title="c16">aux</font></span>)</span>
 <b>by </b><i><a class="txt" href="scm_comp.html#E5:30_2_2_5"><i><font color="Green" title="E44">A44</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><b>then </b><i><font color="Green" title="E48">A48</font></i>: <font color="Maroon" title="c19">s</font> <a href="ami_1.html#K13" title="AMI_1:func.13">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K12" title="AMI_3:func.12">il.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c26">k</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="scm_comp.html#K15" title="SCM_COMP:func.15">SCM-Compile</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c14">tl</font>,<font color="Maroon" title="c15">tr</font></span>)</span>,<font color="Maroon" title="c16">aux</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c26">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="scm_1.html#D1" title="SCM_1:def.1">SCM_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="scm_comp.html#K15" title="SCM_COMP:func.15">SCM-Compile</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c14">tl</font>,<font color="Maroon" title="c15">tr</font></span>)</span>,<font color="Maroon" title="c16">aux</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"><span class="p3">(<span class="default"><font color="Maroon" title="c20">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c22">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>

<br/>.= 
<span class="p1"><a href="lang1.html#K4" title="LANG1:func.4">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="ami_3.html#K7" title="AMI_3:func.7">Divide</a> <span class="p3">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c16">aux</font></span>)</span>,<span class="p3">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c16">aux</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c16">aux</font></span>)</span> <a href="ami_3.html#K3" title="AMI_3:func.3">:=</a> <span class="p3">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <span class="p4">(<span class="default"><font color="Maroon" title="c16">aux</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="lang1.html#K4" title="LANG1:func.4">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2
<b>by </b><i><a class="txt" href="scm_comp.html#E3:30_2"><i><font color="Green" title="E31">A8</font></i></a>, <a class="txt" href="scm_comp.html#E20:30_2"><i><font color="Green" title="E41">A18</font></i></a>, <a class="txt" href="scm_comp.html#E2:30_2_2_5"><i><font color="Green" title="E42">A42</font></i></a>, <a class="txt" href="scm_comp.html#E7:30_2_2_5"><i><font color="Green" title="E45">A45</font></i></a>, <a class="txt" href="scm_comp.html#E8:30_2_2_5"><i><font color="Green" title="E46">A46</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c16">aux</font></span>)</span> <a href="ami_3.html#K3" title="AMI_3:func.3">:=</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="c16">aux</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>
;<br/><a NAME="E14:30_2_2_5"/>
<font color="Maroon" title="c16">aux</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c16">aux</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/><a NAME="E15:30_2_2_5"/><b>then </b><i><font color="Green" title="E49">A49</font></i>: 
 <a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c16">aux</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">aux</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="ami_3.html#T52" title="AMI_3:th.52">AMI_3:52</a></i>;<br/><b>hence </b><i><font color="Green" title="E50">A50</font></i>:  <a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c19">s</font>,<font color="Maroon" title="c26">k</font></span>)</span> = 
 <a href="ami_3.html#K12" title="AMI_3:func.12">il.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c20">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c22">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="scm_comp.html#E15:30_2"><i><font color="Green" title="E36">A13</font></i></a>, <a class="txt" href="scm_comp.html#E16:30_2"><i><font color="Green" title="E37">A14</font></i></a>, <a class="txt" href="scm_comp.html#E11:30_2_2_5"><i><font color="Green" title="E47">A47</font></i></a>, <a class="ref" href="scm_1.html#T22" title="SCM_1:th.22">SCM_1:22</a></i>
<br/>.= 
 <a href="ami_3.html#K12" title="AMI_3:func.12">il.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c26">k</font></span>)</span>

;<br/><br/><b>hence </b> <a href="ami_1.html#K7" title="AMI_1:func.7">IC</a> <font color="Maroon" title="c27">v</font> = 
 <a href="ami_3.html#K12" title="AMI_3:func.12">il.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c26">k</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="scm_comp.html#E13:30_2_2_5"><i><font color="Green" title="E48">A48</font></i></a>, <a class="ref" href="scm_1.html#T18" title="SCM_1:th.18">SCM_1:18</a></i>
<br/>.= 
 <a href="ami_3.html#K12" title="AMI_3:func.12">il.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c26">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>

;<br/><br/><b>thus </b><font color="Maroon" title="c27">v</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> <font color="Maroon" title="c16">aux</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c19">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c20">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c22">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <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="c16">aux</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="scm_comp.html#E13:30_2_2_5"><i><font color="Green" title="E48">A48</font></i></a>, <a class="txt" href="scm_comp.html#E16:30_2_2_5"><i><font color="Green" title="E50">A50</font></i></a>, <a class="ref" href="scm_1.html#T18" title="SCM_1:th.18">SCM_1:18</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c23">u2</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p2">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c16">aux</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c23">u2</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p2">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">aux</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="scm_comp.html#E15:30_2"><i><font color="Green" title="E36">A13</font></i></a>, <a class="txt" href="scm_comp.html#E16:30_2"><i><font color="Green" title="E37">A14</font></i></a>, <a class="txt" href="scm_comp.html#E11:30_2_2_5"><i><font color="Green" title="E47">A47</font></i></a>, <a class="txt" href="scm_comp.html#E15:30_2_2_5"><i><font color="Green" title="E49">A49</font></i></a>, <a class="ref" href="scm_1.html#T22" title="SCM_1:th.22">SCM_1:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c21">u1</font> <a href="ami_3.html#K2" title="AMI_3:func.2">.</a> <span class="p2">(<span class="default"><a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c16">aux</font></span>)</span></span>)</span> <a href="int_1.html#K6" title="INT_1:func.6">mod</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">tr</font> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c21">u1</font></span>)</span>
<b>by </b><i><a class="txt" href="scm_comp.html#E15:30_2"><i><font color="Green" title="E36">A13</font></i></a>, <a class="txt" href="scm_comp.html#E19:30_2"><i><font color="Green" title="E40">A17</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">nt</font> <a href="scm_comp.html#K2" title="SCM_COMP:func.2">-tree</a> <font color="Maroon" title="c14">tl</font>,<font color="Maroon" title="c15">tr</font></span>)</span> <a href="scm_comp.html#K13" title="SCM_COMP:func.13">@</a> <font color="Maroon" title="c19">s</font>
<b>by </b><i><a class="txt" href="scm_comp.html#E11:30_2"><i><font color="Green" title="E35">A12</font></i></a>, <a class="txt" href="scm_comp.html#E17:30_2"><i><font color="Green" title="E38">A15</font></i></a>, <a class="txt" href="scm_comp.html#E18:30_2"><i><font color="Green" title="E39">A16</font></i></a>, <a class="txt" href="scm_comp.html#E2:30_2_2_5"><i><font color="Green" title="E42">A42</font></i></a>, <a class="ref" href="scm_comp.html#T13" target="_self" title="SCM_COMP:th.13">Th13</a></i>
;<br/><b>let </b><font color="Maroon" title="c28">dn</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> ;<br/><b>assume </b><a NAME="E19:30_2_2_5"/><i><font color="Green" title="E51">A51</font></i>: 
<font color="Maroon" title="c28">dn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">aux</font>
 ;<br/><a NAME="E20:30_2_2_5"/><b>then </b><i><font color="Green" title="E52">A52</font></i>: 
 <a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c28">dn</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c16">aux</font>
 <b>by </b><i><a class="ref" href="ami_3.html#T52" title="AMI_3:th.52">AMI_3:52</a></i>;<br/><a NAME="E21:30_2_2_5"/><i><font color="Green" title="E53">A53</font></i>: 
<font color="Maroon" title="c28">dn</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">aux</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="scm_comp.html#E19:30_2_2_5"><i><font color="Green" title="E51">A51</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E22:30_2_2_5"/><b>then </b>
 <a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <font color="Maroon" title="c28">dn</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="ami_3.html#K11" title="AMI_3:func.11">dl.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">aux</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="ami_3.html#T52" title="AMI_3:th.52">AMI_3:52</a></i>;<br/><a NAME="E23:30_2_2_5"/><b>then </b><i><font color="Green" title="E54">A54</font></i>: 
<span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c19">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c20">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c22">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <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> <font color="Maroon" title="c28">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">u2</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> <font color="Maroon" title="c28">dn</font></span>)</span>
 <b>by </b><i><a class="txt" href="scm_comp.html#E15:30_2"><i><font color="Green" title="E36">A13</font></i></a>, <a class="txt" href="scm_comp.html#E16:30_2"><i><font color="Green" title="E37">A14</font></i></a>, <a class="txt" href="scm_comp.html#E11:30_2_2_5"><i><font color="Green" title="E47">A47</font></i></a>, <a class="txt" href="scm_comp.html#E15:30_2_2_5"><i><font color="Green" title="E49">A49</font></i></a>, <a class="txt" href="scm_comp.html#E20:30_2_2_5"><i><font color="Green" title="E52">A52</font></i></a>, <a class="ref" href="scm_1.html#T22" title="SCM_1:th.22">SCM_1:22</a></i>;<br/><a NAME="E24:30_2_2_5"/>
<font color="Maroon" title="c21">u1</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> <font color="Maroon" title="c28">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">u2</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> <font color="Maroon" title="c28">dn</font></span>)</span>
 <b>by </b><i><a class="txt" href="scm_comp.html#E15:30_2"><i><font color="Green" title="E36">A13</font></i></a>, <a class="txt" href="scm_comp.html#E21:30_2_2_5"><i><font color="Green" title="E53">A53</font></i></a></i>;<br/><a NAME="E25:30_2_2_5"/><b>then </b>
<font color="Maroon" title="c19">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> <font color="Maroon" title="c28">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10" title="AMI_1:func.10">Computation</a> <font color="Maroon" title="c19">s</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c20">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c22">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <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> <font color="Maroon" title="c28">dn</font></span>)</span>
 <b>by </b><i><a class="txt" href="scm_comp.html#E11:30_2"><i><font color="Green" title="E35">A12</font></i></a>, <a class="txt" href="scm_comp.html#E19:30_2_2_5"><i><font color="Green" title="E51">A51</font></i></a>, <a class="txt" href="scm_comp.html#E23:30_2_2_5"><i><font color="Green" title="E54">A54</font></i></a></i>;<br/><b>hence </b><a NAME="E26:30_2_2_5"/>
<font color="Maroon" title="c19">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> <font color="Maroon" title="c28">dn</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">v</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> <font color="Maroon" title="c28">dn</font></span>)</span>
 <b>by </b><i><a class="txt" href="scm_comp.html#E13:30_2_2_5"><i><font color="Green" title="E48">A48</font></i></a>, <a class="txt" href="scm_comp.html#E16:30_2_2_5"><i><font color="Green" title="E50">A50</font></i></a>, <a class="txt" href="scm_comp.html#E20:30_2_2_5"><i><font color="Green" title="E52">A52</font></i></a>, <a class="ref" href="scm_1.html#T18" title="SCM_1:th.18">SCM_1:18</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>thus </b><a NAME="E3:30"/>
for <font color="Olive" title="b1">term</font> being  <a href="scm_comp.html#NM1" title="SCM_COMP:NM.1">bin-term</a> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">term</font>]
 <b>from </b><i><a class="ref" href="bintree1.html#S2" title="BINTREE1:sch.2">BINTREE1:sch 2</a>(<a class="txt" href="scm_comp.html#E1:30"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="scm_comp.html#E2:30"><i><font color="Green" title="E28">A5</font></i></a>);</i><br/>


</div>
