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

<a NAME="E1:30"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">t</font> being  <a href="dtconstr.html#NM2">Terminal</a> of <a href="scm_comp.html#K1" target="_self">SCM-AE</a>  holds  <font color="Maroon">S<sub>2</sub></font>[ <a href="scm_comp.html#K3" target="_self">root-tree</a> <font color="Olive">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">t</font> be   <a href="dtconstr.html#NM2">Terminal</a> of <a href="scm_comp.html#K1" target="_self">SCM-AE</a> ;<br/><b>let </b><font color="Maroon">aux</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">s</font> be    <a href="scm_1.html#M1">State-consisting</a> of <font color="Maroon">n</font>,<font color="Maroon">n</font>,<font color="Maroon">k</font>, <a href="scm_comp.html#K15" target="_self">SCM-Compile</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K3" target="_self">root-tree</a> <font color="Maroon">t</font></span>)</span>,<font color="Maroon">aux</font>, <a href="lang1.html#K2">&lt;*&gt;</a> <a href="numbers.html#K4">INT</a> ;<br/>





<b>assume </b><a NAME="E1:30_1"/>
<font color="Maroon">aux</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="scm_comp.html#K17" target="_self">max_Data-Loc_in</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K3" target="_self">root-tree</a> <font color="Maroon">t</font></span>)</span>
 ;<br/>

<b>take </b><font color="Maroon">i</font> = 0;<br/>

<b>take </b><font color="Maroon">u</font> = <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> 1;<br/>

<b>thus </b><a NAME="E2:30_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/>

<a NAME="E3:30_1"/><i><font color="Green">E35</font></i>: 
 <a href="scm_comp.html#K15" target="_self">SCM-Compile</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K3" target="_self">root-tree</a> <font color="Maroon">t</font></span>)</span>,<font color="Maroon">aux</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_3.html#K15">dl.</a> <font color="Maroon">aux</font></span>)</span> <a href="ami_3.html#K3">:=</a> <span class="p3">(<span class="default"><a href="scm_comp.html#K4" target="_self">@</a> <font color="Maroon">t</font></span>)</span></span>)</span></span><a href="lang1.html#K3">*&gt;</a></span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:30_1"/>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K15" target="_self">SCM-Compile</a> <span class="p2">(<span class="default"><a href="scm_comp.html#K3" target="_self">root-tree</a> <font color="Maroon">t</font></span>)</span>,<font color="Maroon">aux</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/>

<a NAME="E5:30_1"/><i><font color="Green">E36</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_3.html#K15">dl.</a> <font color="Maroon">aux</font></span>)</span> <a href="ami_3.html#K3">:=</a> <span class="p3">(<span class="default"><a href="scm_comp.html#K4" target="_self">@</a> <font color="Maroon">t</font></span>)</span></span>)</span></span><a href="lang1.html#K3">*&gt;</a></span> <a href="hidden.html#R1">=</a> 1 &amp; 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 &amp; <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp; <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="ami_3.html#K15">dl.</a> <font color="Maroon">aux</font></span>)</span> <a href="ami_3.html#K3">:=</a> <span class="p3">(<span class="default"><a href="scm_comp.html#K4" target="_self">@</a> <font color="Maroon">t</font></span>)</span></span>)</span></span><a href="lang1.html#K3">*&gt;</a></span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> <font color="Maroon">aux</font></span>)</span> <a href="ami_3.html#K3">:=</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K4" target="_self">@</a> <font color="Maroon">t</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/>
<a NAME="E6:30_1"/><b>then </b><i><font color="Green">E38</font></i>: 
( <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> 0 &amp;  <a href="ami_1.html#K6">IC</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="ami_3.html#K16">il.</a> <font color="Maroon">n</font> &amp; <font color="Maroon">s</font> <a href="ami_1.html#K13">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K16">il.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> <font color="Maroon">aux</font></span>)</span> <a href="ami_3.html#K3">:=</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K4" target="_self">@</a> <font color="Maroon">t</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="ami_1.html#D19">AMI_1:def 19</a>, <a class="ref" href="scm_1.html#D1">SCM_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E7:30_1"/>
 <a href="ami_1.html#K6">IC</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ami_3.html#K16">il.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span>
 ;<br/>

<b>thus </b><a NAME="E8:30_1"/>
 <a href="ami_1.html#K6">IC</a> <font color="Maroon">u</font> <a href="hidden.html#R1">=</a>  <a href="ami_3.html#K16">il.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="scm_1.html#T18">SCM_1:18</a></i>;<br/>

<b>thus </b><font color="Maroon">u</font> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> <font color="Maroon">aux</font></span>)</span> = 
<font color="Maroon">s</font> <a href="scm_comp.html#K11" target="_self">.</a> <font color="Maroon">t</font>
<b>by </b><i>, , <a class="ref" href="scm_1.html#T18">SCM_1:18</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="scm_comp.html#K3" target="_self">root-tree</a> <font color="Maroon">t</font></span>)</span> <a href="scm_comp.html#K13" target="_self">@</a> <font color="Maroon">s</font>
<b>by </b><i/>
;<br/>

<b>let </b><font color="Maroon">dn</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E10:30_1"/>
<font color="Maroon">dn</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">aux</font>
 ;<br/>

<a NAME="E11:30_1"/><b>then </b>
 <a href="ami_3.html#K15">dl.</a> <font color="Maroon">dn</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="ami_3.html#K15">dl.</a> <font color="Maroon">aux</font>
 
<b>by </b><i><a class="ref" href="ami_3.html#T52">AMI_3:52</a></i>;<br/>
<b>hence </b><a NAME="E12:30_1"/>
<font color="Maroon">s</font> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> <font color="Maroon">dn</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> <font color="Maroon">dn</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="scm_1.html#T18">SCM_1:18</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:30"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">nt</font> being  <a href="dtconstr.html#NM3">NonTerminal</a> of <a href="scm_comp.html#K1" target="_self">SCM-AE</a> <br/> for <font color="Olive">tl</font>, <font color="Olive">tr</font> being  <a href="scm_comp.html#NM1">bin-term</a>  st <font color="Olive">nt</font> <a href="lang1.html#R1">==&gt;</a> <span class="p1"><a href="lang1.html#K4">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="bintree1.html#K1">root-label</a> <font color="Olive">tl</font></span>)</span>,<span class="p2">(<span class="default"><a href="bintree1.html#K1">root-label</a> <font color="Olive">tr</font></span>)</span></span><a href="lang1.html#K4">*&gt;</a></span> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">tl</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">tr</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">nt</font> <a href="scm_comp.html#K2" target="_self">-tree</a> <font color="Olive">tl</font>,<font color="Olive">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">nt</font> be   <a href="dtconstr.html#NM3">NonTerminal</a> of <a href="scm_comp.html#K1" target="_self">SCM-AE</a> ;<br/><b>let </b><font color="Maroon">tl</font> be   <a href="scm_comp.html#NM1">bin-term</a>, <font color="Maroon">tr</font> be   <a href="scm_comp.html#NM1">bin-term</a>;<br/>



<b>assume </b><a NAME="E1:30_2"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">nt</font> <a href="lang1.html#R1">==&gt;</a> <span class="p1"><a href="lang1.html#K4">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="bintree1.html#K1">root-label</a> <font color="Maroon">tl</font></span>)</span>,<span class="p2">(<span class="default"><a href="bintree1.html#K1">root-label</a> <font color="Maroon">tr</font></span>)</span></span><a href="lang1.html#K4">*&gt;</a></span> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">tl</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">tr</font>] )
 ;<br/>

<b>let </b><font color="Maroon">aux</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">s</font> be    <a href="scm_1.html#M1">State-consisting</a> of <font color="Maroon">n</font>,<font color="Maroon">n</font>,<font color="Maroon">k</font>, <a href="scm_comp.html#K15" target="_self">SCM-Compile</a> <span class="p1">(<span class="default"><font color="Maroon">nt</font> <a href="scm_comp.html#K2" target="_self">-tree</a> <font color="Maroon">tl</font>,<font color="Maroon">tr</font></span>)</span>,<font color="Maroon">aux</font>, <a href="lang1.html#K2">&lt;*&gt;</a> <a href="numbers.html#K4">INT</a> ;<br/>



<b>assume </b><a NAME="E2:30_2"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">aux</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="scm_comp.html#K17" target="_self">max_Data-Loc_in</a> <span class="p1">(<span class="default"><font color="Maroon">nt</font> <a href="scm_comp.html#K2" target="_self">-tree</a> <font color="Maroon">tl</font>,<font color="Maroon">tr</font></span>)</span>
 ;<br/>

<a NAME="E3:30_2"/><i><font color="Green">E42</font></i>: 
 <a href="scm_comp.html#K15" target="_self">SCM-Compile</a> <span class="p1">(<span class="default"><font color="Maroon">nt</font> <a href="scm_comp.html#K2" target="_self">-tree</a> <font color="Maroon">tl</font>,<font color="Maroon">tr</font></span>)</span>,<font color="Maroon">aux</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scm_comp.html#K15" target="_self">SCM-Compile</a> <font color="Maroon">tl</font>,<font color="Maroon">aux</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="scm_comp.html#K15" target="_self">SCM-Compile</a> <font color="Maroon">tr</font>,<span class="p3">(<span class="default"><font color="Maroon">aux</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K14" target="_self">Selfwork</a> <font color="Maroon">nt</font>,<font color="Maroon">aux</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E4:30_2"/><b>then </b><i><font color="Green">E43</font></i>: 
 <a href="scm_comp.html#K15" target="_self">SCM-Compile</a> <span class="p1">(<span class="default"><font color="Maroon">nt</font> <a href="scm_comp.html#K2" target="_self">-tree</a> <font color="Maroon">tl</font>,<font color="Maroon">tr</font></span>)</span>,<font color="Maroon">aux</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K15" target="_self">SCM-Compile</a> <font color="Maroon">tl</font>,<font color="Maroon">aux</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scm_comp.html#K15" target="_self">SCM-Compile</a> <font color="Maroon">tr</font>,<span class="p3">(<span class="default"><font color="Maroon">aux</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="scm_comp.html#K14" target="_self">Selfwork</a> <font color="Maroon">nt</font>,<font color="Maroon">aux</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T45">FINSEQ_1:45</a></i>;<br/>
<a NAME="E5:30_2"/>
 <a href="scm_comp.html#K17" target="_self">max_Data-Loc_in</a> <span class="p1">(<span class="default"><font color="Maroon">nt</font> <a href="scm_comp.html#K2" target="_self">-tree</a> <font color="Maroon">tl</font>,<font color="Maroon">tr</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="limfunc1.html#K1">max</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K17" target="_self">max_Data-Loc_in</a> <font color="Maroon">tl</font></span>)</span>,<span class="p1">(<span class="default"><a href="scm_comp.html#K17" target="_self">max_Data-Loc_in</a> <font color="Maroon">tr</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E6:30_2"/><b>then </b>
(  <a href="scm_comp.html#K17" target="_self">max_Data-Loc_in</a> <font color="Maroon">tl</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="scm_comp.html#K17" target="_self">max_Data-Loc_in</a> <span class="p1">(<span class="default"><font color="Maroon">nt</font> <a href="scm_comp.html#K2" target="_self">-tree</a> <font color="Maroon">tl</font>,<font color="Maroon">tr</font></span>)</span> &amp;  <a href="scm_comp.html#K17" target="_self">max_Data-Loc_in</a> <font color="Maroon">tr</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="scm_comp.html#K17" target="_self">max_Data-Loc_in</a> <span class="p1">(<span class="default"><font color="Maroon">nt</font> <a href="scm_comp.html#K2" target="_self">-tree</a> <font color="Maroon">tl</font>,<font color="Maroon">tr</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="square_1.html#T46">SQUARE_1:46</a></i>;<br/>
<a NAME="E7:30_2"/><b>then </b><i><font color="Green">E44</font></i>: 
(  <a href="scm_comp.html#K17" target="_self">max_Data-Loc_in</a> <font color="Maroon">tl</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">aux</font> &amp;  <a href="scm_comp.html#K17" target="_self">max_Data-Loc_in</a> <font color="Maroon">tr</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">aux</font> )
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/>
<a NAME="E8:30_2"/><b>then </b><i><font color="Green">E64</font></i>: 
 <a href="scm_comp.html#K17" target="_self">max_Data-Loc_in</a> <font color="Maroon">tr</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">aux</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E9:30_2"/>
<font color="Maroon">s</font> is    <a href="scm_1.html#M1">State-consisting</a> of <font color="Maroon">n</font>,<font color="Maroon">n</font>,<font color="Maroon">k</font>, <a href="scm_comp.html#K15" target="_self">SCM-Compile</a> <font color="Maroon">tl</font>,<font color="Maroon">aux</font>, <a href="lang1.html#K2">&lt;*&gt;</a> <a href="numbers.html#K4">INT</a> 
 
<b>by </b><i>, </i>;<br/>
<b>then consider </b><font color="Maroon">i1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">u1</font> being   <a href="ami_1.html#NM4">State</a> of <a href="ami_3.html#K1">SCM</a> <b> such that </b><br/><a NAME="E11:30_2"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">u1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K15" target="_self">SCM-Compile</a> <font color="Maroon">tl</font>,<font color="Maroon">aux</font></span>)</span> &amp;  <a href="ami_1.html#K6">IC</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i1</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ami_3.html#K16">il.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i1</font></span>)</span> &amp;  <a href="ami_1.html#K6">IC</a> <font color="Maroon">u1</font> <a href="hidden.html#R1">=</a>  <a href="ami_3.html#K16">il.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon">u1</font> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> <font color="Maroon">aux</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">tl</font> <a href="scm_comp.html#K13" target="_self">@</a> <font color="Maroon">s</font> &amp; ( for <font color="Olive">dn</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">dn</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">aux</font> holds <br/><font color="Maroon">s</font> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> <font color="Olive">dn</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">u1</font> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> <font color="Olive">dn</font></span>)</span> ) )
 <b>by </b><i>, </i>;<br/>
<a NAME="E12:30_2"/>
<font color="Maroon">u1</font> is    <a href="scm_1.html#M1">State-consisting</a> of <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font>,<span class="p1">(<span class="default"><a href="scm_comp.html#K15" target="_self">SCM-Compile</a> <font color="Maroon">tr</font>,<span class="p2">(<span class="default"><font color="Maroon">aux</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K14" target="_self">Selfwork</a> <font color="Maroon">nt</font>,<font color="Maroon">aux</font></span>)</span>, <a href="lang1.html#K2">&lt;*&gt;</a> <a href="numbers.html#K4">INT</a> 
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E13:30_2"/><b>then </b>
<font color="Maroon">u1</font> is    <a href="scm_1.html#M1">State-consisting</a> of <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">k</font>, <a href="scm_comp.html#K15" target="_self">SCM-Compile</a> <font color="Maroon">tr</font>,<span class="p1">(<span class="default"><font color="Maroon">aux</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>, <a href="lang1.html#K2">&lt;*&gt;</a> <a href="numbers.html#K4">INT</a> 
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">i2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">u2</font> being   <a href="ami_1.html#NM4">State</a> of <a href="ami_3.html#K1">SCM</a> <b> such that </b><br/><a NAME="E15:30_2"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">u2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">u1</font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K15" target="_self">SCM-Compile</a> <font color="Maroon">tr</font>,<span class="p2">(<span class="default"><font color="Maroon">aux</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp;  <a href="ami_1.html#K6">IC</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">u1</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ami_3.html#K16">il.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i2</font></span>)</span> &amp;  <a href="ami_1.html#K6">IC</a> <font color="Maroon">u2</font> <a href="hidden.html#R1">=</a>  <a href="ami_3.html#K16">il.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon">u2</font> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> <span class="p2">(<span class="default"><font color="Maroon">aux</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">tr</font> <a href="scm_comp.html#K13" target="_self">@</a> <font color="Maroon">u1</font> &amp; ( for <font color="Olive">dn</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">dn</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">aux</font> <a href="nat_1.html#K1">+</a> 1 holds <br/><font color="Maroon">u1</font> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> <font color="Olive">dn</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">u2</font> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> <font color="Olive">dn</font></span>)</span> ) )
 <b>by </b><i>, <a class="txt" href="scm_comp.html#E1:30"><i><font color="Green">E34</font></i></a></i>;<br/>
<a NAME="E16:30_2"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">u2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="ami_1.html#T51">AMI_1:51</a></i>;<br/>
<a NAME="E17:30_2"/><i><font color="Green">E68</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">nt</font> <a href="scm_comp.html#K2" target="_self">-tree</a> <font color="Maroon">tl</font>,<font color="Maroon">tr</font></span>)</span> <a href="scm_comp.html#K13" target="_self">@</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">nt</font> <a href="scm_comp.html#K10" target="_self">-Meaning_on</a> <span class="p1">(<span class="default"><font color="Maroon">tl</font> <a href="scm_comp.html#K13" target="_self">@</a> <font color="Maroon">s</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">tr</font> <a href="scm_comp.html#K13" target="_self">@</a> <font color="Maroon">s</font></span>)</span>
 
<b>by </b><i/>;<br/>
<div><i><font color="Green">E69</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">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:30_2_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="scm_comp.html#K17" target="_self">max_Data-Loc_in</a> <font color="Maroon">tr</font>
 ;<br/><a NAME="E2:30_2_1"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">aux</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T2">XREAL_1:2</a></i>;<br/><b>hence </b><a NAME="E3:30_2_1"/>
<font color="Maroon">s</font> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">u1</font> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E19:30_2"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">aux</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">aux</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E20:30_2"/><i><font color="Green">E71</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="scm_comp.html#K15" target="_self">SCM-Compile</a> <font color="Maroon">tl</font>,<font color="Maroon">aux</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="scm_comp.html#K15" target="_self">SCM-Compile</a> <font color="Maroon">tr</font>,<span class="p3">(<span class="default"><font color="Maroon">aux</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#T35">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">nt</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,0</span><a href="tarski.html#K4">]</a></span> or <font color="Maroon">nt</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,1</span><a href="tarski.html#K4">]</a></span> or <font color="Maroon">nt</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,2</span><a href="tarski.html#K4">]</a></span> or <font color="Maroon">nt</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,3</span><a href="tarski.html#K4">]</a></span> or <font color="Maroon">nt</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,4</span><a href="tarski.html#K4">]</a></span> )
 </span><b>by </b><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">nt</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,0</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><a NAME="E2:30_2_2_1"/><b>then </b><i><font color="Green">E72</font></i>: 
(  <a href="scm_comp.html#K14" target="_self">Selfwork</a> <font color="Maroon">nt</font>,<font color="Maroon">aux</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="ami_3.html#K4">AddTo</a> <span class="p3">(<span class="default"><a href="ami_3.html#K15">dl.</a> <font color="Maroon">aux</font></span>)</span>,<span class="p3">(<span class="default"><a href="ami_3.html#K15">dl.</a> <span class="p4">(<span class="default"><font color="Maroon">aux</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="lang1.html#K3">*&gt;</a></span> &amp; <font color="Maroon">nt</font> <a href="scm_comp.html#K10" target="_self">-Meaning_on</a> <span class="p1">(<span class="default"><font color="Maroon">tl</font> <a href="scm_comp.html#K13" target="_self">@</a> <font color="Maroon">s</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">tr</font> <a href="scm_comp.html#K13" target="_self">@</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">tl</font> <a href="scm_comp.html#K13" target="_self">@</a> <font color="Maroon">s</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">tr</font> <a href="scm_comp.html#K13" target="_self">@</a> <font color="Maroon">s</font></span>)</span> )
 <b>by </b><i>, </i>;<br/><b>take </b><font color="Maroon">i</font> = <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><b>take </b><font color="Maroon">u</font> = <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><b>thus </b><a NAME="E3:30_2_2_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><a NAME="E4:30_2_2_1"/><i><font color="Green">E74</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K14" target="_self">Selfwork</a> <font color="Maroon">nt</font>,<font color="Maroon">aux</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="scm_comp.html#E3:30_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/><a NAME="E5:30_2_2_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K15" target="_self">SCM-Compile</a> <span class="p2">(<span class="default"><font color="Maroon">nt</font> <a href="scm_comp.html#K2" target="_self">-tree</a> <font color="Maroon">tl</font>,<font color="Maroon">tr</font></span>)</span>,<font color="Maroon">aux</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="txt" href="scm_comp.html#E2:30_2"><i><font color="Green">E41</font></i></a>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/><a NAME="E6:30_2_2_1"/><b>then </b>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K15" target="_self">SCM-Compile</a> <span class="p2">(<span class="default"><font color="Maroon">nt</font> <a href="scm_comp.html#K2" target="_self">-tree</a> <font color="Maroon">tl</font>,<font color="Maroon">tr</font></span>)</span>,<font color="Maroon">aux</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>then </b><i><font color="Green">E75</font></i>: <font color="Maroon">s</font> <a href="ami_1.html#K13">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K16">il.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="scm_comp.html#K15" target="_self">SCM-Compile</a> <span class="p2">(<span class="default"><font color="Maroon">nt</font> <a href="scm_comp.html#K2" target="_self">-tree</a> <font color="Maroon">tl</font>,<font color="Maroon">tr</font></span>)</span>,<font color="Maroon">aux</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="scm_1.html#D1">SCM_1:def 1</a></i>
<br/>.= 
 <a href="ami_3.html#K4">AddTo</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> <font color="Maroon">aux</font></span>)</span>,<span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> <span class="p2">(<span class="default"><font color="Maroon">aux</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i>, <a class="txt" href="scm_comp.html#E2:30_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="scm_comp.html#E3:30_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="finseq_1.html#T59">FINSEQ_1:59</a></i>
;<br/><b>thus </b><a NAME="E8:30_2_2_1"/>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K15" target="_self">SCM-Compile</a> <span class="p2">(<span class="default"><font color="Maroon">nt</font> <a href="scm_comp.html#K2" target="_self">-tree</a> <font color="Maroon">tl</font>,<font color="Maroon">tr</font></span>)</span>,<font color="Maroon">aux</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="scm_comp.html#E2:30_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="scm_comp.html#E4:30_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/><b>thus </b><a NAME="E9:30_2_2_1"/>
 <a href="ami_1.html#K6">IC</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ami_3.html#K16">il.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="ami_1.html#T51">AMI_1:51</a></i>;<br/><b>thus </b> <a href="ami_1.html#K6">IC</a> <font color="Maroon">u</font> = 
 <a href="ami_3.html#K16">il.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i>, , <a class="txt" href="scm_comp.html#E7:30_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="scm_1.html#T19">SCM_1:19</a></i>
<br/>.= 
 <a href="ami_3.html#K16">il.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>

;<br/><b>thus </b><font color="Maroon">u</font> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> <font color="Maroon">aux</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">u2</font> <a href="ami_3.html#K2">.</a> <span class="p2">(<span class="default"><a href="ami_3.html#K15">dl.</a> <font color="Maroon">aux</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">u2</font> <a href="ami_3.html#K2">.</a> <span class="p2">(<span class="default"><a href="ami_3.html#K15">dl.</a> <span class="p3">(<span class="default"><font color="Maroon">aux</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i>, , <a class="txt" href="scm_comp.html#E7:30_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="scm_1.html#T19">SCM_1:19</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">u1</font> <a href="ami_3.html#K2">.</a> <span class="p2">(<span class="default"><a href="ami_3.html#K15">dl.</a> <font color="Maroon">aux</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">tr</font> <a href="scm_comp.html#K13" target="_self">@</a> <font color="Maroon">u1</font></span>)</span>
<b>by </b><i>, <a class="txt" href="scm_comp.html#E1:30_2"><i><font color="Green">E40</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">nt</font> <a href="scm_comp.html#K2" target="_self">-tree</a> <font color="Maroon">tl</font>,<font color="Maroon">tr</font></span>)</span> <a href="scm_comp.html#K13" target="_self">@</a> <font color="Maroon">s</font>
<b>by </b><i>, , <a class="txt" href="scm_comp.html#E2:30"><i><font color="Green">E39</font></i></a>, <a class="txt" href="scm_comp.html#E3:30_2"><i><font color="Green">E42</font></i></a>, </i>
;<br/><b>let </b><font color="Maroon">dn</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E12:30_2_2_1"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">dn</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">aux</font>
 ;<br/><a NAME="E13:30_2_2_1"/><b>then </b>
 <a href="ami_3.html#K15">dl.</a> <font color="Maroon">dn</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="ami_3.html#K15">dl.</a> <font color="Maroon">aux</font>
 <b>by </b><i><a class="ref" href="ami_3.html#T52">AMI_3:52</a></i>;<br/><a NAME="E14:30_2_2_1"/><b>then </b><i><font color="Green">E82</font></i>: 
<font color="Maroon">u</font> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> <font color="Maroon">dn</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">u2</font> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> <font color="Maroon">dn</font></span>)</span>
 <b>by </b><i>, , <a class="txt" href="scm_comp.html#E7:30_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="scm_1.html#T19">SCM_1:19</a></i>;<br/><a NAME="E15:30_2_2_1"/>
<font color="Maroon">dn</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">aux</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="scm_comp.html#T2" target="_self">Th2</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E16:30_2_2_1"/><b>then </b>
<font color="Maroon">u1</font> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> <font color="Maroon">dn</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">u2</font> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> <font color="Maroon">dn</font></span>)</span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E17:30_2_2_1"/>
<font color="Maroon">s</font> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> <font color="Maroon">dn</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> <font color="Maroon">dn</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="scm_comp.html#T2" target="_self">Th2</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">nt</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,1</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><a NAME="E2:30_2_2_2"/><b>then </b><i><font color="Green">E83</font></i>: 
(  <a href="scm_comp.html#K14" target="_self">Selfwork</a> <font color="Maroon">nt</font>,<font color="Maroon">aux</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="ami_3.html#K5">SubFrom</a> <span class="p3">(<span class="default"><a href="ami_3.html#K15">dl.</a> <font color="Maroon">aux</font></span>)</span>,<span class="p3">(<span class="default"><a href="ami_3.html#K15">dl.</a> <span class="p4">(<span class="default"><font color="Maroon">aux</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="lang1.html#K3">*&gt;</a></span> &amp; <font color="Maroon">nt</font> <a href="scm_comp.html#K10" target="_self">-Meaning_on</a> <span class="p1">(<span class="default"><font color="Maroon">tl</font> <a href="scm_comp.html#K13" target="_self">@</a> <font color="Maroon">s</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">tr</font> <a href="scm_comp.html#K13" target="_self">@</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">tl</font> <a href="scm_comp.html#K13" target="_self">@</a> <font color="Maroon">s</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">tr</font> <a href="scm_comp.html#K13" target="_self">@</a> <font color="Maroon">s</font></span>)</span> )
 <b>by </b><i>, </i>;<br/><b>take </b><font color="Maroon">i</font> = <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><b>take </b><font color="Maroon">u</font> = <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><b>thus </b><a NAME="E3:30_2_2_2"/>
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><a NAME="E4:30_2_2_2"/><i><font color="Green">E84</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K14" target="_self">Selfwork</a> <font color="Maroon">nt</font>,<font color="Maroon">aux</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="scm_comp.html#E3:30_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/><a NAME="E5:30_2_2_2"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K15" target="_self">SCM-Compile</a> <span class="p2">(<span class="default"><font color="Maroon">nt</font> <a href="scm_comp.html#K2" target="_self">-tree</a> <font color="Maroon">tl</font>,<font color="Maroon">tr</font></span>)</span>,<font color="Maroon">aux</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="txt" href="scm_comp.html#E2:30_2"><i><font color="Green">E41</font></i></a>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/><a NAME="E6:30_2_2_2"/><b>then </b>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K15" target="_self">SCM-Compile</a> <span class="p2">(<span class="default"><font color="Maroon">nt</font> <a href="scm_comp.html#K2" target="_self">-tree</a> <font color="Maroon">tl</font>,<font color="Maroon">tr</font></span>)</span>,<font color="Maroon">aux</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>then </b><i><font color="Green">E89</font></i>: <font color="Maroon">s</font> <a href="ami_1.html#K13">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K16">il.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="scm_comp.html#K15" target="_self">SCM-Compile</a> <span class="p2">(<span class="default"><font color="Maroon">nt</font> <a href="scm_comp.html#K2" target="_self">-tree</a> <font color="Maroon">tl</font>,<font color="Maroon">tr</font></span>)</span>,<font color="Maroon">aux</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="scm_1.html#D1">SCM_1:def 1</a></i>
<br/>.= 
 <a href="ami_3.html#K5">SubFrom</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> <font color="Maroon">aux</font></span>)</span>,<span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> <span class="p2">(<span class="default"><font color="Maroon">aux</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i>, <a class="txt" href="scm_comp.html#E2:30_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="scm_comp.html#E3:30_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="finseq_1.html#T59">FINSEQ_1:59</a></i>
;<br/><b>thus </b><a NAME="E8:30_2_2_2"/>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K15" target="_self">SCM-Compile</a> <span class="p2">(<span class="default"><font color="Maroon">nt</font> <a href="scm_comp.html#K2" target="_self">-tree</a> <font color="Maroon">tl</font>,<font color="Maroon">tr</font></span>)</span>,<font color="Maroon">aux</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="scm_comp.html#E2:30_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="scm_comp.html#E4:30_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/><b>thus </b><a NAME="E9:30_2_2_2"/>
 <a href="ami_1.html#K6">IC</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ami_3.html#K16">il.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="ami_1.html#T51">AMI_1:51</a></i>;<br/><b>thus </b> <a href="ami_1.html#K6">IC</a> <font color="Maroon">u</font> = 
 <a href="ami_3.html#K16">il.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i>, , <a class="txt" href="scm_comp.html#E7:30_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="scm_1.html#T20">SCM_1:20</a></i>
<br/>.= 
 <a href="ami_3.html#K16">il.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>

;<br/><b>thus </b><font color="Maroon">u</font> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> <font color="Maroon">aux</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">u2</font> <a href="ami_3.html#K2">.</a> <span class="p2">(<span class="default"><a href="ami_3.html#K15">dl.</a> <font color="Maroon">aux</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">u2</font> <a href="ami_3.html#K2">.</a> <span class="p2">(<span class="default"><a href="ami_3.html#K15">dl.</a> <span class="p3">(<span class="default"><font color="Maroon">aux</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i>, , <a class="txt" href="scm_comp.html#E7:30_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="scm_1.html#T20">SCM_1:20</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">u1</font> <a href="ami_3.html#K2">.</a> <span class="p2">(<span class="default"><a href="ami_3.html#K15">dl.</a> <font color="Maroon">aux</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">tr</font> <a href="scm_comp.html#K13" target="_self">@</a> <font color="Maroon">u1</font></span>)</span>
<b>by </b><i>, <a class="txt" href="scm_comp.html#E1:30_2"><i><font color="Green">E40</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">nt</font> <a href="scm_comp.html#K2" target="_self">-tree</a> <font color="Maroon">tl</font>,<font color="Maroon">tr</font></span>)</span> <a href="scm_comp.html#K13" target="_self">@</a> <font color="Maroon">s</font>
<b>by </b><i>, , <a class="txt" href="scm_comp.html#E2:30"><i><font color="Green">E39</font></i></a>, <a class="txt" href="scm_comp.html#E3:30_2"><i><font color="Green">E42</font></i></a>, </i>
;<br/><b>let </b><font color="Maroon">dn</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E12:30_2_2_2"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">dn</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">aux</font>
 ;<br/><a NAME="E13:30_2_2_2"/><b>then </b>
 <a href="ami_3.html#K15">dl.</a> <font color="Maroon">dn</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="ami_3.html#K15">dl.</a> <font color="Maroon">aux</font>
 <b>by </b><i><a class="ref" href="ami_3.html#T52">AMI_3:52</a></i>;<br/><a NAME="E14:30_2_2_2"/><b>then </b><i><font color="Green">E93</font></i>: 
<font color="Maroon">u</font> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> <font color="Maroon">dn</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">u2</font> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> <font color="Maroon">dn</font></span>)</span>
 <b>by </b><i>, , <a class="txt" href="scm_comp.html#E7:30_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="scm_1.html#T20">SCM_1:20</a></i>;<br/><a NAME="E15:30_2_2_2"/>
<font color="Maroon">dn</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">aux</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="scm_comp.html#T2" target="_self">Th2</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E16:30_2_2_2"/><b>then </b>
<font color="Maroon">u1</font> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> <font color="Maroon">dn</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">u2</font> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> <font color="Maroon">dn</font></span>)</span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E17:30_2_2_2"/>
<font color="Maroon">s</font> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> <font color="Maroon">dn</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> <font color="Maroon">dn</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="scm_comp.html#T2" target="_self">Th2</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">nt</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,2</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><a NAME="E2:30_2_2_3"/><b>then </b><i><font color="Green">E94</font></i>: 
(  <a href="scm_comp.html#K14" target="_self">Selfwork</a> <font color="Maroon">nt</font>,<font color="Maroon">aux</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="ami_3.html#K6">MultBy</a> <span class="p3">(<span class="default"><a href="ami_3.html#K15">dl.</a> <font color="Maroon">aux</font></span>)</span>,<span class="p3">(<span class="default"><a href="ami_3.html#K15">dl.</a> <span class="p4">(<span class="default"><font color="Maroon">aux</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="lang1.html#K3">*&gt;</a></span> &amp; <font color="Maroon">nt</font> <a href="scm_comp.html#K10" target="_self">-Meaning_on</a> <span class="p1">(<span class="default"><font color="Maroon">tl</font> <a href="scm_comp.html#K13" target="_self">@</a> <font color="Maroon">s</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">tr</font> <a href="scm_comp.html#K13" target="_self">@</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">tl</font> <a href="scm_comp.html#K13" target="_self">@</a> <font color="Maroon">s</font></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">tr</font> <a href="scm_comp.html#K13" target="_self">@</a> <font color="Maroon">s</font></span>)</span> )
 <b>by </b><i>, </i>;<br/><b>take </b><font color="Maroon">i</font> = <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><b>take </b><font color="Maroon">u</font> = <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><b>thus </b><a NAME="E3:30_2_2_3"/>
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><a NAME="E4:30_2_2_3"/><i><font color="Green">E95</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K14" target="_self">Selfwork</a> <font color="Maroon">nt</font>,<font color="Maroon">aux</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="scm_comp.html#E3:30_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/><a NAME="E5:30_2_2_3"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K15" target="_self">SCM-Compile</a> <span class="p2">(<span class="default"><font color="Maroon">nt</font> <a href="scm_comp.html#K2" target="_self">-tree</a> <font color="Maroon">tl</font>,<font color="Maroon">tr</font></span>)</span>,<font color="Maroon">aux</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="txt" href="scm_comp.html#E2:30_2"><i><font color="Green">E41</font></i></a>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/><a NAME="E6:30_2_2_3"/><b>then </b>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K15" target="_self">SCM-Compile</a> <span class="p2">(<span class="default"><font color="Maroon">nt</font> <a href="scm_comp.html#K2" target="_self">-tree</a> <font color="Maroon">tl</font>,<font color="Maroon">tr</font></span>)</span>,<font color="Maroon">aux</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>then </b><i><font color="Green">E96</font></i>: <font color="Maroon">s</font> <a href="ami_1.html#K13">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K16">il.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="scm_comp.html#K15" target="_self">SCM-Compile</a> <span class="p2">(<span class="default"><font color="Maroon">nt</font> <a href="scm_comp.html#K2" target="_self">-tree</a> <font color="Maroon">tl</font>,<font color="Maroon">tr</font></span>)</span>,<font color="Maroon">aux</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="scm_1.html#D1">SCM_1:def 1</a></i>
<br/>.= 
 <a href="ami_3.html#K6">MultBy</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> <font color="Maroon">aux</font></span>)</span>,<span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> <span class="p2">(<span class="default"><font color="Maroon">aux</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i>, <a class="txt" href="scm_comp.html#E2:30_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="scm_comp.html#E3:30_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="finseq_1.html#T59">FINSEQ_1:59</a></i>
;<br/><b>thus </b><a NAME="E8:30_2_2_3"/>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K15" target="_self">SCM-Compile</a> <span class="p2">(<span class="default"><font color="Maroon">nt</font> <a href="scm_comp.html#K2" target="_self">-tree</a> <font color="Maroon">tl</font>,<font color="Maroon">tr</font></span>)</span>,<font color="Maroon">aux</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="scm_comp.html#E2:30_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="scm_comp.html#E4:30_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/><b>thus </b><a NAME="E9:30_2_2_3"/>
 <a href="ami_1.html#K6">IC</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ami_3.html#K16">il.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="ami_1.html#T51">AMI_1:51</a></i>;<br/><b>thus </b> <a href="ami_1.html#K6">IC</a> <font color="Maroon">u</font> = 
 <a href="ami_3.html#K16">il.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i>, , <a class="txt" href="scm_comp.html#E7:30_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="scm_1.html#T21">SCM_1:21</a></i>
<br/>.= 
 <a href="ami_3.html#K16">il.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>

;<br/><b>thus </b><font color="Maroon">u</font> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> <font color="Maroon">aux</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">u2</font> <a href="ami_3.html#K2">.</a> <span class="p2">(<span class="default"><a href="ami_3.html#K15">dl.</a> <font color="Maroon">aux</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">u2</font> <a href="ami_3.html#K2">.</a> <span class="p2">(<span class="default"><a href="ami_3.html#K15">dl.</a> <span class="p3">(<span class="default"><font color="Maroon">aux</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i>, , <a class="txt" href="scm_comp.html#E7:30_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="scm_1.html#T21">SCM_1:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">u1</font> <a href="ami_3.html#K2">.</a> <span class="p2">(<span class="default"><a href="ami_3.html#K15">dl.</a> <font color="Maroon">aux</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">tr</font> <a href="scm_comp.html#K13" target="_self">@</a> <font color="Maroon">u1</font></span>)</span>
<b>by </b><i>, <a class="txt" href="scm_comp.html#E1:30_2"><i><font color="Green">E40</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">nt</font> <a href="scm_comp.html#K2" target="_self">-tree</a> <font color="Maroon">tl</font>,<font color="Maroon">tr</font></span>)</span> <a href="scm_comp.html#K13" target="_self">@</a> <font color="Maroon">s</font>
<b>by </b><i>, , <a class="txt" href="scm_comp.html#E2:30"><i><font color="Green">E39</font></i></a>, <a class="txt" href="scm_comp.html#E3:30_2"><i><font color="Green">E42</font></i></a>, </i>
;<br/><b>let </b><font color="Maroon">dn</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E12:30_2_2_3"/><i><font color="Green">E166</font></i>: 
<font color="Maroon">dn</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">aux</font>
 ;<br/><a NAME="E13:30_2_2_3"/><b>then </b>
 <a href="ami_3.html#K15">dl.</a> <font color="Maroon">dn</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="ami_3.html#K15">dl.</a> <font color="Maroon">aux</font>
 <b>by </b><i><a class="ref" href="ami_3.html#T52">AMI_3:52</a></i>;<br/><a NAME="E14:30_2_2_3"/><b>then </b><i><font color="Green">E167</font></i>: 
<font color="Maroon">u</font> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> <font color="Maroon">dn</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">u2</font> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> <font color="Maroon">dn</font></span>)</span>
 <b>by </b><i>, , <a class="txt" href="scm_comp.html#E7:30_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="scm_1.html#T21">SCM_1:21</a></i>;<br/><a NAME="E15:30_2_2_3"/>
<font color="Maroon">dn</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">aux</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="scm_comp.html#T2" target="_self">Th2</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E16:30_2_2_3"/><b>then </b>
<font color="Maroon">u1</font> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> <font color="Maroon">dn</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">u2</font> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> <font color="Maroon">dn</font></span>)</span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E17:30_2_2_3"/>
<font color="Maroon">s</font> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> <font color="Maroon">dn</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> <font color="Maroon">dn</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="scm_comp.html#T2" target="_self">Th2</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">nt</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,3</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><a NAME="E2:30_2_2_4"/><b>then </b><i><font color="Green">E168</font></i>: 
(  <a href="scm_comp.html#K14" target="_self">Selfwork</a> <font color="Maroon">nt</font>,<font color="Maroon">aux</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="ami_3.html#K7">Divide</a> <span class="p3">(<span class="default"><a href="ami_3.html#K15">dl.</a> <font color="Maroon">aux</font></span>)</span>,<span class="p3">(<span class="default"><a href="ami_3.html#K15">dl.</a> <span class="p4">(<span class="default"><font color="Maroon">aux</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="lang1.html#K3">*&gt;</a></span> &amp; <font color="Maroon">nt</font> <a href="scm_comp.html#K10" target="_self">-Meaning_on</a> <span class="p1">(<span class="default"><font color="Maroon">tl</font> <a href="scm_comp.html#K13" target="_self">@</a> <font color="Maroon">s</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">tr</font> <a href="scm_comp.html#K13" target="_self">@</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">tl</font> <a href="scm_comp.html#K13" target="_self">@</a> <font color="Maroon">s</font></span>)</span> <a href="int_1.html#K5">div</a> <span class="p1">(<span class="default"><font color="Maroon">tr</font> <a href="scm_comp.html#K13" target="_self">@</a> <font color="Maroon">s</font></span>)</span> )
 <b>by </b><i>, </i>;<br/><b>take </b><font color="Maroon">i</font> = <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><b>take </b><font color="Maroon">u</font> = <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><b>thus </b><a NAME="E3:30_2_2_4"/>
<font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><a NAME="E4:30_2_2_4"/><i><font color="Green">E169</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K14" target="_self">Selfwork</a> <font color="Maroon">nt</font>,<font color="Maroon">aux</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="scm_comp.html#E3:30_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/><a NAME="E5:30_2_2_4"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K15" target="_self">SCM-Compile</a> <span class="p2">(<span class="default"><font color="Maroon">nt</font> <a href="scm_comp.html#K2" target="_self">-tree</a> <font color="Maroon">tl</font>,<font color="Maroon">tr</font></span>)</span>,<font color="Maroon">aux</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="txt" href="scm_comp.html#E2:30_2"><i><font color="Green">E41</font></i></a>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/><a NAME="E6:30_2_2_4"/><b>then </b>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K15" target="_self">SCM-Compile</a> <span class="p2">(<span class="default"><font color="Maroon">nt</font> <a href="scm_comp.html#K2" target="_self">-tree</a> <font color="Maroon">tl</font>,<font color="Maroon">tr</font></span>)</span>,<font color="Maroon">aux</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>then </b><i><font color="Green">E170</font></i>: <font color="Maroon">s</font> <a href="ami_1.html#K13">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K16">il.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="scm_comp.html#K15" target="_self">SCM-Compile</a> <span class="p2">(<span class="default"><font color="Maroon">nt</font> <a href="scm_comp.html#K2" target="_self">-tree</a> <font color="Maroon">tl</font>,<font color="Maroon">tr</font></span>)</span>,<font color="Maroon">aux</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="scm_1.html#D1">SCM_1:def 1</a></i>
<br/>.= 
 <a href="ami_3.html#K7">Divide</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> <font color="Maroon">aux</font></span>)</span>,<span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> <span class="p2">(<span class="default"><font color="Maroon">aux</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i>, <a class="txt" href="scm_comp.html#E2:30_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="scm_comp.html#E3:30_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="finseq_1.html#T59">FINSEQ_1:59</a></i>
;<br/><a NAME="E8:30_2_2_4"/>
<font color="Maroon">aux</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">aux</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><a NAME="E9:30_2_2_4"/><b>then </b><i><font color="Green">E171</font></i>: 
 <a href="ami_3.html#K15">dl.</a> <font color="Maroon">aux</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="ami_3.html#K15">dl.</a> <span class="p1">(<span class="default"><font color="Maroon">aux</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="ami_3.html#T52">AMI_3:52</a></i>;<br/><b>thus </b><a NAME="E10:30_2_2_4"/>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K15" target="_self">SCM-Compile</a> <span class="p2">(<span class="default"><font color="Maroon">nt</font> <a href="scm_comp.html#K2" target="_self">-tree</a> <font color="Maroon">tl</font>,<font color="Maroon">tr</font></span>)</span>,<font color="Maroon">aux</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="scm_comp.html#E2:30_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="scm_comp.html#E4:30_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/><b>thus </b><a NAME="E11:30_2_2_4"/>
 <a href="ami_1.html#K6">IC</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="ami_3.html#K16">il.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="ami_1.html#T51">AMI_1:51</a></i>;<br/><b>thus </b> <a href="ami_1.html#K6">IC</a> <font color="Maroon">u</font> = 
 <a href="ami_3.html#K16">il.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i>, , <a class="txt" href="scm_comp.html#E7:30_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="scm_comp.html#T2" target="_self">Th2</a>, <a class="ref" href="scm_1.html#T22">SCM_1:22</a></i>
<br/>.= 
 <a href="ami_3.html#K16">il.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>

;<br/><b>thus </b><font color="Maroon">u</font> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> <font color="Maroon">aux</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">u2</font> <a href="ami_3.html#K2">.</a> <span class="p2">(<span class="default"><a href="ami_3.html#K15">dl.</a> <font color="Maroon">aux</font></span>)</span></span>)</span> <a href="int_1.html#K5">div</a> <span class="p1">(<span class="default"><font color="Maroon">u2</font> <a href="ami_3.html#K2">.</a> <span class="p2">(<span class="default"><a href="ami_3.html#K15">dl.</a> <span class="p3">(<span class="default"><font color="Maroon">aux</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i>, , <a class="txt" href="scm_comp.html#E7:30_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="scm_comp.html#T2" target="_self">Th2</a>, <a class="ref" href="scm_1.html#T22">SCM_1:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">u1</font> <a href="ami_3.html#K2">.</a> <span class="p2">(<span class="default"><a href="ami_3.html#K15">dl.</a> <font color="Maroon">aux</font></span>)</span></span>)</span> <a href="int_1.html#K5">div</a> <span class="p1">(<span class="default"><font color="Maroon">tr</font> <a href="scm_comp.html#K13" target="_self">@</a> <font color="Maroon">u1</font></span>)</span>
<b>by </b><i>, <a class="txt" href="scm_comp.html#E1:30_2"><i><font color="Green">E40</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">nt</font> <a href="scm_comp.html#K2" target="_self">-tree</a> <font color="Maroon">tl</font>,<font color="Maroon">tr</font></span>)</span> <a href="scm_comp.html#K13" target="_self">@</a> <font color="Maroon">s</font>
<b>by </b><i>, , <a class="txt" href="scm_comp.html#E2:30"><i><font color="Green">E39</font></i></a>, <a class="txt" href="scm_comp.html#E3:30_2"><i><font color="Green">E42</font></i></a>, </i>
;<br/><b>let </b><font color="Maroon">dn</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E14:30_2_2_4"/><i><font color="Green">E172</font></i>: 
<font color="Maroon">dn</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">aux</font>
 ;<br/><a NAME="E15:30_2_2_4"/><b>then </b><i><font color="Green">E173</font></i>: 
 <a href="ami_3.html#K15">dl.</a> <font color="Maroon">dn</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="ami_3.html#K15">dl.</a> <font color="Maroon">aux</font>
 <b>by </b><i><a class="ref" href="ami_3.html#T52">AMI_3:52</a></i>;<br/><a NAME="E16:30_2_2_4"/><i><font color="Green">E174</font></i>: 
<font color="Maroon">dn</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">aux</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E17:30_2_2_4"/><b>then </b>
 <a href="ami_3.html#K15">dl.</a> <font color="Maroon">dn</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="ami_3.html#K15">dl.</a> <span class="p1">(<span class="default"><font color="Maroon">aux</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="ami_3.html#T52">AMI_3:52</a></i>;<br/><a NAME="E18:30_2_2_4"/><b>then </b><i><font color="Green">E175</font></i>: 
<font color="Maroon">u</font> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> <font color="Maroon">dn</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">u2</font> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> <font color="Maroon">dn</font></span>)</span>
 <b>by </b><i>, , <a class="txt" href="scm_comp.html#E7:30_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="scm_comp.html#T2" target="_self">Th2</a>, , <a class="ref" href="scm_1.html#T22">SCM_1:22</a></i>;<br/><a NAME="E19:30_2_2_4"/>
<font color="Maroon">u1</font> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> <font color="Maroon">dn</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">u2</font> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> <font color="Maroon">dn</font></span>)</span>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E20:30_2_2_4"/>
<font color="Maroon">s</font> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> <font color="Maroon">dn</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> <font color="Maroon">dn</font></span>)</span>
 <b>by </b><i>, , <a class="txt" href="scm_comp.html#E3"><i><font color="Green">Lemma49</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">nt</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default">0,4</span><a href="tarski.html#K4">]</a></span>
 ;<br/><div class="add"><a NAME="E2:30_2_2_5"/><b>then </b><i><font color="Green">E176</font></i>: 
(  <a href="scm_comp.html#K14" target="_self">Selfwork</a> <font color="Maroon">nt</font>,<font color="Maroon">aux</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="lang1.html#K4">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="ami_3.html#K7">Divide</a> <span class="p3">(<span class="default"><a href="ami_3.html#K15">dl.</a> <font color="Maroon">aux</font></span>)</span>,<span class="p3">(<span class="default"><a href="ami_3.html#K15">dl.</a> <span class="p4">(<span class="default"><font color="Maroon">aux</font> <a href="nat_1.html#K1">+</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#K15">dl.</a> <font color="Maroon">aux</font></span>)</span> <a href="ami_3.html#K3">:=</a> <span class="p3">(<span class="default"><a href="ami_3.html#K15">dl.</a> <span class="p4">(<span class="default"><font color="Maroon">aux</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="lang1.html#K4">*&gt;</a></span> &amp; <font color="Maroon">nt</font> <a href="scm_comp.html#K10" target="_self">-Meaning_on</a> <span class="p1">(<span class="default"><font color="Maroon">tl</font> <a href="scm_comp.html#K13" target="_self">@</a> <font color="Maroon">s</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">tr</font> <a href="scm_comp.html#K13" target="_self">@</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">tl</font> <a href="scm_comp.html#K13" target="_self">@</a> <font color="Maroon">s</font></span>)</span> <a href="int_1.html#K6">mod</a> <span class="p1">(<span class="default"><font color="Maroon">tr</font> <a href="scm_comp.html#K13" target="_self">@</a> <font color="Maroon">s</font></span>)</span> )
 <b>by </b><i>, </i>;<br/><b>set </b><font color="Maroon">i</font> = <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><b>set </b><font color="Maroon">u</font> = <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><b>take </b><font color="Maroon">k</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1;<br/><b>take </b><font color="Maroon">v</font> = <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><b>thus </b><a NAME="E3:30_2_2_5"/>
<font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><a NAME="E4:30_2_2_5"/><i><font color="Green">E178</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K14" target="_self">Selfwork</a> <font color="Maroon">nt</font>,<font color="Maroon">aux</font></span>)</span> <a href="hidden.html#R1">=</a> 2
 <b>by </b><i><a class="txt" href="scm_comp.html#E3:30_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/><a NAME="E5:30_2_2_5"/><b>then </b><i><font color="Green">E179</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K15" target="_self">SCM-Compile</a> <span class="p2">(<span class="default"><font color="Maroon">nt</font> <a href="scm_comp.html#K2" target="_self">-tree</a> <font color="Maroon">tl</font>,<font color="Maroon">tr</font></span>)</span>,<font color="Maroon">aux</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="txt" href="scm_comp.html#E2:30_2"><i><font color="Green">E41</font></i></a>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/><b>hence </b><a NAME="E6:30_2_2_5"/>
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K15" target="_self">SCM-Compile</a> <span class="p2">(<span class="default"><font color="Maroon">nt</font> <a href="scm_comp.html#K2" target="_self">-tree</a> <font color="Maroon">tl</font>,<font color="Maroon">tr</font></span>)</span>,<font color="Maroon">aux</font></span>)</span>
 ;<br/><a NAME="E7:30_2_2_5"/><i><font color="Green">E180</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K14" target="_self">Selfwork</a> <font color="Maroon">nt</font>,<font color="Maroon">aux</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> 2
 <b>by </b><i><a class="txt" href="scm_comp.html#E4:30_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E8:30_2_2_5"/><i><font color="Green">E181</font></i>: 
( 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 2 &amp; 2 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 2 )
 <b>by </b><i><a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E9:30_2_2_5"/>
( 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 2 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 ;<br/><a NAME="E10:30_2_2_5"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K15" target="_self">SCM-Compile</a> <span class="p2">(<span class="default"><font color="Maroon">nt</font> <a href="scm_comp.html#K2" target="_self">-tree</a> <font color="Maroon">tl</font>,<font color="Maroon">tr</font></span>)</span>,<font color="Maroon">aux</font></span>)</span>
 <b>by </b><i><a class="txt" href="scm_comp.html#E7:30_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><b>then </b><i><font color="Green">E182</font></i>: <font color="Maroon">s</font> <a href="ami_1.html#K13">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K16">il.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="scm_comp.html#K15" target="_self">SCM-Compile</a> <span class="p2">(<span class="default"><font color="Maroon">nt</font> <a href="scm_comp.html#K2" target="_self">-tree</a> <font color="Maroon">tl</font>,<font color="Maroon">tr</font></span>)</span>,<font color="Maroon">aux</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="scm_1.html#D1">SCM_1:def 1</a></i>
<br/>.= 
<span class="p1"><a href="lang1.html#K4">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="ami_3.html#K7">Divide</a> <span class="p3">(<span class="default"><a href="ami_3.html#K15">dl.</a> <font color="Maroon">aux</font></span>)</span>,<span class="p3">(<span class="default"><a href="ami_3.html#K15">dl.</a> <span class="p4">(<span class="default"><font color="Maroon">aux</font> <a href="nat_1.html#K1">+</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#K15">dl.</a> <font color="Maroon">aux</font></span>)</span> <a href="ami_3.html#K3">:=</a> <span class="p3">(<span class="default"><a href="ami_3.html#K15">dl.</a> <span class="p4">(<span class="default"><font color="Maroon">aux</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="lang1.html#K4">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 1
<b>by </b><i>, <a class="txt" href="scm_comp.html#E2:30_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="scm_comp.html#E3:30_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="scm_comp.html#T2" target="_self">Th2</a>, , <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
 <a href="ami_3.html#K7">Divide</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> <font color="Maroon">aux</font></span>)</span>,<span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> <span class="p2">(<span class="default"><font color="Maroon">aux</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
;<br/><a NAME="E12:30_2_2_5"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="scm_comp.html#K15" target="_self">SCM-Compile</a> <span class="p2">(<span class="default"><font color="Maroon">nt</font> <a href="scm_comp.html#K2" target="_self">-tree</a> <font color="Maroon">tl</font>,<font color="Maroon">tr</font></span>)</span>,<font color="Maroon">aux</font></span>)</span>
 <b>by </b><i><a class="txt" href="scm_comp.html#E7:30_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><b>then </b><i><font color="Green">E183</font></i>: <font color="Maroon">s</font> <a href="ami_1.html#K13">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K16">il.</a> <span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="scm_comp.html#K15" target="_self">SCM-Compile</a> <span class="p2">(<span class="default"><font color="Maroon">nt</font> <a href="scm_comp.html#K2" target="_self">-tree</a> <font color="Maroon">tl</font>,<font color="Maroon">tr</font></span>)</span>,<font color="Maroon">aux</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="scm_1.html#D1">SCM_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="scm_comp.html#K15" target="_self">SCM-Compile</a> <span class="p2">(<span class="default"><font color="Maroon">nt</font> <a href="scm_comp.html#K2" target="_self">-tree</a> <font color="Maroon">tl</font>,<font color="Maroon">tr</font></span>)</span>,<font color="Maroon">aux</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>

<br/>.= 
<span class="p1"><a href="lang1.html#K4">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="ami_3.html#K7">Divide</a> <span class="p3">(<span class="default"><a href="ami_3.html#K15">dl.</a> <font color="Maroon">aux</font></span>)</span>,<span class="p3">(<span class="default"><a href="ami_3.html#K15">dl.</a> <span class="p4">(<span class="default"><font color="Maroon">aux</font> <a href="nat_1.html#K1">+</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#K15">dl.</a> <font color="Maroon">aux</font></span>)</span> <a href="ami_3.html#K3">:=</a> <span class="p3">(<span class="default"><a href="ami_3.html#K15">dl.</a> <span class="p4">(<span class="default"><font color="Maroon">aux</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="lang1.html#K4">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 2
<b>by </b><i>, <a class="txt" href="scm_comp.html#E2:30_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="scm_comp.html#E3:30_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="scm_comp.html#T2" target="_self">Th2</a>, , <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> <font color="Maroon">aux</font></span>)</span> <a href="ami_3.html#K3">:=</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> <span class="p2">(<span class="default"><font color="Maroon">aux</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>
;<br/><a NAME="E14:30_2_2_5"/>
<font color="Maroon">aux</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">aux</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><a NAME="E15:30_2_2_5"/><b>then </b><i><font color="Green">E184</font></i>: 
 <a href="ami_3.html#K15">dl.</a> <font color="Maroon">aux</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="ami_3.html#K15">dl.</a> <span class="p1">(<span class="default"><font color="Maroon">aux</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="ami_3.html#T52">AMI_3:52</a></i>;<br/><b>hence </b><i><font color="Green">E185</font></i>:  <a href="ami_1.html#K6">IC</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <font color="Maroon">k</font></span>)</span> = 
 <a href="ami_3.html#K16">il.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i>, , , <a class="ref" href="scm_1.html#T22">SCM_1:22</a></i>
<br/>.= 
 <a href="ami_3.html#K16">il.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span>

;<br/><br/><b>hence </b> <a href="ami_1.html#K6">IC</a> <font color="Maroon">v</font> = 
 <a href="ami_3.html#K16">il.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i>, <a class="ref" href="scm_1.html#T18">SCM_1:18</a></i>
<br/>.= 
 <a href="ami_3.html#K16">il.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>

;<br/><br/><b>thus </b><font color="Maroon">v</font> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> <font color="Maroon">aux</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> <span class="p2">(<span class="default"><font color="Maroon">aux</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="scm_1.html#T18">SCM_1:18</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">u2</font> <a href="ami_3.html#K2">.</a> <span class="p2">(<span class="default"><a href="ami_3.html#K15">dl.</a> <font color="Maroon">aux</font></span>)</span></span>)</span> <a href="int_1.html#K6">mod</a> <span class="p1">(<span class="default"><font color="Maroon">u2</font> <a href="ami_3.html#K2">.</a> <span class="p2">(<span class="default"><a href="ami_3.html#K15">dl.</a> <span class="p3">(<span class="default"><font color="Maroon">aux</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i>, , , <a class="txt" href="scm_comp.html#E3"><i><font color="Green">Lemma49</font></i></a>, <a class="ref" href="scm_1.html#T22">SCM_1:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">u1</font> <a href="ami_3.html#K2">.</a> <span class="p2">(<span class="default"><a href="ami_3.html#K15">dl.</a> <font color="Maroon">aux</font></span>)</span></span>)</span> <a href="int_1.html#K6">mod</a> <span class="p1">(<span class="default"><font color="Maroon">tr</font> <a href="scm_comp.html#K13" target="_self">@</a> <font color="Maroon">u1</font></span>)</span>
<b>by </b><i>, <a class="txt" href="scm_comp.html#E1:30_2"><i><font color="Green">E40</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">nt</font> <a href="scm_comp.html#K2" target="_self">-tree</a> <font color="Maroon">tl</font>,<font color="Maroon">tr</font></span>)</span> <a href="scm_comp.html#K13" target="_self">@</a> <font color="Maroon">s</font>
<b>by </b><i>, , <a class="txt" href="scm_comp.html#E2:30"><i><font color="Green">E39</font></i></a>, <a class="txt" href="scm_comp.html#E3:30_2"><i><font color="Green">E42</font></i></a>, </i>
;<br/><b>let </b><font color="Maroon">dn</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E19:30_2_2_5"/><i><font color="Green">E186</font></i>: 
<font color="Maroon">dn</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">aux</font>
 ;<br/><a NAME="E20:30_2_2_5"/><b>then </b><i><font color="Green">E187</font></i>: 
 <a href="ami_3.html#K15">dl.</a> <font color="Maroon">dn</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="ami_3.html#K15">dl.</a> <font color="Maroon">aux</font>
 <b>by </b><i><a class="ref" href="ami_3.html#T52">AMI_3:52</a></i>;<br/><a NAME="E21:30_2_2_5"/><i><font color="Green">E188</font></i>: 
<font color="Maroon">dn</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">aux</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="scm_comp.html#E4"><i><font color="Green">Lemma51</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E22:30_2_2_5"/><b>then </b>
 <a href="ami_3.html#K15">dl.</a> <font color="Maroon">dn</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="ami_3.html#K15">dl.</a> <span class="p1">(<span class="default"><font color="Maroon">aux</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="ami_3.html#T52">AMI_3:52</a></i>;<br/><a NAME="E23:30_2_2_5"/><b>then </b><i><font color="Green">E189</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> <font color="Maroon">dn</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">u2</font> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> <font color="Maroon">dn</font></span>)</span>
 <b>by </b><i>, , , <a class="txt" href="scm_comp.html#E3"><i><font color="Green">Lemma49</font></i></a>, <a class="ref" href="scm_comp.html#D1" target="_self">Def1</a>, <a class="ref" href="scm_1.html#T22">SCM_1:22</a></i>;<br/><a NAME="E24:30_2_2_5"/>
<font color="Maroon">u1</font> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> <font color="Maroon">dn</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">u2</font> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> <font color="Maroon">dn</font></span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E25:30_2_2_5"/><b>then </b>
<font color="Maroon">s</font> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> <font color="Maroon">dn</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="ami_1.html#K10">Computation</a> <font color="Maroon">s</font></span>)</span> <a href="ami_1.html#K11">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> <font color="Maroon">dn</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="scm_comp.html#E4"><i><font color="Green">Lemma51</font></i></a>, </i>;<br/><b>hence </b><a NAME="E26:30_2_2_5"/>
<font color="Maroon">s</font> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> <font color="Maroon">dn</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font> <a href="ami_3.html#K2">.</a> <span class="p1">(<span class="default"><a href="ami_3.html#K15">dl.</a> <font color="Maroon">dn</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="scm_comp.html#D1" target="_self">Def1</a>, <a class="ref" href="scm_1.html#T18">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">term</font> being  <a href="scm_comp.html#NM1">bin-term</a> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">term</font>]
 <b>from </b><i><a class="ref" href="bintree1.html#S2">BINTREE1:sch 2</a>(, );</i><br/>


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