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

<b>let </b><font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#V2">reflexive</a> <a href="orders_2.html#V3">transitive</a> non <a href="abcmiz_0.html#V4" target="_self">void</a>  <a href="abcmiz_0.html#L2" target="_self">TA-structure</a> ;<br/><b>let </b><font color="Maroon">t</font> be   <a href="abcmiz_0.html#NM1">type</a> of <font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">v1</font> be    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="abcmiz_0.html#U1" target="_self">adjectives</a> of <font color="Maroon">T</font>, <font color="Maroon">v2</font> be    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="abcmiz_0.html#U1" target="_self">adjectives</a> of <font color="Maroon">T</font>;<br/>





<a NAME="E1:69"/><i><font color="Green">E32</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <font color="Maroon">v1</font>,<font color="Maroon">t</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">v1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <font color="Maroon">v2</font>,<span class="p2">(<span class="default"><font color="Maroon">v1</font> <a href="abcmiz_0.html#K8" target="_self">ast</a> <font color="Maroon">t</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">v2</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">tt</font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>, <font color="Maroon">q</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E3:69"/><i><font color="Green">E35</font></i>: 
 <a href="abcmiz_0.html#K7" target="_self">apply</a> <font color="Maroon">v1</font>,<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">tt</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">q</font></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="hilbert2.html#T4">HILBERT2:4</a></i>;<br/>
<a NAME="E4:69"/><i><font color="Green">E43</font></i>: 
<span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <font color="Maroon">v1</font>,<font color="Maroon">t</font></span>)</span> <a href="rewrite1.html#K1">$^</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <font color="Maroon">v2</font>,<span class="p2">(<span class="default"><font color="Maroon">v1</font> <a href="abcmiz_0.html#K8" target="_self">ast</a> <font color="Maroon">t</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">tt</font> <a href="finseq_1.html#K8">^</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <font color="Maroon">v2</font>,<span class="p2">(<span class="default"><font color="Maroon">v1</font> <a href="abcmiz_0.html#K8" target="_self">ast</a> <font color="Maroon">t</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="rewrite1.html#T2">REWRITE1:2</a></i>;<br/>
<a NAME="E5:69"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">q</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/>
<a NAME="E6:69"/><b>then </b><i><font color="Green">E46</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">v1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">tt</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>;<br/>
<a NAME="E7:69"/><b>then </b><i><font color="Green">E47</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">tt</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">v1</font>
 
;<br/>
<b>set </b><font color="Maroon">s</font> = <span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <font color="Maroon">v1</font>,<font color="Maroon">t</font></span>)</span> <a href="rewrite1.html#K1">$^</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <font color="Maroon">v2</font>,<span class="p2">(<span class="default"><font color="Maroon">v1</font> <a href="abcmiz_0.html#K8" target="_self">ast</a> <font color="Maroon">t</font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon">p</font> = <font color="Maroon">v1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">v2</font>;<br/>
<i><font color="Green">E48</font></i>:  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <font color="Maroon">v1</font>,<font color="Maroon">t</font></span>)</span> <a href="rewrite1.html#K1">$^</a> <span class="p2">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <font color="Maroon">v2</font>,<span class="p3">(<span class="default"><font color="Maroon">v1</font> <a href="abcmiz_0.html#K8" target="_self">ast</a> <font color="Maroon">t</font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">tt</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">v2</font></span>)</span> <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/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">v1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">v2</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">v1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">v2</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
;<br/>
<a NAME="E9:69"/><i><font color="Green">E49</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <font color="Maroon">v1</font>,<font color="Maroon">t</font></span>)</span> <a href="rewrite1.html#K1">$^</a> <span class="p2">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <font color="Maroon">v2</font>,<span class="p3">(<span class="default"><font color="Maroon">v1</font> <a href="abcmiz_0.html#K8" target="_self">ast</a> <font color="Maroon">t</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">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">len</a> <font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> 0 or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">v1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </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">len</a> <font color="Maroon">v1</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:69_2_1_1"/><b>then </b>
( <font color="Maroon">v1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K6">&lt;*&gt;</a> the <a href="abcmiz_0.html#U1" target="_self">adjectives</a> of <font color="Maroon">T</font> &amp; <font color="Maroon">tt</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T25">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" target="_self">apply</a> <font color="Maroon">v1</font>,<font color="Maroon">t</font></span>)</span> <a href="rewrite1.html#K1">$^</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <font color="Maroon">v2</font>,<span class="p2">(<span class="default"><font color="Maroon">v1</font> <a href="abcmiz_0.html#K8" target="_self">ast</a> <font color="Maroon">t</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="abcmiz_0.html#K7" target="_self">apply</a> <font color="Maroon">v2</font>,<span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="abcmiz_0.html#K8" target="_self">ast</a> <font color="Maroon">t</font></span>)</span> &amp; <font color="Maroon">v1</font> <a href="abcmiz_0.html#K8" target="_self">ast</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">t</font> )
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>;<br/><b>hence </b><a NAME="E4:69_2_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <font color="Maroon">v1</font>,<font color="Maroon">t</font></span>)</span> <a href="rewrite1.html#K1">$^</a> <span class="p2">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <font color="Maroon">v2</font>,<span class="p3">(<span class="default"><font color="Maroon">v1</font> <a href="abcmiz_0.html#K8" target="_self">ast</a> <font color="Maroon">t</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">t</font>
 <b>by </b><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">len</a> <font color="Maroon">v1</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:69_2_1_2"/><b>then </b>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">tt</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1 &amp; 0 <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> 1 &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:69_2_1_2"/><b>then </b>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">tt</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">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" target="_self">apply</a> <font color="Maroon">v1</font>,<font color="Maroon">t</font></span>)</span> <a href="rewrite1.html#K1">$^</a> <span class="p2">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <font color="Maroon">v2</font>,<span class="p3">(<span class="default"><font color="Maroon">v1</font> <a href="abcmiz_0.html#K8" target="_self">ast</a> <font color="Maroon">t</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">tt</font> <a href="funct_1.html#K1">.</a> 1 &amp; <font color="Maroon">tt</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <font color="Maroon">v1</font>,<font color="Maroon">t</font></span>)</span> <a href="funct_1.html#K1">.</a> 1 )
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/><b>hence </b><a NAME="E5:69_2_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <font color="Maroon">v1</font>,<font color="Maroon">t</font></span>)</span> <a href="rewrite1.html#K1">$^</a> <span class="p2">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <font color="Maroon">v2</font>,<span class="p3">(<span class="default"><font color="Maroon">v1</font> <a href="abcmiz_0.html#K8" target="_self">ast</a> <font color="Maroon">t</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">t</font>
 <b>by </b><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">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">a</font> be   <a href="abcmiz_0.html#NM2">adjective</a> of <font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">t'</font> be   <a href="abcmiz_0.html#NM1">type</a> of <font color="Maroon">T</font>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:69_3"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">v2</font></span>)</span>
 <b>and </b><br/><a NAME="E2:69_3"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">v2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">t'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <font color="Maroon">v1</font>,<font color="Maroon">t</font></span>)</span> <a href="rewrite1.html#K1">$^</a> <span class="p2">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <font color="Maroon">v2</font>,<span class="p3">(<span class="default"><font color="Maroon">v1</font> <a href="abcmiz_0.html#K8" target="_self">ast</a> <font color="Maroon">t</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> )
 ;<br/><a NAME="E3:69_3"/><i><font color="Green">E52</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">v2</font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">v2</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">v1</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">v2</font></span>)</span> )
 <b>by </b><i><a class="ref" href="abcmiz_0.html#D2" target="_self">Def2</a>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E4:69_3"/><i><font color="Green">E53</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <font color="Maroon">v2</font>,<span class="p2">(<span class="default"><font color="Maroon">v1</font> <a href="abcmiz_0.html#K8" target="_self">ast</a> <font color="Maroon">t</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">v2</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><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">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">v1</font> or <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">v1</font> or <font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">v1</font> )
 </span><b>by </b><i><a class="ref" href="xxreal_0.html#T1">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">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">v1</font>
 ;<br/><div class="add"><a NAME="E2:69_3_1_1"/><b>then </b>
( <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">v1</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">v1</font> &amp; <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 )
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:69_3_1_1"/><b>then </b><i><font color="Green">E110</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">tt</font> &amp; <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">tt</font> &amp; <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">v1</font> )
 <b>by </b><i>, <a class="txt" href="abcmiz_0.html#E6:69"><i><font color="Green">E46</font></i></a>, <a class="ref" href="finseq_3.html#T27">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" target="_self">apply</a> <font color="Maroon">v1</font>,<font color="Maroon">t</font></span>)</span> <a href="rewrite1.html#K1">$^</a> <span class="p2">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <font color="Maroon">v2</font>,<span class="p3">(<span class="default"><font color="Maroon">v1</font> <a href="abcmiz_0.html#K8" target="_self">ast</a> <font color="Maroon">t</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">tt</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">tt</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <font color="Maroon">v1</font>,<font color="Maroon">t</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <font color="Maroon">v1</font>,<font color="Maroon">t</font></span>)</span> <a href="rewrite1.html#K1">$^</a> <span class="p2">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <font color="Maroon">v2</font>,<span class="p3">(<span class="default"><font color="Maroon">v1</font> <a href="abcmiz_0.html#K8" target="_self">ast</a> <font color="Maroon">t</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">tt</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">v2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">tt</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <font color="Maroon">v1</font>,<font color="Maroon">t</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i>, , <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/><b>hence </b><a NAME="E5:69_3_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <font color="Maroon">v1</font>,<font color="Maroon">t</font></span>)</span> <a href="rewrite1.html#K1">$^</a> <span class="p2">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <font color="Maroon">v2</font>,<span class="p3">(<span class="default"><font color="Maroon">v1</font> <a href="abcmiz_0.html#K8" target="_self">ast</a> <font color="Maroon">t</font></span>)</span></span>)</span></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> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="abcmiz_0.html#K5" target="_self">ast</a> <font color="Maroon">t'</font>
 <b>by </b><i>, <a class="txt" href="abcmiz_0.html#E8:69"><i><font color="Green">E48</font></i></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">E111</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">v1</font>
 ;<br/><div class="add"><a NAME="E2:69_3_1_2"/>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <font color="Maroon">v2</font>,<span class="p2">(<span class="default"><font color="Maroon">v1</font> <a href="abcmiz_0.html#K8" target="_self">ast</a> <font color="Maroon">t</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="abcmiz_0.html#E7:69"><i><font color="Green">E47</font></i></a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E3:69_3_1_2"/><b>then </b><i><font color="Green">E112</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">tt</font> &amp; <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">v1</font> &amp; 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <font color="Maroon">v2</font>,<span class="p2">(<span class="default"><font color="Maroon">v1</font> <a href="abcmiz_0.html#K8" target="_self">ast</a> <font color="Maroon">t</font></span>)</span></span>)</span> )
 <b>by </b><i>, <a class="txt" href="abcmiz_0.html#E6:69"><i><font color="Green">E46</font></i></a>, <a class="txt" href="abcmiz_0.html#E8:69"><i><font color="Green">E48</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E4:69_3_1_2"/><b>then </b><i><font color="Green">E115</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <font color="Maroon">v1</font>,<font color="Maroon">t</font></span>)</span> <a href="rewrite1.html#K1">$^</a> <span class="p2">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <font color="Maroon">v2</font>,<span class="p3">(<span class="default"><font color="Maroon">v1</font> <a href="abcmiz_0.html#K8" target="_self">ast</a> <font color="Maroon">t</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">tt</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">tt</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <font color="Maroon">v1</font>,<font color="Maroon">t</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <font color="Maroon">v1</font>,<font color="Maroon">t</font></span>)</span> <a href="rewrite1.html#K1">$^</a> <span class="p2">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <font color="Maroon">v2</font>,<span class="p3">(<span class="default"><font color="Maroon">v1</font> <a href="abcmiz_0.html#K8" target="_self">ast</a> <font color="Maroon">t</font></span>)</span></span>)</span></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> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <font color="Maroon">v2</font>,<span class="p2">(<span class="default"><font color="Maroon">v1</font> <a href="abcmiz_0.html#K8" target="_self">ast</a> <font color="Maroon">t</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">v2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> )
 <b>by </b><i>, , <a class="ref" href="abcmiz_0.html#D1" target="_self">Def1</a>, <a class="txt" href="abcmiz_0.html#E8:69"><i><font color="Green">E48</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/><a NAME="E5:69_3_1_2"/><b>then </b>
<font color="Maroon">a</font> <a href="abcmiz_0.html#K5" target="_self">ast</a> <font color="Maroon">t'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v1</font> <a href="abcmiz_0.html#K8" target="_self">ast</a> <font color="Maroon">t</font>
 <b>by </b><i>, <a class="txt" href="abcmiz_0.html#E9:69"><i><font color="Green">E49</font></i></a>, , <a class="txt" href="abcmiz_0.html#E8:69"><i><font color="Green">E48</font></i></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" target="_self">apply</a> <font color="Maroon">v1</font>,<font color="Maroon">t</font></span>)</span> <a href="rewrite1.html#K1">$^</a> <span class="p2">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <font color="Maroon">v2</font>,<span class="p3">(<span class="default"><font color="Maroon">v1</font> <a href="abcmiz_0.html#K8" target="_self">ast</a> <font color="Maroon">t</font></span>)</span></span>)</span></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> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="abcmiz_0.html#K5" target="_self">ast</a> <font color="Maroon">t'</font>
 <b>by </b><i><a class="txt" href="abcmiz_0.html#E1:69_3"><i><font color="Green">E50</font></i></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">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">v1</font>
 ;<br/><div class="add"><a NAME="E2:69_3_1_3"/><b>then </b>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">v1</font></span>)</span> <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/><b>then consider </b><font color="Maroon">j</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:69_3_1_3"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">v1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/><a NAME="E5:69_3_1_3"/><i><font color="Green">E117</font></i>: 
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">v1</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">v1</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">v1</font></span>)</span> )
 <b>by </b><i><a class="txt" href="abcmiz_0.html#E8:69"><i><font color="Green">E48</font></i></a></i>;<br/><a NAME="E6:69_3_1_3"/><b>then </b><i><font color="Green">E118</font></i>: 
( 1 <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; 1 <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">v2</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i><a class="txt" href="abcmiz_0.html#E6:69"><i><font color="Green">E46</font></i></a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a>, <a class="ref" href="xreal_1.html#T8">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="xcmplx_0.html#K2">+</a> <font color="Maroon">j</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">v2</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">j</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; <span class="p1">(<span class="default">1 <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">j</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">v2</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E8:69_3_1_3"/><b>then </b><i><font color="Green">E119</font></i>: 
( <font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">v2</font> &amp; <font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <font color="Maroon">v2</font>,<span class="p2">(<span class="default"><font color="Maroon">v1</font> <a href="abcmiz_0.html#K8" target="_self">ast</a> <font color="Maroon">t</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <font color="Maroon">v2</font>,<span class="p2">(<span class="default"><font color="Maroon">v1</font> <a href="abcmiz_0.html#K8" target="_self">ast</a> <font color="Maroon">t</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="abcmiz_0.html#E7:69"><i><font color="Green">E47</font></i></a>, <a class="txt" href="abcmiz_0.html#E1:69_3"><i><font color="Green">E50</font></i></a>, <a class="ref" href="finseq_3.html#T27">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" target="_self">apply</a> <font color="Maroon">v1</font>,<font color="Maroon">t</font></span>)</span> <a href="rewrite1.html#K1">$^</a> <span class="p2">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <font color="Maroon">v2</font>,<span class="p3">(<span class="default"><font color="Maroon">v1</font> <a href="abcmiz_0.html#K8" target="_self">ast</a> <font color="Maroon">t</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <font color="Maroon">v2</font>,<span class="p2">(<span class="default"><font color="Maroon">v1</font> <a href="abcmiz_0.html#K8" target="_self">ast</a> <font color="Maroon">t</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">v2</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">v2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <font color="Maroon">v1</font>,<font color="Maroon">t</font></span>)</span> <a href="rewrite1.html#K1">$^</a> <span class="p2">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <font color="Maroon">v2</font>,<span class="p3">(<span class="default"><font color="Maroon">v1</font> <a href="abcmiz_0.html#K8" target="_self">ast</a> <font color="Maroon">t</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <font color="Maroon">v2</font>,<span class="p2">(<span class="default"><font color="Maroon">v1</font> <a href="abcmiz_0.html#K8" target="_self">ast</a> <font color="Maroon">t</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1</span>)</span> )
 <b>by </b><i>, , <a class="txt" href="abcmiz_0.html#E9:69"><i><font color="Green">E49</font></i></a>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/><b>hence </b><a NAME="E10:69_3_1_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <font color="Maroon">v1</font>,<font color="Maroon">t</font></span>)</span> <a href="rewrite1.html#K1">$^</a> <span class="p2">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <font color="Maroon">v2</font>,<span class="p3">(<span class="default"><font color="Maroon">v1</font> <a href="abcmiz_0.html#K8" target="_self">ast</a> <font color="Maroon">t</font></span>)</span></span>)</span></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> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="abcmiz_0.html#K5" target="_self">ast</a> <font color="Maroon">t'</font>
 <b>by </b><i>, <a class="txt" href="abcmiz_0.html#E2:69_3"><i><font color="Green">E51</font></i></a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E11:69"/>
 <a href="abcmiz_0.html#K7" target="_self">apply</a> <span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">v2</font></span>)</span>,<font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <font color="Maroon">v1</font>,<font color="Maroon">t</font></span>)</span> <a href="rewrite1.html#K1">$^</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" target="_self">apply</a> <font color="Maroon">v2</font>,<span class="p2">(<span class="default"><font color="Maroon">v1</font> <a href="abcmiz_0.html#K8" target="_self">ast</a> <font color="Maroon">t</font></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="txt" href="abcmiz_0.html#E4:69"><i><font color="Green">E43</font></i></a>, </i>;<br/>


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