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

<span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">t</font> being  <a href="abcmiz_0.html#NM1" title="ABCMIZ_0:NM.1">type</a> of <font color="Maroon" title="c1">T</font><br/> for <font color="Olive" title="b2">v1</font>, <font color="Olive" title="b3">v2</font> being   <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> holds   <a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">v1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b3">v2</font></span>)</span>,<font color="Olive" title="b1">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="Olive" title="b2">v1</font>,<font color="Olive" title="b1">t</font></span>)</span> <a href="rewrite1.html#K1" title="REWRITE1:func.1">$^</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> <font color="Olive" title="b3">v2</font>,<span class="p2">(<span class="default"><font color="Olive" title="b2">v1</font> <a href="abcmiz_0.html#K8" title="ABCMIZ_0:func.8">ast</a> <font color="Olive" title="b1">t</font></span>)</span></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">t</font> be   <a href="abcmiz_0.html#NM1" title="ABCMIZ_0:NM.1">type</a> of <font color="Maroon" title="c1">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">v1</font>, <font color="Olive" title="b2">v2</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="abcmiz_0.html#U1" title="ABCMIZ_0:sel.1">adjectives</a> of <font color="Maroon" title="c1">T</font> holds   <a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">v1</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b2">v2</font></span>)</span>,<font color="Maroon" title="c2">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> <font color="Olive" title="b1">v1</font>,<font color="Maroon" title="c2">t</font></span>)</span> <a href="rewrite1.html#K1" title="REWRITE1:func.1">$^</a> <span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> <font color="Olive" title="b2">v2</font>,<span class="p2">(<span class="default"><font color="Olive" title="b1">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><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <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></span><br/>





<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E2:69"/><span class="lab"><font color="Green" title="E36">A1</font></span>: 
 <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="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"><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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="hilbert2.html#T4" title="HILBERT2:th.4">HILBERT2:4</a></span>;<br/>
<span class="kw">set </span><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/>
<span class="kw">set </span><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/>
<a NAME="E3:69"/><span class="lab"><font color="Green" title="E37">A2</font></span>: 
 <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
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="abcmiz_0.html#D19" target="_self" title="ABCMIZ_0:def.19">Def19</a></span>;<br/>
<a NAME="E4:69"/><span class="lab"><font color="Green" title="E38">A3</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:69"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="ref" href="rewrite1.html#T2" title="REWRITE1:th.2">REWRITE1:2</a></span>;<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
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T56" title="FINSEQ_1:th.56">FINSEQ_1:56</a></span>;<br/>
<a NAME="E6:69"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A4</font></span>: 
<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
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:69"><span class="lab"><font color="Green" title="E37">A2</font></span></a>, <a class="txt" href="#E2:69"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></span>;<br/>
<a NAME="E7:69"/><span class="lab"><font color="Green" title="E40">A5</font></span>: 
<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_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:69_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>  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_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:69_1_1_1"/><span class="lab"><font color="Green" title="E41">A6</font></span>: 
 <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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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">$^</a> <span class="p2">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> <font color="Maroon" title="c4">v2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">v1</font> <a href="abcmiz_0.html#K8" title="ABCMIZ_0:func.8">ast</a> <font color="Maroon" title="c2">t</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">t</font></span><br/><div class="add"><a NAME="E2:69_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">tt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:69"><span class="lab"><font color="Green" title="E39">A4</font></span></a></span>;<br/><a NAME="E3:69_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A7</font></span>: 
<span class="p1">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> <font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c2">t</font></span>)</span> <a href="rewrite1.html#K1" title="REWRITE1:func.1">$^</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:69"><span class="lab"><font color="Green" title="E38">A3</font></span></a>, <a class="ref" href="finseq_1.html#T47" title="FINSEQ_1:th.47">FINSEQ_1:47</a></span>;<br/><a NAME="E4:69_1_1_1"/>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69_1_1_1"><span class="lab"><font color="Green" title="E41">A6</font></span></a>, <a class="ref" href="abcmiz_0.html#D19" target="_self" title="ABCMIZ_0:def.19">Def19</a></span>;<br/><span class="kw">hence </span><a NAME="E5:69_1_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">$^</a> <span class="p2">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> <font color="Maroon" title="c4">v2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">v1</font> <a href="abcmiz_0.html#K8" title="ABCMIZ_0:func.8">ast</a> <font color="Maroon" title="c2">t</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">t</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:69_1_1_1"><span class="lab"><font color="Green" title="E42">A7</font></span></a>, <a class="ref" href="abcmiz_0.html#D19" target="_self" title="ABCMIZ_0:def.19">Def19</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:69_1_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">v1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> <font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c2">t</font></span>)</span> <a href="rewrite1.html#K1" title="REWRITE1:func.1">$^</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></span><br/><div class="add"><a NAME="E2:69_1_1_2"/><span class="kw">then </span>
 <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
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:69"><span class="lab"><font color="Green" title="E39">A4</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E3:69_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A8</font></span>: 
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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E4:69_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A9</font></span>: 
<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
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:69"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>;<br/><a NAME="E5:69_1_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">$^</a> <span class="p2">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> <font color="Maroon" title="c4">v2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">v1</font> <a href="abcmiz_0.html#K8" title="ABCMIZ_0:func.8">ast</a> <font color="Maroon" title="c2">t</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">tt</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:69"><span class="lab"><font color="Green" title="E38">A3</font></span></a>, <a class="txt" href="#E3:69_1_1_2"><span class="lab"><font color="Green" title="E41">A8</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>;<br/><span class="kw">hence </span><a NAME="E6:69_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> <font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c2">t</font></span>)</span> <a href="rewrite1.html#K1" title="REWRITE1:func.1">$^</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:69_1_1_2"><span class="lab"><font color="Green" title="E42">A9</font></span></a>, <a class="ref" href="abcmiz_0.html#D19" target="_self" title="ABCMIZ_0:def.19">Def19</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E41">A10</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="69_2"><span class="kw">now </span></a><div class="add"><a NAME="E1:69_2"/><span class="lab"><font color="Green" title="E42">A11</font></span>: 
 <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></span>;<br/><a NAME="E2:69_2"/><span class="lab"><font color="Green" title="E43">A12</font></span>: 
 <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
 <span class="kw">by</span> <span class="lab"><a class="ref" href="abcmiz_0.html#D19" target="_self" title="ABCMIZ_0:def.19">Def19</a></span>;<br/><span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">a</font> being  <a href="abcmiz_0.html#NM2" title="ABCMIZ_0:NM.2">adjective</a> of <font color="Maroon" title="c1">T</font><br/> for <font color="Olive" title="b2">t9</font> being  <a href="abcmiz_0.html#NM1" title="ABCMIZ_0:NM.1">type</a> of <font color="Maroon" title="c1">T</font>  st <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> &amp; <font color="Olive" title="b1">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="Olive" title="b2">t9</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">$^</a> <span class="p2">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> <font color="Maroon" title="c4">v2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">v1</font> <a href="abcmiz_0.html#K8" title="ABCMIZ_0:func.8">ast</a> <font color="Maroon" title="c2">t</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> <font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c2">t</font></span>)</span> <a href="rewrite1.html#K1" title="REWRITE1:func.1">$^</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="Olive">b<sub>3</sub></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="Olive">b<sub>4</sub></font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">ast</a> <font color="Olive">b<sub>5</sub></font></span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">t9</font> being  <a href="abcmiz_0.html#NM1" title="ABCMIZ_0:NM.1">type</a> of <font color="Maroon" title="c1">T</font>  st <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> &amp; <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="Olive" title="b1">t9</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">$^</a> <span class="p2">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> <font color="Maroon" title="c4">v2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">v1</font> <a href="abcmiz_0.html#K8" title="ABCMIZ_0:func.8">ast</a> <font color="Maroon" title="c2">t</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> <font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c2">t</font></span>)</span> <a href="rewrite1.html#K1" title="REWRITE1:func.1">$^</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="Olive">b<sub>2</sub></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="Olive">b<sub>3</sub></font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">ast</a> <font color="Olive">b<sub>4</sub></font></span><br/><span class="kw">let </span><font color="Maroon" title="c11">t9</font> be   <a href="abcmiz_0.html#NM1" title="ABCMIZ_0:NM.1">type</a> of <font color="Maroon" title="c1">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="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> &amp; <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">t9</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">$^</a> <span class="p2">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> <font color="Maroon" title="c4">v2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">v1</font> <a href="abcmiz_0.html#K8" title="ABCMIZ_0:func.8">ast</a> <font color="Maroon" title="c2">t</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> <font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c2">t</font></span>)</span> <a href="rewrite1.html#K1" title="REWRITE1:func.1">$^</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="Olive">b<sub>1</sub></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="Olive">b<sub>2</sub></font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">ast</a> <font color="Olive">b<sub>3</sub></font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E3:69_2"/><span class="lab"><font color="Green" title="E44">A13</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E4:69_2"/><span class="lab"><font color="Green" title="E45">A14</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E5:69_2"/><span class="lab"><font color="Green" title="E46">A15</font></span>: 
<font color="Maroon" title="c11">t9</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">$^</a> <span class="p2">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> <font color="Maroon" title="c4">v2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">v1</font> <a href="abcmiz_0.html#K8" title="ABCMIZ_0:func.8">ast</a> <font color="Maroon" title="c2">t</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="abcmiz_0.html#K7" title="ABCMIZ_0:func.7">apply</a> <font color="Maroon" title="c3">v1</font>,<font color="Maroon" title="c2">t</font></span>)</span> <a href="rewrite1.html#K1" title="REWRITE1:func.1">$^</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="Olive">b<sub>1</sub></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="Olive">b<sub>2</sub></font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">ast</a> <font color="Olive">b<sub>3</sub></font></span><br/><a NAME="E6:69_2"/><span class="lab"><font color="Green" title="E47">A16</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:69_2"><span class="lab"><font color="Green" title="E44">A13</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E7:69_2"/><span class="lab"><font color="Green" title="E48">A17</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:69_2"><span class="lab"><font color="Green" title="E44">A13</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:69_2_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><span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:69_2_1_1"/><span class="lab"><font color="Green" title="E49">A18</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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="Olive">b<sub>1</sub></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="Olive">b<sub>2</sub></font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">ast</a> <font color="Olive">b<sub>3</sub></font></span><br/><div class="add"><a NAME="E2:69_2_1_1"/><span class="lab"><font color="Green" title="E50">A19</font></span>: 
<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
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/><a NAME="E3:69_2_1_1"/>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69_2_1_1"><span class="lab"><font color="Green" title="E49">A18</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E4:69_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A20</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:69"><span class="lab"><font color="Green" title="E39">A4</font></span></a>, <a class="txt" href="#E2:69_2_1_1"><span class="lab"><font color="Green" title="E50">A19</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E5:69_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A21</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:69"><span class="lab"><font color="Green" title="E38">A3</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>;<br/><a NAME="E6:69_2_1_1"/><span class="lab"><font color="Green" title="E53">A22</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:69"><span class="lab"><font color="Green" title="E39">A4</font></span></a>, <a class="txt" href="#E6:69_2"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="txt" href="#E1:69_2_1_1"><span class="lab"><font color="Green" title="E49">A18</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E7:69_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A23</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:69"><span class="lab"><font color="Green" title="E38">A3</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>;<br/><a NAME="E8:69_2_1_1"/><span class="lab"><font color="Green" title="E55">A24</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:69"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="txt" href="#E4:69_2_1_1"><span class="lab"><font color="Green" title="E51">A20</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>;<br/><a NAME="E9:69_2_1_1"/><span class="lab"><font color="Green" title="E56">A25</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:69"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="txt" href="#E6:69_2_1_1"><span class="lab"><font color="Green" title="E53">A22</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>;<br/><a NAME="E10:69_2_1_1"/><span class="lab"><font color="Green" title="E57">A26</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:69_2"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="txt" href="#E1:69_2_1_1"><span class="lab"><font color="Green" title="E49">A18</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E11:69_2_1_1"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>;<br/><span class="kw">hence </span><a NAME="E12:69_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> <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">t9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:69_2"><span class="lab"><font color="Green" title="E45">A14</font></span></a>, <a class="txt" href="#E5:69_2"><span class="lab"><font color="Green" title="E46">A15</font></span></a>, <a class="txt" href="#E10:69_2_1_1"><span class="lab"><font color="Green" title="E57">A26</font></span></a>, <a class="txt" href="#E7:69_2_1_1"><span class="lab"><font color="Green" title="E54">A23</font></span></a>, <a class="txt" href="#E9:69_2_1_1"><span class="lab"><font color="Green" title="E56">A25</font></span></a>, <a class="txt" href="#E5:69_2_1_1"><span class="lab"><font color="Green" title="E52">A21</font></span></a>, <a class="txt" href="#E8:69_2_1_1"><span class="lab"><font color="Green" title="E55">A24</font></span></a>, <a class="ref" href="abcmiz_0.html#D19" target="_self" title="ABCMIZ_0:def.19">Def19</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:69_2_1_2"/><span class="lab"><font color="Green" title="E49">A27</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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="Olive">b<sub>1</sub></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="Olive">b<sub>2</sub></font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">ast</a> <font color="Olive">b<sub>3</sub></font></span><br/><div class="add"><a NAME="E2:69_2_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:69_2"><span class="lab"><font color="Green" title="E43">A12</font></span></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/><a NAME="E3:69_2_1_2"/><span class="kw">then </span>
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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E4:69_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A28</font></span>: 
<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
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:69"><span class="lab"><font color="Green" title="E38">A3</font></span></a>, <a class="txt" href="#E6:69"><span class="lab"><font color="Green" title="E39">A4</font></span></a>, <a class="txt" href="#E1:69_2_1_2"><span class="lab"><font color="Green" title="E49">A27</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>;<br/><a NAME="E5:69_2_1_2"/><span class="lab"><font color="Green" title="E51">A29</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:69"><span class="lab"><font color="Green" title="E39">A4</font></span></a>, <a class="txt" href="#E6:69_2"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="txt" href="#E1:69_2_1_2"><span class="lab"><font color="Green" title="E49">A27</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E6:69_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A30</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:69"><span class="lab"><font color="Green" title="E38">A3</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>;<br/><a NAME="E7:69_2_1_2"/><span class="lab"><font color="Green" title="E53">A31</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:69"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="txt" href="#E5:69_2_1_2"><span class="lab"><font color="Green" title="E51">A29</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>;<br/><a NAME="E8:69_2_1_2"/><span class="lab"><font color="Green" title="E54">A32</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:69_2"><span class="lab"><font color="Green" title="E47">A16</font></span></a>, <a class="txt" href="#E1:69_2_1_2"><span class="lab"><font color="Green" title="E49">A27</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E9:69_2_1_2"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>;<br/><a NAME="E10:69_2_1_2"/><span class="kw">then </span>
<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">t9</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:69_2"><span class="lab"><font color="Green" title="E45">A14</font></span></a>, <a class="txt" href="#E5:69_2"><span class="lab"><font color="Green" title="E46">A15</font></span></a>, <a class="txt" href="#E1:69_2_1_2"><span class="lab"><font color="Green" title="E49">A27</font></span></a>, <a class="txt" href="#E8:69_2_1_2"><span class="lab"><font color="Green" title="E54">A32</font></span></a>, <a class="txt" href="#E6:69_2_1_2"><span class="lab"><font color="Green" title="E52">A30</font></span></a>, <a class="txt" href="#E7:69_2_1_2"><span class="lab"><font color="Green" title="E53">A31</font></span></a>, <a class="ref" href="abcmiz_0.html#D19" target="_self" title="ABCMIZ_0:def.19">Def19</a></span>;<br/><span class="kw">hence </span><a NAME="E11:69_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> <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">t9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:69_2_1_2"><span class="lab"><font color="Green" title="E50">A28</font></span></a>, <a class="ref" href="abcmiz_0.html#D19" target="_self" title="ABCMIZ_0:def.19">Def19</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_2_1_3"><span class="kw">suppose </span></a><a NAME="E1:69_2_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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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="Olive">b<sub>1</sub></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="Olive">b<sub>2</sub></font> <a href="abcmiz_0.html#K5" title="ABCMIZ_0:func.5">ast</a> <font color="Olive">b<sub>3</sub></font></span><br/><div class="add"><a NAME="E2:69_2_1_3"/><span class="kw">then </span>
<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
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E4:69_2_1_3"/><span class="lab"><font color="Green" title="E49">A33</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></span>;<br/><a NAME="E5:69_2_1_3"/><span class="lab"><font color="Green" title="E50">A34</font></span>: 
<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
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/><a NAME="E6:69_2_1_3"/><span class="lab"><font color="Green" title="E51">A35</font></span>: 
<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>
 ;<br/><a NAME="E7:69_2_1_3"/><span class="lab"><font color="Green" title="E52">A36</font></span>: 
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
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/><a NAME="E8:69_2_1_3"/><span class="lab"><font color="Green" title="E53">A37</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:69_2_1_3"><span class="lab"><font color="Green" title="E49">A33</font></span></a></span>;<br/><a NAME="E9:69_2_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A38</font></span>: 
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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:69_2"><span class="lab"><font color="Green" title="E48">A17</font></span></a>, <a class="txt" href="#E1:69_2"><span class="lab"><font color="Green" title="E42">A11</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E10:69_2_1_3"/><span class="kw">then </span>
<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
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/><a NAME="E11:69_2_1_3"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:69_2"><span class="lab"><font color="Green" title="E43">A12</font></span></a>, <a class="txt" href="#E7:69_2_1_3"><span class="lab"><font color="Green" title="E52">A36</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E12:69_2_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A39</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:69"><span class="lab"><font color="Green" title="E38">A3</font></span></a>, <a class="txt" href="#E6:69"><span class="lab"><font color="Green" title="E39">A4</font></span></a>, <a class="txt" href="#E8:69_2_1_3"><span class="lab"><font color="Green" title="E53">A37</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>;<br/><a NAME="E13:69_2_1_3"/>
<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
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:69_2_1_3"><span class="lab"><font color="Green" title="E54">A38</font></span></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/><a NAME="E14:69_2_1_3"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:69_2"><span class="lab"><font color="Green" title="E43">A12</font></span></a>, <a class="txt" href="#E5:69_2_1_3"><span class="lab"><font color="Green" title="E50">A34</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E15:69_2_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A40</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:69"><span class="lab"><font color="Green" title="E38">A3</font></span></a>, <a class="txt" href="#E6:69"><span class="lab"><font color="Green" title="E39">A4</font></span></a>, <a class="txt" href="#E4:69_2_1_3"><span class="lab"><font color="Green" title="E49">A33</font></span></a>, <a class="txt" href="#E6:69_2_1_3"><span class="lab"><font color="Green" title="E51">A35</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>;<br/><a NAME="E16:69_2_1_3"/><span class="lab"><font color="Green" title="E57">A41</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:69_2_1_3"><span class="lab"><font color="Green" title="E52">A36</font></span></a>, <a class="txt" href="#E9:69_2_1_3"><span class="lab"><font color="Green" title="E54">A38</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E17:69_2_1_3"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:69_2_1_3"><span class="lab"><font color="Green" title="E53">A37</font></span></a>, <a class="ref" href="finseq_1.html#D7" title="FINSEQ_1:def.7">FINSEQ_1:def 7</a></span>;<br/><span class="kw">hence </span><a NAME="E18:69_2_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">t9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:69_2"><span class="lab"><font color="Green" title="E45">A14</font></span></a>, <a class="txt" href="#E5:69_2"><span class="lab"><font color="Green" title="E46">A15</font></span></a>, <a class="txt" href="#E16:69_2_1_3"><span class="lab"><font color="Green" title="E57">A41</font></span></a>, <a class="txt" href="#E12:69_2_1_3"><span class="lab"><font color="Green" title="E55">A39</font></span></a>, <a class="txt" href="#E15:69_2_1_3"><span class="lab"><font color="Green" title="E56">A40</font></span></a>, <a class="ref" href="abcmiz_0.html#D19" target="_self" title="ABCMIZ_0:def.19">Def19</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<a NAME="E9:69"/>
 <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
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="abcmiz_0.html#D19" target="_self" title="ABCMIZ_0:def.19">Def19</a></span>;<br/>
<span class="kw">then </span> <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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:69"><span class="lab"><font color="Green" title="E38">A3</font></span></a>, <a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></span>
<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
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:69"><span class="lab"><font color="Green" title="E39">A4</font></span></a></span>
<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
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T35" title="FINSEQ_1:th.35">FINSEQ_1:35</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E11:69"/>
 <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:69"><span class="lab"><font color="Green" title="E38">A3</font></span></a>, <a class="txt" href="#E7:69"><span class="lab"><font color="Green" title="E40">A5</font></span></a>, <a class="txt" href="#E8:69"><span class="lab"><font color="Green" title="E41">A10</font></span></a>, <a class="ref" href="abcmiz_0.html#D19" target="_self" title="ABCMIZ_0:def.19">Def19</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
