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

<span class="kw">let </span><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></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">3,<font color="Maroon" title="c1">n</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> is   <a href="modal_1.html#NM3" title="MODAL_1:NM.3">MP-wff</a></span><br/>

<span class="kw">set </span><font color="Maroon" title="c2">x</font> = <span class="p1">(<span class="default"><a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></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">3,<font color="Maroon" title="c1">n</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>;<br/>
<a NAME="E1:42"/><span class="lab"><font color="Green" title="E28">A1</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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></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">3,<font color="Maroon" title="c1">n</font></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="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a></span><a href="tarski.html#K1" title="TARSKI: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#T29" title="TREES_1:th.29">TREES_1:29</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c3">x</font> = <span class="p1">(<span class="default"><a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></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">3,<font color="Maroon" title="c1">n</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> 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> ;<br/>
<a NAME="E3:42"/><span class="lab"><font color="Green" title="E29">A2</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
 
<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="E4:42"/>
 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="c3">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="c3">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="c3">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="c3">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="c3">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="c3">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="42_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">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="c3">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="c4">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="c4">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="c3">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c4">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="c3">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c4">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="c4">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c3">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c4">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="c3">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c4">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="c4">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 implies <font color="Maroon" title="c3">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c4">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="E1:42_1"/><span class="lab"><font color="Green" title="E30">A3</font></span>: 
 <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c4">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c5">y</font> =  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c4">v</font>;<br/>
<span class="kw">assume </span><a NAME="E1:42_1_1"/>
 not  <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c4">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <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="E2:42_1_1"/><span class="kw">then </span>
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c4">v</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="c4">v</font>
 
;<br/>
<a NAME="E3:42_1_1"/><span class="kw">then </span>
 the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c4">v</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="c4">v</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">m</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> where <font color="Olive" title="b1">m</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="c4">v</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">m</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="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">x</font> </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/>
<a NAME="E4:42_1_1"/><span class="kw">then </span>
 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 <br/>(  the    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c4">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v</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">m</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> &amp; <font color="Maroon" title="c4">v</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">m</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="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">x</font> )
 
;<br/>
<span class="kw">hence </span><a NAME="E5:42_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42"><span class="lab"><font color="Green" title="E28">A1</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 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="E2:42_1"/>
 <a href="modal_1.html#K5" title="MODAL_1:func.5">branchdeg</a> <font color="Maroon" title="c4">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="card_1.html#T27" title="CARD_1:th.27">CARD_1:27</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"> ( (  not  <a href="modal_1.html#K5" title="MODAL_1:func.5">branchdeg</a> <font color="Maroon" title="c4">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="c3">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c4">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="c3">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c4">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="c4">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c3">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c4">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="c3">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c4">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="c4">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 implies <font color="Maroon" title="c3">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c4">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/>

<span class="kw">thus </span><a NAME="E3:42_1"/>
(  not  <a href="modal_1.html#K5" title="MODAL_1:func.5">branchdeg</a> <font color="Maroon" title="c4">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="c3">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c4">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">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="c3">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c4">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">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="txt" href="#E3:42"><span class="lab"><font color="Green" title="E29">A2</font></span></a>, <a class="ref" href="funcop_1.html#T7" title="FUNCOP_1:th.7">FUNCOP_1:7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  not  <a href="modal_1.html#K5" title="MODAL_1:func.5">branchdeg</a> <font color="Maroon" title="c4">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c3">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c4">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="c3">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c4">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="c4">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 implies <font color="Maroon" title="c3">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c4">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/>

<span class="kw">thus </span><a NAME="E4:42_1"/>
( (  not  <a href="modal_1.html#K5" title="MODAL_1:func.5">branchdeg</a> <font color="Maroon" title="c4">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c3">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c4">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="c3">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c4">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="c4">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 implies <font color="Maroon" title="c3">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c4">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="#E1:42_1"><span class="lab"><font color="Green" title="E30">A3</font></span></a>, <a class="ref" href="card_1.html#T27" title="CARD_1:th.27">CARD_1:27</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>
<span class="kw">hence </span><a NAME="E5:42"/>
<span class="p1">(<span class="default"><a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></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">3,<font color="Maroon" title="c1">n</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> 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>
