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

<b>set </b><font color="Maroon" title="c1">A</font> = <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/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>( $1 is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="trees_2.html#NM6" title="TREES_2:NM.6">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> &amp; ( for <font color="Olive" title="b1">y</font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="trees_2.html#NM6" title="TREES_2:NM.6">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>  st <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 holds <br/>(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Olive" title="b1">y</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; ( for <font color="Olive" title="b2">v</font> being   <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Olive" title="b1">y</font> holds <br/> (  <a href="modal_1.html#K5" title="MODAL_1:func.5">branchdeg</a> <font color="Olive" title="b2">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="b2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Olive" title="b1">y</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Olive" title="b2">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">0,0</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> or  ex <font color="Olive" title="b3">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="Olive" title="b1">y</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Olive" title="b2">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="b3">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="b2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Olive" title="b1">y</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Olive" title="b2">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">1,0</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> or <font color="Olive" title="b1">y</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Olive" title="b2">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="b2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 implies <font color="Olive" title="b1">y</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Olive" title="b2">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">2,0</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> ) ) ) ) ) );<br/>
<b>consider </b><font color="Maroon" title="c2">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E2:33_1_1"/><i><font color="Green" title="E23">A1</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> iff ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <span class="p1">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span>)</span>,<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> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>] ) )
 <b>from </b><i><a class="ref" href="xboole_0.html#S1" title="XBOOLE_0:sch.1">XBOOLE_0:sch 1</a>();<br/></i>
<a NAME="E3:33_1_1"/><i><font color="Green" title="E24">A2</font></i>: 
for <font color="Olive" title="b1">x</font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="trees_2.html#NM6" title="TREES_2:NM.6">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> holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> iff (  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Olive" title="b1">x</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; ( for <font color="Olive" title="b2">v</font> being   <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Olive" title="b1">x</font> holds <br/> (  <a href="modal_1.html#K5" title="MODAL_1:func.5">branchdeg</a> <font color="Olive" title="b2">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="b2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Olive" title="b1">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Olive" title="b2">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">0,0</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> or  ex <font color="Olive" title="b3">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="Olive" title="b1">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Olive" title="b2">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="b3">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="b2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Olive" title="b1">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Olive" title="b2">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">1,0</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> or <font color="Olive" title="b1">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Olive" title="b2">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="b2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 implies <font color="Olive" title="b1">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Olive" title="b2">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">2,0</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> ) ) ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x</font> be  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="trees_2.html#NM6" title="TREES_2:NM.6">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/>

<b>thus </b><a NAME="E1:33_1_1_1"/>
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font> implies (  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">x</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; ( 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#K1" title="RELAT_1:func.1">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> 0 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">0,0</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">1,0</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">2,0</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> ) ) ) ) )
 <b>by </b><i><a class="txt" href="modal_1.html#E2:33_1_1"><i><font color="Green" title="E23">A1</font></i></a></i>;<br/>

<b>assume </b><a NAME="E2:33_1_1_1"/>
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c3">x</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; ( 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#K1" title="RELAT_1:func.1">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> 0 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">0,0</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">1,0</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">2,0</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> ) ) ) )
 ;<br/>

<a NAME="E3:33_1_1_1"/><b>then </b><i><font color="Green" title="E25">A3</font></i>: 
for <font color="Olive" title="b1">y</font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="trees_2.html#NM6" title="TREES_2:NM.6">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>  st <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> holds <br/>(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Olive" title="b1">y</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; ( for <font color="Olive" title="b2">v</font> being   <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Olive" title="b1">y</font> holds <br/> (  <a href="modal_1.html#K5" title="MODAL_1:func.5">branchdeg</a> <font color="Olive" title="b2">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="b2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Olive" title="b1">y</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Olive" title="b2">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">0,0</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> or  ex <font color="Olive" title="b3">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="Olive" title="b1">y</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Olive" title="b2">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="b3">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="b2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Olive" title="b1">y</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Olive" title="b2">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">1,0</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> or <font color="Olive" title="b1">y</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Olive" title="b2">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="b2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 implies <font color="Olive" title="b1">y</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Olive" title="b2">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">2,0</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> ) ) ) )
 
;<br/>
<a NAME="E4:33_1_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <span class="p1">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span>)</span>,<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>
 
<b>by </b><i><a class="ref" href="modal_1.html#T13" target="_self" title="MODAL_1:th.13">Th13</a></i>;<br/>
<b>hence </b><a NAME="E5:33_1_1_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Y</font>
 <b>by </b><i><a class="txt" href="modal_1.html#E2:33_1_1"><i><font color="Green" title="E23">A1</font></i></a>, <a class="txt" href="modal_1.html#E3:33_1_1_1"><i><font color="Green" title="E25">A3</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>set </b><font color="Maroon" title="c3">t</font> =  <a href="trees_1.html#K2" title="TREES_1:func.2">elementary_tree</a> 0;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</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="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> = <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default">0,0</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>;<br/>
<b>consider </b><font color="Maroon" title="c4">T</font> being   <a href="trees_2.html#NM4" title="TREES_2:NM.4">DecoratedTree</a><b> such that </b><br/><a NAME="E5:33_1_1"/><i><font color="Green" title="E25">A4</font></i>: 
(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">T</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> 0 &amp; ( for <font color="Olive" title="b1">p</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 <font color="Olive" title="b1">p</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> 0 holds <br/><font color="Maroon" title="c4">T</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">p</font>) ) )
 <b>from </b><i><a class="ref" href="trees_2.html#S7" title="TREES_2:sch.7">TREES_2:sch 7</a>();<br/></i>
<a NAME="E6:33_1_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">T</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:33_1_1_2"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">T</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c6">z</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:33_1_1_2"/><i><font color="Green" title="E26">A5</font></i>: 
( <font color="Maroon" title="c6">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">T</font> &amp; <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">T</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">z</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:33_1_1_2"/>
<font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="modal_1.html#E5:33_1_1"><i><font color="Green" title="E25">A4</font></i></a>, <a class="txt" href="modal_1.html#E3:33_1_1_2"><i><font color="Green" title="E26">A5</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a>, <a class="ref" href="trees_1.html#T56" title="TREES_1:th.56">TREES_1:56</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c7">z</font> = <font color="Maroon" title="c6">z</font> as    <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>  <b>by </b><i><a class="ref" href="finseq_1.html#T29" title="FINSEQ_1:th.29">FINSEQ_1:29</a></i>;<br/>
<a NAME="E6:33_1_1_2"/>
<font color="Maroon" title="c4">T</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">z</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">0,0</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 
<b>by </b><i><a class="txt" href="modal_1.html#E5:33_1_1"><i><font color="Green" title="E25">A4</font></i></a>, <a class="txt" href="modal_1.html#E3:33_1_1_2"><i><font color="Green" title="E26">A5</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:33_1_1_2"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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>
 <b>by </b><i><a class="txt" href="modal_1.html#E3:33_1_1_2"><i><font color="Green" title="E26">A5</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c5">T</font> = <font color="Maroon" title="c4">T</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="trees_2.html#NM6" title="TREES_2:NM.6">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> <b>by </b><i><a class="txt" href="modal_1.html#E5:33_1_1"><i><font color="Green" title="E25">A4</font></i></a>, <a class="txt" href="modal_1.html#E31"><i><font color="Green" title="E17">Lm2</font></i></a>, <a class="ref" href="trees_2.html#D9" title="TREES_2:def.9">TREES_2:def 9</a></i>;<br/>
<a NAME="E8:33_1_1"/><i><font color="Green" title="E26">A6</font></i>: 
<font color="Maroon" title="c5">T</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="partfun1.html#K4" title="PARTFUN1:func.4">PFuncs</a> <span class="p1">(<span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span>)</span>,<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>
 
<b>by </b><i><a class="ref" href="modal_1.html#T13" target="_self" title="MODAL_1:th.13">Th13</a></i>;<br/>
<a NAME="E9:33_1_1"/>
for <font color="Olive" title="b1">y</font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="trees_2.html#NM6" title="TREES_2:NM.6">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>  st <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">T</font> holds <br/>(  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Olive" title="b1">y</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp; ( for <font color="Olive" title="b2">v</font> being   <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Olive" title="b1">y</font> holds <br/> (  <a href="modal_1.html#K5" title="MODAL_1:func.5">branchdeg</a> <font color="Olive" title="b2">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="b2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Olive" title="b1">y</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Olive" title="b2">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">0,0</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> or  ex <font color="Olive" title="b3">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="Olive" title="b1">y</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Olive" title="b2">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="b3">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="b2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Olive" title="b1">y</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Olive" title="b2">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">1,0</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> or <font color="Olive" title="b1">y</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Olive" title="b2">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="b2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 implies <font color="Olive" title="b1">y</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Olive" title="b2">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">2,0</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> ) ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">y</font> be  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="trees_2.html#NM6" title="TREES_2:NM.6">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/>

<b>assume </b><a NAME="E1:33_1_1_3"/><i><font color="Green" title="E27">A7</font></i>: 
<font color="Maroon" title="c6">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">T</font>
 ;<br/>

<b>thus </b><a NAME="E2:33_1_1_3"/>
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c6">y</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c7">v</font> be    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c6">y</font>;<br/>

<a NAME="E3:33_1_1_3"/><i><font color="Green" title="E28">A8</font></i>: 
 <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c7">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="33_1_1_3_1"><b>proof </b></a><div class="add">

<a NAME="E1:33_1_1_3_1"/><i><font color="Green" title="E29">A9</font></i>: 
 <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c7">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c7">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">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="c7">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">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="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c6">y</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="trees_2.html#D5" title="TREES_2:def.5">TREES_2:def 5</a></i>;<br/>
<b>assume </b><a NAME="E2:33_1_1_3_1"/><i><font color="Green" title="E30">A10</font></i>: 
not  <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c7">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/>

<b>consider </b><font color="Maroon" title="c8">x</font> being    <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="c7">v</font>;<br/>
<a NAME="E4:33_1_1_3_1"/>
<font color="Maroon" title="c8">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="c7">v</font>
 
<b>by </b><i><a class="txt" href="modal_1.html#E2:33_1_1_3_1"><i><font color="Green" title="E30">A10</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c9">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> <b> such that </b><br/><a NAME="E6:33_1_1_3_1"/><i><font color="Green" title="E31">A11</font></i>: 
( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">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="Maroon" title="c9">n</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> &amp; <font color="Maroon" title="c7">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="Maroon" title="c9">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="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c6">y</font> )
 <b>by </b><i><a class="txt" href="modal_1.html#E1:33_1_1_3_1"><i><font color="Green" title="E29">A9</font></i></a></i>;<br/>
<b>thus </b><a NAME="E7:33_1_1_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="modal_1.html#E5:33_1_1"><i><font color="Green" title="E25">A4</font></i></a>, <a class="txt" href="modal_1.html#E1:33_1_1_3"><i><font color="Green" title="E27">A7</font></i></a>, <a class="txt" href="modal_1.html#E6:33_1_1_3_1"><i><font color="Green" title="E31">A11</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a>, <a class="ref" href="trees_1.html#T56" title="TREES_1:th.56">TREES_1:56</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:33_1_1_3"/>
 <a href="modal_1.html#K5" title="MODAL_1:func.5">branchdeg</a> <font color="Maroon" title="c7">v</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2
 <b>by </b><i><a class="ref" href="card_1.html#T78" title="CARD_1:th.78">CARD_1:78</a>, <a class="ref" href="trees_2.html#D13" title="TREES_2:def.13">TREES_2:def 13</a></i>;<br/>

<b>thus </b><a NAME="E5:33_1_1_3"/>
( (  not  <a href="modal_1.html#K5" title="MODAL_1:func.5">branchdeg</a> <font color="Maroon" title="c7">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Maroon" title="c6">y</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c7">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">0,0</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="c6">y</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c7">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="c7">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c6">y</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c7">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">1,0</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> or <font color="Maroon" title="c6">y</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c7">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="c7">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 implies <font color="Maroon" title="c6">y</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c7">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">2,0</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> ) )
 <b>by </b><i><a class="txt" href="modal_1.html#E5:33_1_1"><i><font color="Green" title="E25">A4</font></i></a>, <a class="txt" href="modal_1.html#E1:33_1_1_3"><i><font color="Green" title="E27">A7</font></i></a>, <a class="txt" href="modal_1.html#E3:33_1_1_3"><i><font color="Green" title="E28">A8</font></i></a>, <a class="ref" href="card_1.html#T78" title="CARD_1:th.78">CARD_1:78</a>, <a class="ref" href="trees_2.html#D13" title="TREES_2:def.13">TREES_2:def 13</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c6">Y</font> = <font color="Maroon" title="c2">Y</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="txt" href="modal_1.html#E2:33_1_1"><i><font color="Green" title="E23">A1</font></i></a>, <a class="txt" href="modal_1.html#E8:33_1_1"><i><font color="Green" title="E26">A6</font></i></a></i>;<br/>
<a NAME="E11:33_1_1"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Y</font> holds <br/><font color="Olive" title="b1">x</font> is   <a href="trees_2.html#NM6" title="TREES_2:NM.6">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>
 
<b>by </b><i><a class="txt" href="modal_1.html#E2:33_1_1"><i><font color="Green" title="E23">A1</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c7">Y</font> = <font color="Maroon" title="c6">Y</font> as    <a href="modal_1.html#M1" title="MODAL_1:mode.1">DOMAIN_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> <b>by </b><i><a class="ref" href="modal_1.html#D5" target="_self" title="MODAL_1:def.5">Def5</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c7">Y</font>
;<br/>

<b>thus </b><a NAME="E13:33_1_1"/>
( ( for <font color="Olive" title="b1">x</font> being  <a href="trees_2.html#NM6" title="TREES_2:NM.6">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>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Y</font> holds <br/><font color="Olive" title="b1">x</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> ) &amp; ( for <font color="Olive" title="b1">x</font> being <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="trees_2.html#NM6" title="TREES_2:NM.6">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> holds <br/> ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Y</font> iff for <font color="Olive" title="b2">v</font> being   <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Olive" title="b1">x</font> holds <br/> (  <a href="modal_1.html#K5" title="MODAL_1:func.5">branchdeg</a> <font color="Olive" title="b2">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="b2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Olive" title="b1">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Olive" title="b2">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">0,0</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> or  ex <font color="Olive" title="b3">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="Olive" title="b1">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Olive" title="b2">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="b3">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="b2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Olive" title="b1">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Olive" title="b2">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">1,0</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> or <font color="Olive" title="b1">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Olive" title="b2">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="b2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 implies <font color="Olive" title="b1">x</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Olive" title="b2">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">2,0</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> ) ) ) ) )
 <b>by </b><i><a class="txt" href="modal_1.html#E2:33_1_1"><i><font color="Green" title="E23">A1</font></i></a>, <a class="txt" href="modal_1.html#E3:33_1_1"><i><font color="Green" title="E24">A2</font></i></a></i>;<br/>


</div>
