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

<b>let </b><font color="Maroon" title="c1">T</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="orders_2.html#V3" title="ORDERS_2:attr.3">reflexive</a> <a href="orders_2.html#V4" title="ORDERS_2:attr.4">transitive</a> non <a href="abcmiz_0.html#V4" title="ABCMIZ_0:attr.4">void</a>  <a href="abcmiz_0.html#L2" title="ABCMIZ_0:struct.2">TA-structure</a> ;<br/><b>let </b><font color="Maroon" title="c2">t</font> be   <a href="abcmiz_0.html#NM1" title="ABCMIZ_0:NM.1">type</a> of <font color="Maroon" title="c1">T</font>;<br/><b>let </b><font color="Maroon" title="c3">v1</font>, <font color="Maroon" title="c4">v2</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="abcmiz_0.html#U1" title="ABCMIZ_0:sel.1">adjectives</a> of <font color="Maroon" title="c1">T</font>;<br/>





<a NAME="E1:69"/><i><font color="Green" title="E37">A1</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> <font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c2">t</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">v1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> <font color="Maroon" title="c4">v2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">v1</font> <a href="abcmiz_0.html#K8" title="ABCMIZ_0:func.8">ast</a> <font color="Maroon" title="c2">t</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">v2</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 
<b>by </b><i><a class="ref" href="abcmiz_0.html#D19" target="_self" title="ABCMIZ_0:def.19">Def19</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">tt</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>, <font color="Maroon" title="c6">q</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E3:69"/><i><font color="Green" title="E38">A2</font></i>: 
 <a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> <font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c2">t</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c5">tt</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c6">q</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="hilbert2.html#T4" title="HILBERT2:th.4">HILBERT2:4</a></i>;<br/>
<a NAME="E4:69"/><i><font color="Green" title="E39">A3</font></i>: 
<span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> <font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c2">t</font></span>)</span> <a href="rewrite1.html#K1" title="REWRITE1:func.1">$^</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> <font color="Maroon" title="c4">v2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">v1</font> <a href="abcmiz_0.html#K8" title="ABCMIZ_0:func.8">ast</a> <font color="Maroon" title="c2">t</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">tt</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> <font color="Maroon" title="c4">v2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">v1</font> <a href="abcmiz_0.html#K8" title="ABCMIZ_0:func.8">ast</a> <font color="Maroon" title="c2">t</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="abcmiz_0.html#E3:69"><i><font color="Green" title="E38">A2</font></i></a>, <a class="ref" href="rewrite1.html#T2" title="REWRITE1:th.2">REWRITE1:2</a></i>;<br/>
<a NAME="E5:69"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c6">q</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T56" title="FINSEQ_1:th.56">FINSEQ_1:56</a></i>;<br/>
<a NAME="E6:69"/><b>then </b><i><font color="Green" title="E40">A4</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">v1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">tt</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="txt" href="abcmiz_0.html#E1:69"><i><font color="Green" title="E37">A1</font></i></a>, <a class="txt" href="abcmiz_0.html#E3:69"><i><font color="Green" title="E38">A2</font></i></a>, <a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>;<br/>
<b>set </b><font color="Maroon" title="c7">s</font> = <span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> <font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c2">t</font></span>)</span> <a href="rewrite1.html#K1" title="REWRITE1:func.1">$^</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> <font color="Maroon" title="c4">v2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">v1</font> <a href="abcmiz_0.html#K8" title="ABCMIZ_0:func.8">ast</a> <font color="Maroon" title="c2">t</font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c8">p</font> = <font color="Maroon" title="c3">v1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">v2</font>;<br/>
<i><font color="Green" title="E41">A5</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> <font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c2">t</font></span>)</span> <a href="rewrite1.html#K1" title="REWRITE1:func.1">$^</a> <span class="p2">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> <font color="Maroon" title="c4">v2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">v1</font> <a href="abcmiz_0.html#K8" title="ABCMIZ_0:func.8">ast</a> <font color="Maroon" title="c2">t</font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">tt</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">v2</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="abcmiz_0.html#E1:69"><i><font color="Green" title="E37">A1</font></i></a>, <a class="txt" href="abcmiz_0.html#E4:69"><i><font color="Green" title="E39">A3</font></i></a>, <a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">v1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">v2</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
<b>by </b><i><a class="txt" href="abcmiz_0.html#E6:69"><i><font color="Green" title="E40">A4</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">v1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">v2</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
<b>by </b><i><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></i>
;<br/>
<a NAME="E8:69"/><i><font color="Green" title="E42">A6</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> <font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c2">t</font></span>)</span> <a href="rewrite1.html#K1" title="REWRITE1:func.1">$^</a> <span class="p2">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> <font color="Maroon" title="c4">v2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">v1</font> <a href="abcmiz_0.html#K8" title="ABCMIZ_0:func.8">ast</a> <font color="Maroon" title="c2">t</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">t</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_2"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:69_2_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">v1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_2_1_1"><b>suppose </b></a><a NAME="E1:69_2_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><a NAME="E2:69_2_1_1"/><b>then </b>
( <font color="Maroon" title="c3">v1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a>  <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> the <a href="abcmiz_0.html#U1" title="ABCMIZ_0:sel.1">adjectives</a> of <font color="Maroon" title="c1">T</font> &amp; <font color="Maroon" title="c5">tt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 <b>by </b><i><a class="txt" href="abcmiz_0.html#E6:69"><i><font color="Green" title="E40">A4</font></i></a>, <a class="ref" href="finseq_1.html#T25" title="FINSEQ_1:th.25">FINSEQ_1:25</a></i>;<br/><a NAME="E3:69_2_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> <font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c2">t</font></span>)</span> <a href="rewrite1.html#K1" title="REWRITE1:func.1">$^</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> <font color="Maroon" title="c4">v2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">v1</font> <a href="abcmiz_0.html#K8" title="ABCMIZ_0:func.8">ast</a> <font color="Maroon" title="c2">t</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> <font color="Maroon" title="c4">v2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">v1</font> <a href="abcmiz_0.html#K8" title="ABCMIZ_0:func.8">ast</a> <font color="Maroon" title="c2">t</font></span>)</span> &amp; <font color="Maroon" title="c3">v1</font> <a href="abcmiz_0.html#K8" title="ABCMIZ_0:func.8">ast</a> <font color="Maroon" title="c2">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">t</font> )
 <b>by </b><i><a class="txt" href="abcmiz_0.html#E4:69"><i><font color="Green" title="E39">A3</font></i></a>, <a class="ref" href="abcmiz_0.html#T31" target="_self" title="ABCMIZ_0:th.31">Th31</a>, <a class="ref" href="finseq_1.html#T47" title="FINSEQ_1:th.47">FINSEQ_1:47</a></i>;<br/><b>hence </b><a NAME="E4:69_2_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> <font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c2">t</font></span>)</span> <a href="rewrite1.html#K1" title="REWRITE1:func.1">$^</a> <span class="p2">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> <font color="Maroon" title="c4">v2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">v1</font> <a href="abcmiz_0.html#K8" title="ABCMIZ_0:func.8">ast</a> <font color="Maroon" title="c2">t</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">t</font>
 <b>by </b><i><a class="ref" href="abcmiz_0.html#D19" target="_self" title="ABCMIZ_0:def.19">Def19</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_2_1_2"><b>suppose </b></a><a NAME="E1:69_2_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">v1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><a NAME="E2:69_2_1_2"/><b>then </b>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">tt</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 <b>by </b><i><a class="txt" href="abcmiz_0.html#E6:69"><i><font color="Green" title="E40">A4</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:69_2_1_2"/><b>then </b>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">tt</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E4:69_2_1_2"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> <font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c2">t</font></span>)</span> <a href="rewrite1.html#K1" title="REWRITE1:func.1">$^</a> <span class="p2">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> <font color="Maroon" title="c4">v2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">v1</font> <a href="abcmiz_0.html#K8" title="ABCMIZ_0:func.8">ast</a> <font color="Maroon" title="c2">t</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">tt</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 &amp; <font color="Maroon" title="c5">tt</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> <font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c2">t</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 )
 <b>by </b><i><a class="txt" href="abcmiz_0.html#E3:69"><i><font color="Green" title="E38">A2</font></i></a>, <a class="txt" href="abcmiz_0.html#E4:69"><i><font color="Green" title="E39">A3</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>;<br/><b>hence </b><a NAME="E5:69_2_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> <font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c2">t</font></span>)</span> <a href="rewrite1.html#K1" title="REWRITE1:func.1">$^</a> <span class="p2">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> <font color="Maroon" title="c4">v2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">v1</font> <a href="abcmiz_0.html#K8" title="ABCMIZ_0:func.8">ast</a> <font color="Maroon" title="c2">t</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">t</font>
 <b>by </b><i><a class="ref" href="abcmiz_0.html#D19" target="_self" title="ABCMIZ_0:def.19">Def19</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c10">a</font> be   <a href="abcmiz_0.html#NM2" title="ABCMIZ_0:NM.2">adjective</a> of <font color="Maroon" title="c1">T</font>;<br/><b>let </b><font color="Maroon" title="c11">t'</font> be   <a href="abcmiz_0.html#NM1" title="ABCMIZ_0:NM.1">type</a> of <font color="Maroon" title="c1">T</font>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:69_3"/><i><font color="Green" title="E43">A7</font></i>: 
<font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">v1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">v2</font></span>)</span>
 <b>and </b><br/><a NAME="E2:69_3"/><i><font color="Green" title="E44">A8</font></i>: 
( <font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">v1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">v2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> &amp; <font color="Maroon" title="c11">t'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> <font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c2">t</font></span>)</span> <a href="rewrite1.html#K1" title="REWRITE1:func.1">$^</a> <span class="p2">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> <font color="Maroon" title="c4">v2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">v1</font> <a href="abcmiz_0.html#K8" title="ABCMIZ_0:func.8">ast</a> <font color="Maroon" title="c2">t</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> )
 ;<br/><a NAME="E3:69_3"/><i><font color="Green" title="E45">A9</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">i</font> &amp; <font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">v1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">v2</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">v1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">v2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">v1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">v2</font></span>)</span> )
 <b>by </b><i><a class="txt" href="abcmiz_0.html#E1:69_3"><i><font color="Green" title="E43">A7</font></i></a>, <a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E4:69_3"/><i><font color="Green" title="E46">A10</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> <font color="Maroon" title="c4">v2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">v1</font> <a href="abcmiz_0.html#K8" title="ABCMIZ_0:func.8">ast</a> <font color="Maroon" title="c2">t</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">v2</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="abcmiz_0.html#D19" target="_self" title="ABCMIZ_0:def.19">Def19</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:69_3_1"/>
( <font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">v1</font> or <font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">v1</font> or <font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">v1</font> )
 </span><b>by </b><i><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_3_1_1"><b>suppose </b></a><a NAME="E1:69_3_1_1"/>
<font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">v1</font>
 ;<br/><div class="add"><a NAME="E2:69_3_1_1"/><b>then </b>
( <font color="Maroon" title="c9">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">v1</font> &amp; <font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">v1</font> &amp; <font color="Maroon" title="c9">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 )
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:69_3_1_1"/><b>then </b><i><font color="Green" title="E47">A11</font></i>: 
( <font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">tt</font> &amp; <font color="Maroon" title="c9">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">tt</font> &amp; <font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">v1</font> )
 <b>by </b><i><a class="txt" href="abcmiz_0.html#E6:69"><i><font color="Green" title="E40">A4</font></i></a>, <a class="txt" href="abcmiz_0.html#E3:69_3"><i><font color="Green" title="E45">A9</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E4:69_3_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> <font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c2">t</font></span>)</span> <a href="rewrite1.html#K1" title="REWRITE1:func.1">$^</a> <span class="p2">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> <font color="Maroon" title="c4">v2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">v1</font> <a href="abcmiz_0.html#K8" title="ABCMIZ_0:func.8">ast</a> <font color="Maroon" title="c2">t</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">tt</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> &amp; <font color="Maroon" title="c5">tt</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> <font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c2">t</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> <font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c2">t</font></span>)</span> <a href="rewrite1.html#K1" title="REWRITE1:func.1">$^</a> <span class="p2">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> <font color="Maroon" title="c4">v2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">v1</font> <a href="abcmiz_0.html#K8" title="ABCMIZ_0:func.8">ast</a> <font color="Maroon" title="c2">t</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">tt</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">v1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">v2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> &amp; <font color="Maroon" title="c5">tt</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> <font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c2">t</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="abcmiz_0.html#E3:69"><i><font color="Green" title="E38">A2</font></i></a>, <a class="txt" href="abcmiz_0.html#E4:69"><i><font color="Green" title="E39">A3</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>;<br/><b>hence </b><a NAME="E5:69_3_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> <font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c2">t</font></span>)</span> <a href="rewrite1.html#K1" title="REWRITE1:func.1">$^</a> <span class="p2">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> <font color="Maroon" title="c4">v2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">v1</font> <a href="abcmiz_0.html#K8" title="ABCMIZ_0:func.8">ast</a> <font color="Maroon" title="c2">t</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">a</font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">ast</a> <font color="Maroon" title="c11">t'</font>
 <b>by </b><i><a class="txt" href="abcmiz_0.html#E2:69_3"><i><font color="Green" title="E44">A8</font></i></a>, <a class="txt" href="abcmiz_0.html#E3:69_3_1_1"><i><font color="Green" title="E47">A11</font></i></a>, <a class="ref" href="abcmiz_0.html#D19" target="_self" title="ABCMIZ_0:def.19">Def19</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_3_1_2"><b>suppose </b></a><a NAME="E1:69_3_1_2"/><i><font color="Green" title="E47">A12</font></i>: 
<font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">v1</font>
 ;<br/><div class="add"><a NAME="E2:69_3_1_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> <font color="Maroon" title="c4">v2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">v1</font> <a href="abcmiz_0.html#K8" title="ABCMIZ_0:func.8">ast</a> <font color="Maroon" title="c2">t</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="abcmiz_0.html#E4:69_3"><i><font color="Green" title="E46">A10</font></i></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/><a NAME="E3:69_3_1_2"/><b>then </b><i><font color="Green" title="E48">A13</font></i>: 
( <font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">tt</font> &amp; <font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">v1</font> &amp; 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> <font color="Maroon" title="c4">v2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">v1</font> <a href="abcmiz_0.html#K8" title="ABCMIZ_0:func.8">ast</a> <font color="Maroon" title="c2">t</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="abcmiz_0.html#E6:69"><i><font color="Green" title="E40">A4</font></i></a>, <a class="txt" href="abcmiz_0.html#E3:69_3"><i><font color="Green" title="E45">A9</font></i></a>, <a class="txt" href="abcmiz_0.html#E1:69_3_1_2"><i><font color="Green" title="E47">A12</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E4:69_3_1_2"/><b>then </b><i><font color="Green" title="E49">A14</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> <font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c2">t</font></span>)</span> <a href="rewrite1.html#K1" title="REWRITE1:func.1">$^</a> <span class="p2">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> <font color="Maroon" title="c4">v2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">v1</font> <a href="abcmiz_0.html#K8" title="ABCMIZ_0:func.8">ast</a> <font color="Maroon" title="c2">t</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">tt</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> &amp; <font color="Maroon" title="c5">tt</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> <font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c2">t</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> <font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c2">t</font></span>)</span> <a href="rewrite1.html#K1" title="REWRITE1:func.1">$^</a> <span class="p2">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> <font color="Maroon" title="c4">v2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">v1</font> <a href="abcmiz_0.html#K8" title="ABCMIZ_0:func.8">ast</a> <font color="Maroon" title="c2">t</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> <font color="Maroon" title="c4">v2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">v1</font> <a href="abcmiz_0.html#K8" title="ABCMIZ_0:func.8">ast</a> <font color="Maroon" title="c2">t</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">v1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">v2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> )
 <b>by </b><i><a class="txt" href="abcmiz_0.html#E3:69"><i><font color="Green" title="E38">A2</font></i></a>, <a class="txt" href="abcmiz_0.html#E4:69"><i><font color="Green" title="E39">A3</font></i></a>, <a class="txt" href="abcmiz_0.html#E6:69"><i><font color="Green" title="E40">A4</font></i></a>, <a class="txt" href="abcmiz_0.html#E1:69_3_1_2"><i><font color="Green" title="E47">A12</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>;<br/><a NAME="E5:69_3_1_2"/><b>then </b>
<font color="Maroon" title="c10">a</font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">ast</a> <font color="Maroon" title="c11">t'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v1</font> <a href="abcmiz_0.html#K8" title="ABCMIZ_0:func.8">ast</a> <font color="Maroon" title="c2">t</font>
 <b>by </b><i><a class="txt" href="abcmiz_0.html#E2:69_3"><i><font color="Green" title="E44">A8</font></i></a>, <a class="txt" href="abcmiz_0.html#E1:69_3_1_2"><i><font color="Green" title="E47">A12</font></i></a>, <a class="txt" href="abcmiz_0.html#E3:69_3_1_2"><i><font color="Green" title="E48">A13</font></i></a>, <a class="ref" href="abcmiz_0.html#D19" target="_self" title="ABCMIZ_0:def.19">Def19</a></i>;<br/><b>hence </b><a NAME="E6:69_3_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> <font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c2">t</font></span>)</span> <a href="rewrite1.html#K1" title="REWRITE1:func.1">$^</a> <span class="p2">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> <font color="Maroon" title="c4">v2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">v1</font> <a href="abcmiz_0.html#K8" title="ABCMIZ_0:func.8">ast</a> <font color="Maroon" title="c2">t</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">a</font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">ast</a> <font color="Maroon" title="c11">t'</font>
 <b>by </b><i><a class="txt" href="abcmiz_0.html#E4:69_3_1_2"><i><font color="Green" title="E49">A14</font></i></a>, <a class="ref" href="abcmiz_0.html#D19" target="_self" title="ABCMIZ_0:def.19">Def19</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_3_1_3"><b>suppose </b></a><a NAME="E1:69_3_1_3"/>
<font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">v1</font>
 ;<br/><div class="add"><a NAME="E2:69_3_1_3"/><b>then </b>
<font color="Maroon" title="c9">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">v1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>then consider </b><font color="Maroon" title="c12">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E4:69_3_1_3"/><i><font color="Green" title="E47">A15</font></i>: 
<font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">v1</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c12">j</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></i>;<br/><a NAME="E5:69_3_1_3"/><i><font color="Green" title="E48">A16</font></i>: 
( <font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">v1</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">v1</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">v1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="abcmiz_0.html#E4:69_3_1_3"><i><font color="Green" title="E47">A15</font></i></a></i>;<br/><a NAME="E6:69_3_1_3"/><b>then </b><i><font color="Green" title="E49">A17</font></i>: 
( 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c12">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c12">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">v2</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 <b>by </b><i><a class="txt" href="abcmiz_0.html#E3:69_3"><i><font color="Green" title="E45">A9</font></i></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E7:69_3_1_3"/><b>then </b>
( <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c12">j</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">v2</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c12">j</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp; <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c12">j</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">v2</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><a NAME="E8:69_3_1_3"/><b>then </b><i><font color="Green" title="E50">A18</font></i>: 
( <font color="Maroon" title="c12">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">v2</font> &amp; <font color="Maroon" title="c12">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> <font color="Maroon" title="c4">v2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">v1</font> <a href="abcmiz_0.html#K8" title="ABCMIZ_0:func.8">ast</a> <font color="Maroon" title="c2">t</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> <font color="Maroon" title="c4">v2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">v1</font> <a href="abcmiz_0.html#K8" title="ABCMIZ_0:func.8">ast</a> <font color="Maroon" title="c2">t</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="abcmiz_0.html#E4:69_3"><i><font color="Green" title="E46">A10</font></i></a>, <a class="txt" href="abcmiz_0.html#E6:69_3_1_3"><i><font color="Green" title="E49">A17</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E9:69_3_1_3"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> <font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c2">t</font></span>)</span> <a href="rewrite1.html#K1" title="REWRITE1:func.1">$^</a> <span class="p2">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> <font color="Maroon" title="c4">v2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">v1</font> <a href="abcmiz_0.html#K8" title="ABCMIZ_0:func.8">ast</a> <font color="Maroon" title="c2">t</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> <font color="Maroon" title="c4">v2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">v1</font> <a href="abcmiz_0.html#K8" title="ABCMIZ_0:func.8">ast</a> <font color="Maroon" title="c2">t</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">v1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">v2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> <font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c2">t</font></span>)</span> <a href="rewrite1.html#K1" title="REWRITE1:func.1">$^</a> <span class="p2">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> <font color="Maroon" title="c4">v2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">v1</font> <a href="abcmiz_0.html#K8" title="ABCMIZ_0:func.8">ast</a> <font color="Maroon" title="c2">t</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> <font color="Maroon" title="c4">v2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">v1</font> <a href="abcmiz_0.html#K8" title="ABCMIZ_0:func.8">ast</a> <font color="Maroon" title="c2">t</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="abcmiz_0.html#E4:69"><i><font color="Green" title="E39">A3</font></i></a>, <a class="txt" href="abcmiz_0.html#E6:69"><i><font color="Green" title="E40">A4</font></i></a>, <a class="txt" href="abcmiz_0.html#E5:69_3_1_3"><i><font color="Green" title="E48">A16</font></i></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></i>;<br/><b>hence </b><a NAME="E10:69_3_1_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> <font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c2">t</font></span>)</span> <a href="rewrite1.html#K1" title="REWRITE1:func.1">$^</a> <span class="p2">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> <font color="Maroon" title="c4">v2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">v1</font> <a href="abcmiz_0.html#K8" title="ABCMIZ_0:func.8">ast</a> <font color="Maroon" title="c2">t</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">a</font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">ast</a> <font color="Maroon" title="c11">t'</font>
 <b>by </b><i><a class="txt" href="abcmiz_0.html#E2:69_3"><i><font color="Green" title="E44">A8</font></i></a>, <a class="txt" href="abcmiz_0.html#E8:69_3_1_3"><i><font color="Green" title="E50">A18</font></i></a>, <a class="ref" href="abcmiz_0.html#D19" target="_self" title="ABCMIZ_0:def.19">Def19</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E10:69"/>
 <a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">v1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c4">v2</font></span>)</span>,<font color="Maroon" title="c2">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> <font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c2">t</font></span>)</span> <a href="rewrite1.html#K1" title="REWRITE1:func.1">$^</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> <font color="Maroon" title="c4">v2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">v1</font> <a href="abcmiz_0.html#K8" title="ABCMIZ_0:func.8">ast</a> <font color="Maroon" title="c2">t</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="abcmiz_0.html#E4:69"><i><font color="Green" title="E39">A3</font></i></a>, <a class="txt" href="abcmiz_0.html#E7:69"><i><font color="Green" title="E41">A5</font></i></a>, <a class="txt" href="abcmiz_0.html#E8:69"><i><font color="Green" title="E42">A6</font></i></a>, <a class="ref" href="abcmiz_0.html#D19" target="_self" title="ABCMIZ_0:def.19">Def19</a></i>;<br/>


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