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

<span class="kw">let </span><font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c2">B</font> be   <a href="modal_1.html#NM3" title="MODAL_1:NM.3">MP-wff</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 2</span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">2,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="trees_2.html#K7" title="TREES_2:func.7">with-replacement</a> (<span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>,<font color="Maroon" title="c1">A</font>)</span>)</span> <a href="trees_2.html#K7" title="TREES_2:func.7">with-replacement</a> (<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>,<font color="Maroon" title="c2">B</font>) is   <a href="modal_1.html#NM3" title="MODAL_1:NM.3">MP-wff</a></span><br/>



<span class="kw">reconsider </span><font color="Maroon" title="c3">d</font> = <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> as    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 2 <span class="kw">by</span> <span class="lab"><a class="ref" href="trees_1.html#T28" title="TREES_1:th.28">TREES_1:28</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">y</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 2</span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">2,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="trees_2.html#K7" title="TREES_2:func.7">with-replacement</a> (<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>,<font color="Maroon" title="c1">A</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c5">x</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 2</span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">2,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="trees_2.html#K7" title="TREES_2:func.7">with-replacement</a> (<span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>,<font color="Maroon" title="c1">A</font>)</span>)</span> <a href="trees_2.html#K7" title="TREES_2:func.7">with-replacement</a> (<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>,<font color="Maroon" title="c2">B</font>);<br/>
<a NAME="E2:39"/><span class="lab"><font color="Green" title="E27">A1</font></span>: 
 not <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="trees_1.html#NR3" title="TREES_1:NR.3">is_a_proper_prefix_of</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:39_1"/>
<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="trees_1.html#NR3" title="TREES_1:NR.3">is_a_proper_prefix_of</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:39_1"/><span class="kw">then </span>
<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">is_a_prefix_of</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</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="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:39_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="ref" href="trees_1.html#T3" title="TREES_1:th.3">TREES_1:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">reconsider </span><font color="Maroon" title="c6">db</font> =  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">B</font> as   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">da</font> =  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">A</font> as   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a> ;<br/>
<a NAME="E5:39"/>
<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="trees_1.html#T28" title="TREES_1:th.28">TREES_1:28</a></span>;<br/>
<a NAME="E6:39"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A2</font></span>: 
<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 2</span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">2,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>;<br/>
<a NAME="E7:39"/><span class="kw">then </span>
 <a href="modal_1.html#K9" title="MODAL_1:func.9">@</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 2</span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">2,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span>,<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>,<font color="Maroon" title="c1">A</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="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 2</span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">2,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="trees_2.html#K7" title="TREES_2:func.7">with-replacement</a> (<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>,<font color="Maroon" title="c1">A</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="modal_1.html#D7" target="_self" title="MODAL_1:def.7">Def8</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">y</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 2</span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">2,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="trees_2.html#K7" title="TREES_2:func.7">with-replacement</a> (<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>,<font color="Maroon" title="c1">A</font>) as   <a href="trees_2.html#NM5" title="TREES_2:NM.5">DecoratedTree</a> of <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> ;<br/>
<a NAME="E9:39"/><span class="lab"><font color="Green" title="E29">A3</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 2</span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">2,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span></span>)</span> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</a> (<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>,<span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">A</font></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:39"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="ref" href="trees_2.html#D11" title="TREES_2:def.11">TREES_2:def 11</a></span>;<br/>
<a NAME="E10:39"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 2</span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">2,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>;<br/>
<a NAME="E11:39"/><span class="kw">then </span>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 2</span>)</span> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</a> (<font color="Maroon" title="c3">d</font>,<font color="Maroon" title="c7">da</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="trees_2.html#D11" title="TREES_2:def.11">TREES_2:def 11</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">dy</font> =  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">y</font> as   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</a> ;<br/>
<a NAME="E13:39"/>
<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="trees_1.html#T28" title="TREES_1:th.28">TREES_1:28</a></span>;<br/>
<a NAME="E14:39"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A4</font></span>: 
<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 2</span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">2,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></span>;<br/>
<a NAME="E15:39"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A5</font></span>: 
<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:39"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="txt" href="#E9:39"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="txt" href="#E2:39"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="ref" href="trees_1.html#D9" title="TREES_1:def.9">TREES_1:def 9</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">d1</font> = <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> as    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon" title="c9">dy</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:39"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="txt" href="#E9:39"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="txt" href="#E14:39"><span class="lab"><font color="Green" title="E30">A4</font></span></a>, <a class="txt" href="#E2:39"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="ref" href="trees_1.html#D9" title="TREES_1:def.9">TREES_1:def 9</a></span>;<br/>
<a NAME="E17:39"/>
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 2</span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p4"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">2,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="trees_2.html#K7" title="TREES_2:func.7">with-replacement</a> (<span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>,<font color="Maroon" title="c1">A</font>)</span>)</span> <a href="trees_2.html#K7" title="TREES_2:func.7">with-replacement</a> (<span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>,<font color="Maroon" title="c2">B</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">dy</font> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</a> (<font color="Maroon" title="c10">d1</font>,<font color="Maroon" title="c6">db</font>) &amp;  <a href="modal_1.html#K9" title="MODAL_1:func.9">@</a> (<font color="Maroon" title="c8">y</font>,<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>,<font color="Maroon" title="c2">B</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 2</span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">2,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="trees_2.html#K7" title="TREES_2:func.7">with-replacement</a> (<span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>,<font color="Maroon" title="c1">A</font>)</span>)</span> <a href="trees_2.html#K7" title="TREES_2:func.7">with-replacement</a> (<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>,<font color="Maroon" title="c2">B</font>) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:39"><span class="lab"><font color="Green" title="E31">A5</font></span></a>, <a class="ref" href="modal_1.html#D7" target="_self" title="MODAL_1:def.7">Def8</a>, <a class="ref" href="trees_2.html#D11" title="TREES_2:def.11">TREES_2:def 11</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">x</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 2</span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">2,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="trees_2.html#K7" title="TREES_2:func.7">with-replacement</a> (<span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>,<font color="Maroon" title="c1">A</font>)</span>)</span> <a href="trees_2.html#K7" title="TREES_2:func.7">with-replacement</a> (<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>,<font color="Maroon" title="c2">B</font>) as   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="trees_2.html#NM5" title="TREES_2:NM.5">DecoratedTree</a> of <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> <span class="kw">by</span> <span class="lab"><a class="txt" href="modal_1.html#E26"><span class="lab"><font color="Green" title="E17">Lm2</font></span></a></span>;<br/>
<a NAME="E19:39"/><span class="lab"><font color="Green" title="E32">A6</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">y</font></span>)</span> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</a> (<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>,<span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">B</font></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:39"><span class="lab"><font color="Green" title="E31">A5</font></span></a>, <a class="ref" href="trees_2.html#D11" title="TREES_2:def.11">TREES_2:def 11</a></span>;<br/>
<a NAME="E20:39"/>
 for <font color="Olive" title="b1">v</font> being    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c11">x</font> holds <br/> (  <a href="modal_1.html#K5" title="MODAL_1:func.5">branchdeg</a> <font color="Olive" title="b1">v</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 &amp; (  not  <a href="modal_1.html#K5" title="MODAL_1:func.5">branchdeg</a> <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c11">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> or  ex <font color="Olive" title="b2">k</font> being    <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>  st <font color="Maroon" title="c11">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">3,<font color="Olive" title="b2">k</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> ) &amp; (  not  <a href="modal_1.html#K5" title="MODAL_1:func.5">branchdeg</a> <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c11">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> or <font color="Maroon" title="c11">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">1,1</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> ) &amp; (  <a href="modal_1.html#K5" title="MODAL_1:func.5">branchdeg</a> <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 implies <font color="Maroon" title="c11">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Olive" title="b1">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">2,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c12">e</font> = <span class="p1">(<span class="default"><a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 2</span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">2,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c13">v</font> be    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c11">x</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="modal_1.html#K5" title="MODAL_1:func.5">branchdeg</a> <font color="Maroon" title="c13">v</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 &amp; (  not  <a href="modal_1.html#K5" title="MODAL_1:func.5">branchdeg</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c11">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> or  ex <font color="Olive" title="b1">k</font> being    <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>  st <font color="Maroon" title="c11">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">3,<font color="Olive" title="b1">k</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> ) &amp; (  not  <a href="modal_1.html#K5" title="MODAL_1:func.5">branchdeg</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c11">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> or <font color="Maroon" title="c11">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">1,1</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> ) &amp; (  <a href="modal_1.html#K5" title="MODAL_1:func.5">branchdeg</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 implies <font color="Maroon" title="c11">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">2,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> ) )</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_2_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="E1:39_2_1_1"/>
( (  not <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">is_a_prefix_of</a> <font color="Maroon" title="c13">v</font> &amp; <font color="Maroon" title="c11">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">v</font> ) or  ex <font color="Olive" title="b1">s</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">B</font> &amp; <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b1">s</font> &amp; <font color="Maroon" title="c11">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:39"><span class="lab"><font color="Green" title="E31">A5</font></span></a>, <a class="txt" href="#E19:39"><span class="lab"><font color="Green" title="E32">A6</font></span></a>, <a class="ref" href="trees_2.html#D11" title="TREES_2:def.11">TREES_2:def 11</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:39_2_1_1_1"/><span class="lab"><font color="Green" title="E33">A7</font></span>: 
(  not <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">is_a_prefix_of</a> <font color="Maroon" title="c13">v</font> &amp; <font color="Maroon" title="c11">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">v</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="modal_1.html#K5" title="MODAL_1:func.5">branchdeg</a> <font color="Maroon" title="c13">v</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 &amp; (  not  <a href="modal_1.html#K5" title="MODAL_1:func.5">branchdeg</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c11">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> or  ex <font color="Olive" title="b1">k</font> being    <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>  st <font color="Maroon" title="c11">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">3,<font color="Olive" title="b1">k</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> ) &amp; (  not  <a href="modal_1.html#K5" title="MODAL_1:func.5">branchdeg</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c11">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> or <font color="Maroon" title="c11">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">1,1</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> ) &amp; (  <a href="modal_1.html#K5" title="MODAL_1:func.5">branchdeg</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 implies <font color="Maroon" title="c11">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">2,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> ) )</span><br/><div class="add"><a NAME="E2:39_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A8</font></span>: 
 for <font color="Olive" title="b1">s</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   holds <br/>(  not <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">B</font> or  not <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b1">s</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="trees_1.html#T1" title="TREES_1:th.1">TREES_1:1</a></span>;<br/><a NAME="E3:39_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A9</font></span>: 
<font color="Maroon" title="c13">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 2</span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">2,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span></span>)</span> <a href="trees_1.html#K5" title="TREES_1:func.5">with-replacement</a> (<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>,<span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">A</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:39"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="txt" href="#E15:39"><span class="lab"><font color="Green" title="E31">A5</font></span></a>, <a class="txt" href="#E19:39"><span class="lab"><font color="Green" title="E32">A6</font></span></a>, <a class="ref" href="trees_1.html#D9" title="TREES_1:def.9">TREES_1:def 9</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_2_1_1_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_2_1_1_1_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="E1:39_2_1_1_1_1_1"/>
( (  not <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">is_a_prefix_of</a> <font color="Maroon" title="c13">v</font> &amp; <font color="Maroon" title="c8">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">v</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="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 2</span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">2,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">v</font> ) or  ex <font color="Olive" title="b1">s</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b1">s</font> &amp; <font color="Maroon" title="c8">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:39"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="txt" href="#E3:39_2_1_1_1"><span class="lab"><font color="Green" title="E35">A9</font></span></a>, <a class="ref" href="trees_2.html#D11" title="TREES_2:def.11">TREES_2:def 11</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_2_1_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:39_2_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E36">A10</font></span>: 
(  not <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">is_a_prefix_of</a> <font color="Maroon" title="c13">v</font> &amp; <font color="Maroon" title="c8">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">v</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="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 2</span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">2,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">v</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="modal_1.html#K5" title="MODAL_1:func.5">branchdeg</a> <font color="Maroon" title="c13">v</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 &amp;  <a href="modal_1.html#K5" title="MODAL_1:func.5">branchdeg</a> <font color="Maroon" title="c13">v</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 &amp; (  not  <a href="modal_1.html#K5" title="MODAL_1:func.5">branchdeg</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c11">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> or  ex <font color="Olive" title="b1">k</font> being    <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>  st <font color="Maroon" title="c11">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">3,<font color="Olive" title="b1">k</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> ) &amp; (  not  <a href="modal_1.html#K5" title="MODAL_1:func.5">branchdeg</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c11">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> or <font color="Maroon" title="c11">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">1,1</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> ) &amp; (  <a href="modal_1.html#K5" title="MODAL_1:func.5">branchdeg</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 implies <font color="Maroon" title="c11">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">2,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> ) )</span><br/><div class="add"><a NAME="E2:39_2_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E37">A11</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 2</span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">2,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>,<span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>,<span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a>, <a class="ref" href="trees_1.html#T53" title="TREES_1:th.53">TREES_1:53</a></span>;<br/><a NAME="E3:39_2_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E38">A12</font></span>: 
 for <font color="Olive" title="b1">s</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   holds <br/>(  not <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">A</font> or  not <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b1">s</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E36">A10</font></span></a>, <a class="ref" href="trees_1.html#T1" title="TREES_1:th.1">TREES_1:1</a></span>;<br/><a NAME="E4:39_2_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A13</font></span>: 
<font color="Maroon" title="c13">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 2</span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">2,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:39"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="txt" href="#E3:39_2_1_1_1"><span class="lab"><font color="Green" title="E35">A9</font></span></a>, <a class="ref" href="trees_1.html#D9" title="TREES_1:def.9">TREES_1:def 9</a></span>;<br/><a NAME="E5:39_2_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A14</font></span>: 
<font color="Maroon" title="c13">v</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="#E1:39_2_1_1_1"><span class="lab"><font color="Green" title="E33">A7</font></span></a>, <a class="txt" href="#E1:39_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E36">A10</font></span></a>, <a class="txt" href="#E2:39_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E37">A11</font></span></a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c14">v9</font> = <font color="Maroon" title="c13">v</font> as    <a href="trees_3.html#M2" title="TREES_3:mode.2">Element</a> of  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 2</span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">2,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:39"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="txt" href="#E3:39_2_1_1_1"><span class="lab"><font color="Green" title="E35">A9</font></span></a>, <a class="txt" href="#E3:39_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E38">A12</font></span></a>, <a class="ref" href="trees_1.html#D9" title="TREES_1:def.9">TREES_1:def 9</a></span>;<br/><a NAME="E7:39_2_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E41">A15</font></span>: 
 <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c14">v9</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_2_1_1_1_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c15">v99</font> = <font color="Maroon" title="c13">v</font> as    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">y</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:39"><span class="lab"><font color="Green" title="E31">A5</font></span></a>, <a class="txt" href="#E19:39"><span class="lab"><font color="Green" title="E32">A6</font></span></a>, <a class="txt" href="#E2:39_2_1_1_1"><span class="lab"><font color="Green" title="E34">A8</font></span></a>, <a class="ref" href="trees_1.html#D9" title="TREES_1:def.9">TREES_1:def 9</a></span>;<br/>
<a NAME="E2:39_2_1_1_1_1_1_1_1"/>
 <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c15">v99</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c14">v9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:39"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="txt" href="#E9:39"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="txt" href="#E5:39_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E40">A14</font></span></a>, <a class="txt" href="modal_1.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a>, <a class="ref" href="modal_1.html#T13" target="_self" title="MODAL_1:th.13">Th17</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:39_2_1_1_1_1_1_1_1"/>
 <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c14">v9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:39"><span class="lab"><font color="Green" title="E31">A5</font></span></a>, <a class="txt" href="#E19:39"><span class="lab"><font color="Green" title="E32">A6</font></span></a>, <a class="txt" href="#E5:39_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E40">A14</font></span></a>, <a class="txt" href="modal_1.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a>, <a class="ref" href="modal_1.html#T13" target="_self" title="MODAL_1:th.13">Th17</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_2_1_1_1_1_1_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c15">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c14">v9</font> implies <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>,<span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> ) &amp; ( <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>,<span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> implies <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c14">v9</font> ) )</span><br/><span class="kw">thus </span><a NAME="E1:39_2_1_1_1_1_1_1_2"/>
( <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c14">v9</font> implies <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>,<span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>,<span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> implies <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c14">v9</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_2_1_1_1_1_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:39_2_1_1_1_1_1_1_2_1"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c14">v9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>,<span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span></span><br/>

<a NAME="E2:39_2_1_1_1_1_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c14">v9</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Olive" title="b1">n</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> where <font color="Olive" title="b1">n</font> is    <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>  : <font color="Maroon" title="c14">v9</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="Olive" title="b1">n</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 2</span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">2,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="trees_2.html#D5" title="TREES_2:def.5">TREES_2:def 5</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">n</font> being    <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> <span class="kw"> such that </span><br/><a NAME="E4:39_2_1_1_1_1_1_1_2_1"/><span class="lab"><font color="Green" title="E42">A16</font></span>: 
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">v9</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="c16">n</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">and </span><br/><a NAME="E5:39_2_1_1_1_1_1_1_2_1"/><span class="lab"><font color="Green" title="E43">A17</font></span>: 
<font color="Maroon" title="c14">v9</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="c16">n</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 2</span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">2,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span>
 ;<br/>
<a NAME="E6:39_2_1_1_1_1_1_1_2_1"/>
<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c16">n</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 2</span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">2,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:39_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E40">A14</font></span></a>, <a class="txt" href="#E5:39_2_1_1_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E43">A17</font></span></a>, <a class="ref" href="finseq_1.html#T34" title="FINSEQ_1:th.34">FINSEQ_1:34</a></span>;<br/>
<a NAME="E7:39_2_1_1_1_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A18</font></span>: 
( <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c16">n</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>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c16">n</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> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> or <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c16">n</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> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</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="txt" href="#E2:39_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E37">A11</font></span></a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></span>;<br/>
<a NAME="E8:39_2_1_1_1_1_1_1_2_1"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c16">n</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="txt" href="#E5:39_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E40">A14</font></span></a>, <a class="txt" href="#E4:39_2_1_1_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E42">A16</font></span></a>, <a class="ref" href="finseq_1.html#T34" title="FINSEQ_1:th.34">FINSEQ_1:34</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:39_2_1_1_1_1_1_1_2_1"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>,<span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:39_2_1_1_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E44">A18</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">assume </span><a NAME="E2:39_2_1_1_1_1_1_1_2"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>,<span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c14">v9</font></span><br/><a NAME="E3:39_2_1_1_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A19</font></span>: 
( <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> or <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</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="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_2_1_1_1_1_1_1_2_2"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_2_1_1_1_1_1_1_2_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="E1:39_2_1_1_1_1_1_1_2_2_1"/>
( <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">v9</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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> or <font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">v9</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">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:39_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E40">A14</font></span></a>, <a class="txt" href="#E3:39_2_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E42">A19</font></span></a>, <a class="ref" href="finseq_1.html#T34" title="FINSEQ_1:th.34">FINSEQ_1:34</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_2_1_1_1_1_1_1_2_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:39_2_1_1_1_1_1_1_2_2_1_1"/><span class="lab"><font color="Green" title="E43">A20</font></span>: 
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">v9</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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c14">v9</font></span><br/><div class="add"><a NAME="E2:39_2_1_1_1_1_1_1_2_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">v9</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"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 2</span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">2,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:39_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E37">A11</font></span></a>, <a class="txt" href="#E3:39_2_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E42">A19</font></span></a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></span>;<br/><a NAME="E3:39_2_1_1_1_1_1_1_2_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c14">v9</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Olive" title="b1">n</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> where <font color="Olive" title="b1">n</font> is    <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>  : <font color="Maroon" title="c14">v9</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="Olive" title="b1">n</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 2</span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">2,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_2_1_1_1_1_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E43">A20</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:39_2_1_1_1_1_1_1_2_2_1_1"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c14">v9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="trees_2.html#D5" title="TREES_2:def.5">TREES_2:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="39_2_1_1_1_1_1_1_2_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:39_2_1_1_1_1_1_1_2_2_1_2"/><span class="lab"><font color="Green" title="E43">A21</font></span>: 
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">v9</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">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c14">v9</font></span><br/><div class="add"><a NAME="E2:39_2_1_1_1_1_1_1_2_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c14">v9</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">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 2</span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">2,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:39_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E37">A11</font></span></a>, <a class="txt" href="#E3:39_2_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E42">A19</font></span></a>, <a class="ref" href="enumset1.html#D1" title="ENUMSET1:def.1">ENUMSET1:def 1</a></span>;<br/><a NAME="E3:39_2_1_1_1_1_1_1_2_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c14">v9</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Olive" title="b1">n</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> where <font color="Olive" title="b1">n</font> is    <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>  : <font color="Maroon" title="c14">v9</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="Olive" title="b1">n</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 2</span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">2,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> </span> } </span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_2_1_1_1_1_1_1_2_2_1_2"><span class="lab"><font color="Green" title="E43">A21</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:39_2_1_1_1_1_1_1_2_2_1_2"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c14">v9</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="trees_2.html#D5" title="TREES_2:def.5">TREES_2:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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><span class="kw">hence </span><a NAME="E5:39_2_1_1_1_1_1_1_2"/>
<font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c14">v9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E9:39_2_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A22</font></span>: 
 <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c14">v9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>,<span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#T1" title="TARSKI:th.1">TARSKI:1</a></span>;<br/><a NAME="E10:39_2_1_1_1_1_1_1"/>
<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</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="trees_1.html#T3" title="TREES_1:th.3">TREES_1:3</a></span>;<br/><a NAME="E11:39_2_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A23</font></span>: 
2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c13">v</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:39_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E42">A22</font></span></a>, <a class="txt" href="#E7:39_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E41">A15</font></span></a>, <a class="ref" href="card_2.html#T57" title="CARD_2:th.57">CARD_2:57</a></span>;<br/><span class="kw">hence </span><a NAME="E12:39_2_1_1_1_1_1_1"/>
 <a href="modal_1.html#K5" title="MODAL_1:func.5">branchdeg</a> <font color="Maroon" title="c13">v</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="trees_2.html#D12" title="TREES_2:def.12">TREES_2:def 12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="modal_1.html#K5" title="MODAL_1:func.5">branchdeg</a> <font color="Maroon" title="c13">v</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 &amp; (  not  <a href="modal_1.html#K5" title="MODAL_1:func.5">branchdeg</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c11">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> or  ex <font color="Olive" title="b1">k</font> being    <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>  st <font color="Maroon" title="c11">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">3,<font color="Olive" title="b1">k</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> ) &amp; (  not  <a href="modal_1.html#K5" title="MODAL_1:func.5">branchdeg</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c11">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> or <font color="Maroon" title="c11">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">1,1</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> ) &amp; (  <a href="modal_1.html#K5" title="MODAL_1:func.5">branchdeg</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 implies <font color="Maroon" title="c11">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">2,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> ) )</span><br/><a NAME="E13:39_2_1_1_1_1_1_1"/>
<font color="Maroon" title="c13">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:39_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E39">A13</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E14:39_2_1_1_1_1_1_1"/>
(  <a href="modal_1.html#K5" title="MODAL_1:func.5">branchdeg</a> <font color="Maroon" title="c13">v</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 &amp; (  not  <a href="modal_1.html#K5" title="MODAL_1:func.5">branchdeg</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c11">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> or  ex <font color="Olive" title="b1">k</font> being    <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>  st <font color="Maroon" title="c11">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">3,<font color="Olive" title="b1">k</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> ) &amp; (  not  <a href="modal_1.html#K5" title="MODAL_1:func.5">branchdeg</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c11">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> or <font color="Maroon" title="c11">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">1,1</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> ) &amp; (  <a href="modal_1.html#K5" title="MODAL_1:func.5">branchdeg</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 implies <font color="Maroon" title="c11">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">2,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_2_1_1_1"><span class="lab"><font color="Green" title="E33">A7</font></span></a>, <a class="txt" href="#E1:39_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E36">A10</font></span></a>, <a class="txt" href="#E11:39_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E43">A23</font></span></a>, <a class="ref" href="funcop_1.html#T7" title="FUNCOP_1:th.7">FUNCOP_1:7</a>, <a class="ref" href="trees_2.html#D12" title="TREES_2:def.12">TREES_2:def 12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="39_2_1_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:39_2_1_1_1_1_1_2"/>
 ex <font color="Olive" title="b1">s</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">A</font> &amp; <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b1">s</font> &amp; <font color="Maroon" title="c8">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="modal_1.html#K5" title="MODAL_1:func.5">branchdeg</a> <font color="Maroon" title="c13">v</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 &amp;  <a href="modal_1.html#K5" title="MODAL_1:func.5">branchdeg</a> <font color="Maroon" title="c13">v</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 &amp; (  not  <a href="modal_1.html#K5" title="MODAL_1:func.5">branchdeg</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c11">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> or  ex <font color="Olive" title="b1">k</font> being    <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>  st <font color="Maroon" title="c11">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">3,<font color="Olive" title="b1">k</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> ) &amp; (  not  <a href="modal_1.html#K5" title="MODAL_1:func.5">branchdeg</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c11">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> or <font color="Maroon" title="c11">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">1,1</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> ) &amp; (  <a href="modal_1.html#K5" title="MODAL_1:func.5">branchdeg</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 implies <font color="Maroon" title="c11">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">2,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> ) )</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">s</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E3:39_2_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E36">A24</font></span>: 
<font color="Maroon" title="c14">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">A</font>
 <span class="kw">and </span><br/><a NAME="E4:39_2_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E37">A25</font></span>: 
<font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c14">s</font>
 <span class="kw">and </span><br/><a NAME="E5:39_2_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E38">A26</font></span>: 
<font color="Maroon" title="c8">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">s</font>
 ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c15">s</font> = <font color="Maroon" title="c14">s</font> as    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">A</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:39_2_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E36">A24</font></span></a></span>;<br/><a NAME="E7:39_2_1_1_1_1_1_2"/>
 <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c13">v</font>, <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c15">s</font> <a href="wellord2.html#R2" title="WELLORD2:pred.2">are_equipotent</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_2_1_1_1_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c16">v9</font> = <font color="Maroon" title="c13">v</font> as    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c8">y</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:39"><span class="lab"><font color="Green" title="E31">A5</font></span></a>, <a class="txt" href="#E19:39"><span class="lab"><font color="Green" title="E32">A6</font></span></a>, <a class="txt" href="#E2:39_2_1_1_1"><span class="lab"><font color="Green" title="E34">A8</font></span></a>, <a class="ref" href="trees_1.html#D9" title="TREES_1:def.9">TREES_1:def 9</a></span>;<br/>
<a NAME="E2:39_2_1_1_1_1_1_2_1"/>
 <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c16">v9</font>, <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c15">s</font> <a href="wellord2.html#R2" title="WELLORD2:pred.2">are_equipotent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:39"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="txt" href="#E9:39"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="txt" href="#E4:39_2_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E37">A25</font></span></a>, <a class="ref" href="trees_2.html#T37" title="TREES_2:th.37">TREES_2:37</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:39_2_1_1_1_1_1_2_1"/>
 <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c13">v</font>, <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c15">s</font> <a href="wellord2.html#R2" title="WELLORD2:pred.2">are_equipotent</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:39"><span class="lab"><font color="Green" title="E31">A5</font></span></a>, <a class="txt" href="#E19:39"><span class="lab"><font color="Green" title="E32">A6</font></span></a>, <a class="txt" href="#E4:39_2_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E37">A25</font></span></a>, <a class="ref" href="modal_1.html#T1" target="_self" title="MODAL_1:th.1">Th3</a>, <a class="ref" href="modal_1.html#T14" target="_self" title="MODAL_1:th.14">Th18</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E8:39_2_1_1_1_1_1_2"/><span class="kw">then </span>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c13">v</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c15">s</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T5" title="CARD_1:th.5">CARD_1:5</a></span>;<br/><a NAME="E9:39_2_1_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A27</font></span>: 
 <a href="modal_1.html#K5" title="MODAL_1:func.5">branchdeg</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c15">s</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="trees_2.html#D12" title="TREES_2:def.12">TREES_2:def 12</a></span>;<br/><a NAME="E10:39_2_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E40">A28</font></span>: 
 <a href="modal_1.html#K5" title="MODAL_1:func.5">branchdeg</a> <font color="Maroon" title="c15">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="modal_1.html#D5" target="_self" title="MODAL_1:def.5">Def6</a></span>;<br/><span class="kw">hence </span><a NAME="E11:39_2_1_1_1_1_1_2"/>
 <a href="modal_1.html#K5" title="MODAL_1:func.5">branchdeg</a> <font color="Maroon" title="c13">v</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:39_2_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E39">A27</font></span></a>, <a class="ref" href="trees_2.html#D12" title="TREES_2:def.12">TREES_2:def 12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="modal_1.html#K5" title="MODAL_1:func.5">branchdeg</a> <font color="Maroon" title="c13">v</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 &amp; (  not  <a href="modal_1.html#K5" title="MODAL_1:func.5">branchdeg</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c11">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> or  ex <font color="Olive" title="b1">k</font> being    <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>  st <font color="Maroon" title="c11">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">3,<font color="Olive" title="b1">k</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> ) &amp; (  not  <a href="modal_1.html#K5" title="MODAL_1:func.5">branchdeg</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c11">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> or <font color="Maroon" title="c11">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">1,1</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> ) &amp; (  <a href="modal_1.html#K5" title="MODAL_1:func.5">branchdeg</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 implies <font color="Maroon" title="c11">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">2,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> ) )</span><br/><a NAME="E12:39_2_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E41">A29</font></span>: 
(  not  <a href="modal_1.html#K5" title="MODAL_1:func.5">branchdeg</a> <font color="Maroon" title="c15">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c1">A</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c15">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> or <font color="Maroon" title="c1">A</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c15">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">1,1</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="modal_1.html#D5" target="_self" title="MODAL_1:def.5">Def6</a></span>;<br/><a NAME="E13:39_2_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E42">A30</font></span>: 
(  <a href="modal_1.html#K5" title="MODAL_1:func.5">branchdeg</a> <font color="Maroon" title="c15">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 implies <font color="Maroon" title="c1">A</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c15">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">2,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="modal_1.html#D5" target="_self" title="MODAL_1:def.5">Def6</a></span>;<br/><a NAME="E14:39_2_1_1_1_1_1_2"/>
(  not  <a href="modal_1.html#K5" title="MODAL_1:func.5">branchdeg</a> <font color="Maroon" title="c15">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c1">A</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c15">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> or  ex <font color="Olive" title="b1">m</font> being    <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>  st <font color="Maroon" title="c1">A</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c15">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">3,<font color="Olive" title="b1">m</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="modal_1.html#D5" target="_self" title="MODAL_1:def.5">Def6</a></span>;<br/><span class="kw">hence </span><a NAME="E15:39_2_1_1_1_1_1_2"/>
(  <a href="modal_1.html#K5" title="MODAL_1:func.5">branchdeg</a> <font color="Maroon" title="c13">v</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 &amp; (  not  <a href="modal_1.html#K5" title="MODAL_1:func.5">branchdeg</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c11">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> or  ex <font color="Olive" title="b1">k</font> being    <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>  st <font color="Maroon" title="c11">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">3,<font color="Olive" title="b1">k</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> ) &amp; (  not  <a href="modal_1.html#K5" title="MODAL_1:func.5">branchdeg</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c11">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> or <font color="Maroon" title="c11">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">1,1</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> ) &amp; (  <a href="modal_1.html#K5" title="MODAL_1:func.5">branchdeg</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 implies <font color="Maroon" title="c11">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">2,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:39_2_1_1_1"><span class="lab"><font color="Green" title="E33">A7</font></span></a>, <a class="txt" href="#E5:39_2_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E38">A26</font></span></a>, <a class="txt" href="#E10:39_2_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E40">A28</font></span></a>, <a class="txt" href="#E12:39_2_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E41">A29</font></span></a>, <a class="txt" href="#E13:39_2_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E42">A30</font></span></a>, <a class="txt" href="#E9:39_2_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E39">A27</font></span></a>, <a class="ref" href="trees_2.html#D12" title="TREES_2:def.12">TREES_2:def 12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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><span class="kw">hence </span><a NAME="E5:39_2_1_1_1"/>
(  <a href="modal_1.html#K5" title="MODAL_1:func.5">branchdeg</a> <font color="Maroon" title="c13">v</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 &amp; (  not  <a href="modal_1.html#K5" title="MODAL_1:func.5">branchdeg</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c11">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> or  ex <font color="Olive" title="b1">k</font> being    <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>  st <font color="Maroon" title="c11">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">3,<font color="Olive" title="b1">k</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> ) &amp; (  not  <a href="modal_1.html#K5" title="MODAL_1:func.5">branchdeg</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c11">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> or <font color="Maroon" title="c11">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">1,1</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> ) &amp; (  <a href="modal_1.html#K5" title="MODAL_1:func.5">branchdeg</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 implies <font color="Maroon" title="c11">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">2,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="39_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:39_2_1_1_2"/>
 ex <font color="Olive" title="b1">s</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">B</font> &amp; <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b1">s</font> &amp; <font color="Maroon" title="c11">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="modal_1.html#K5" title="MODAL_1:func.5">branchdeg</a> <font color="Maroon" title="c13">v</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 &amp; (  not  <a href="modal_1.html#K5" title="MODAL_1:func.5">branchdeg</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c11">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> or  ex <font color="Olive" title="b1">k</font> being    <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>  st <font color="Maroon" title="c11">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">3,<font color="Olive" title="b1">k</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> ) &amp; (  not  <a href="modal_1.html#K5" title="MODAL_1:func.5">branchdeg</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c11">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> or <font color="Maroon" title="c11">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">1,1</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> ) &amp; (  <a href="modal_1.html#K5" title="MODAL_1:func.5">branchdeg</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 implies <font color="Maroon" title="c11">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">2,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> ) )</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">s</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E3:39_2_1_1_2"/><span class="lab"><font color="Green" title="E33">A31</font></span>: 
<font color="Maroon" title="c14">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">B</font>
 <span class="kw">and </span><br/><a NAME="E4:39_2_1_1_2"/><span class="lab"><font color="Green" title="E34">A32</font></span>: 
<font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c14">s</font>
 <span class="kw">and </span><br/><a NAME="E5:39_2_1_1_2"/><span class="lab"><font color="Green" title="E35">A33</font></span>: 
<font color="Maroon" title="c11">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">B</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">s</font>
 ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c15">s</font> = <font color="Maroon" title="c14">s</font> as    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">B</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:39_2_1_1_2"><span class="lab"><font color="Green" title="E33">A31</font></span></a></span>;<br/><a NAME="E7:39_2_1_1_2"/>
 <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c13">v</font>, <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c15">s</font> <a href="wellord2.html#R2" title="WELLORD2:pred.2">are_equipotent</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:39"><span class="lab"><font color="Green" title="E31">A5</font></span></a>, <a class="txt" href="#E19:39"><span class="lab"><font color="Green" title="E32">A6</font></span></a>, <a class="txt" href="#E4:39_2_1_1_2"><span class="lab"><font color="Green" title="E34">A32</font></span></a>, <a class="ref" href="trees_2.html#T37" title="TREES_2:th.37">TREES_2:37</a></span>;<br/><a NAME="E8:39_2_1_1_2"/><span class="kw">then </span>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c13">v</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c15">s</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T5" title="CARD_1:th.5">CARD_1:5</a></span>;<br/><a NAME="E9:39_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A34</font></span>: 
 <a href="modal_1.html#K5" title="MODAL_1:func.5">branchdeg</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c15">s</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="trees_2.html#D12" title="TREES_2:def.12">TREES_2:def 12</a></span>;<br/><a NAME="E10:39_2_1_1_2"/><span class="lab"><font color="Green" title="E37">A35</font></span>: 
(  <a href="modal_1.html#K5" title="MODAL_1:func.5">branchdeg</a> <font color="Maroon" title="c15">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 implies <font color="Maroon" title="c2">B</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c15">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">2,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="modal_1.html#D5" target="_self" title="MODAL_1:def.5">Def6</a></span>;<br/><a NAME="E11:39_2_1_1_2"/><span class="lab"><font color="Green" title="E38">A36</font></span>: 
(  not  <a href="modal_1.html#K5" title="MODAL_1:func.5">branchdeg</a> <font color="Maroon" title="c15">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c2">B</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c15">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> or <font color="Maroon" title="c2">B</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c15">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">1,1</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="modal_1.html#D5" target="_self" title="MODAL_1:def.5">Def6</a></span>;<br/><a NAME="E12:39_2_1_1_2"/><span class="lab"><font color="Green" title="E39">A37</font></span>: 
(  not  <a href="modal_1.html#K5" title="MODAL_1:func.5">branchdeg</a> <font color="Maroon" title="c15">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c2">B</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c15">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> or  ex <font color="Olive" title="b1">m</font> being    <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>  st <font color="Maroon" title="c2">B</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c15">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">3,<font color="Olive" title="b1">m</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="modal_1.html#D5" target="_self" title="MODAL_1:def.5">Def6</a></span>;<br/><a NAME="E13:39_2_1_1_2"/>
 <a href="modal_1.html#K5" title="MODAL_1:func.5">branchdeg</a> <font color="Maroon" title="c15">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2
 <span class="kw">by</span> <span class="lab"><a class="ref" href="modal_1.html#D5" target="_self" title="MODAL_1:def.5">Def6</a></span>;<br/><span class="kw">hence </span><a NAME="E14:39_2_1_1_2"/>
(  <a href="modal_1.html#K5" title="MODAL_1:func.5">branchdeg</a> <font color="Maroon" title="c13">v</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 &amp; (  not  <a href="modal_1.html#K5" title="MODAL_1:func.5">branchdeg</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c11">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> or  ex <font color="Olive" title="b1">k</font> being    <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>  st <font color="Maroon" title="c11">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">3,<font color="Olive" title="b1">k</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> ) &amp; (  not  <a href="modal_1.html#K5" title="MODAL_1:func.5">branchdeg</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c11">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> or <font color="Maroon" title="c11">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">1,1</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> ) &amp; (  <a href="modal_1.html#K5" title="MODAL_1:func.5">branchdeg</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 implies <font color="Maroon" title="c11">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">2,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:39_2_1_1_2"><span class="lab"><font color="Green" title="E35">A33</font></span></a>, <a class="txt" href="#E12:39_2_1_1_2"><span class="lab"><font color="Green" title="E39">A37</font></span></a>, <a class="txt" href="#E11:39_2_1_1_2"><span class="lab"><font color="Green" title="E38">A36</font></span></a>, <a class="txt" href="#E10:39_2_1_1_2"><span class="lab"><font color="Green" title="E37">A35</font></span></a>, <a class="txt" href="#E9:39_2_1_1_2"><span class="lab"><font color="Green" title="E36">A34</font></span></a>, <a class="ref" href="trees_2.html#D12" title="TREES_2:def.12">TREES_2:def 12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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>
<span class="kw">hence </span><a NAME="E2:39_2"/>
(  <a href="modal_1.html#K5" title="MODAL_1:func.5">branchdeg</a> <font color="Maroon" title="c13">v</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 &amp; (  not  <a href="modal_1.html#K5" title="MODAL_1:func.5">branchdeg</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c11">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> or  ex <font color="Olive" title="b1">k</font> being    <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>  st <font color="Maroon" title="c11">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">3,<font color="Olive" title="b1">k</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> ) &amp; (  not  <a href="modal_1.html#K5" title="MODAL_1:func.5">branchdeg</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c11">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> or <font color="Maroon" title="c11">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">1,1</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> ) &amp; (  <a href="modal_1.html#K5" title="MODAL_1:func.5">branchdeg</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 implies <font color="Maroon" title="c11">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c13">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">2,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E21:39"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 2</span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">2,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="trees_2.html#K7" title="TREES_2:func.7">with-replacement</a> (<span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>,<font color="Maroon" title="c1">A</font>)</span>)</span> <a href="trees_2.html#K7" title="TREES_2:func.7">with-replacement</a> (<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default">1</span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>,<font color="Maroon" title="c2">B</font>) is   <a href="modal_1.html#NM3" title="MODAL_1:NM.3">MP-wff</a>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="modal_1.html#D5" target="_self" title="MODAL_1:def.5">Def6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
