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

<b>let </b><font color="Maroon" title="c4">T1</font>, <font color="Maroon" title="c5">T2</font> be   <a href="trees_2.html#NM4" title="TREES_2:NM.4">DecoratedTree</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:43_1_2"/><i><font color="Green" title="E15">A16</font></i>: 
for <font color="Olive" title="b1">p</font> being  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> holds <br/> ( <font color="Olive" title="b1">p</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">T1</font> iff ( <font color="Olive" title="b1">p</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="c1">T</font> or  ex <font color="Olive" title="b2">q</font> being  <a href="trees_4.html#NM1" title="TREES_4:NM.1">Node</a> of <font color="Maroon" title="c1">T</font> ex <font color="Olive" title="b3">r</font> being  <a href="trees_4.html#NM1" title="TREES_4:NM.1">Node</a> of <font color="Maroon" title="c2">T'</font> st <br/>( <font color="Olive" title="b2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_1.html#K3" title="TREES_1:func.3">Leaves</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c1">T</font></span>)</span> &amp; <font color="Maroon" title="c1">T</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">q</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b3">r</font> ) ) )
 <b>and </b><br/><a NAME="E2:43_1_2"/><i><font color="Green" title="E16">A17</font></i>: 
for <font color="Olive" title="b1">p</font> being  <a href="trees_4.html#NM1" title="TREES_4:NM.1">Node</a> of <font color="Maroon" title="c1">T</font>  st (  not <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_1.html#K3" title="TREES_1:func.3">Leaves</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c1">T</font></span>)</span> or <font color="Maroon" title="c1">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#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">x</font> ) holds <br/><font color="Maroon" title="c4">T1</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" title="c1">T</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">p</font>
 <b>and </b><br/><a NAME="E3:43_1_2"/><i><font color="Green" title="E17">A18</font></i>: 
for <font color="Olive" title="b1">p</font> being  <a href="trees_4.html#NM1" title="TREES_4:NM.1">Node</a> of <font color="Maroon" title="c1">T</font><br/> for <font color="Olive" title="b2">q</font> being  <a href="trees_4.html#NM1" title="TREES_4:NM.1">Node</a> of <font color="Maroon" title="c2">T'</font>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_1.html#K3" title="TREES_1:func.3">Leaves</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c1">T</font></span>)</span> &amp; <font color="Maroon" title="c1">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" title="c3">x</font> holds <br/><font color="Maroon" title="c4">T1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b2">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">T'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">q</font>
 <b>and </b><br/><a NAME="E4:43_1_2"/><i><font color="Green" title="E18">A19</font></i>: 
for <font color="Olive" title="b1">p</font> being  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> holds <br/> ( <font color="Olive" title="b1">p</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="c5">T2</font> iff ( <font color="Olive" title="b1">p</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="c1">T</font> or  ex <font color="Olive" title="b2">q</font> being  <a href="trees_4.html#NM1" title="TREES_4:NM.1">Node</a> of <font color="Maroon" title="c1">T</font> ex <font color="Olive" title="b3">r</font> being  <a href="trees_4.html#NM1" title="TREES_4:NM.1">Node</a> of <font color="Maroon" title="c2">T'</font> st <br/>( <font color="Olive" title="b2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_1.html#K3" title="TREES_1:func.3">Leaves</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c1">T</font></span>)</span> &amp; <font color="Maroon" title="c1">T</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">q</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b3">r</font> ) ) )
 <b>and </b><br/><a NAME="E5:43_1_2"/><i><font color="Green" title="E19">A20</font></i>: 
for <font color="Olive" title="b1">p</font> being  <a href="trees_4.html#NM1" title="TREES_4:NM.1">Node</a> of <font color="Maroon" title="c1">T</font>  st (  not <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_1.html#K3" title="TREES_1:func.3">Leaves</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c1">T</font></span>)</span> or <font color="Maroon" title="c1">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#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">x</font> ) holds <br/><font color="Maroon" title="c5">T2</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" title="c1">T</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">p</font>
 <b>and </b><br/><a NAME="E6:43_1_2"/><i><font color="Green" title="E20">A21</font></i>: 
for <font color="Olive" title="b1">p</font> being  <a href="trees_4.html#NM1" title="TREES_4:NM.1">Node</a> of <font color="Maroon" title="c1">T</font><br/> for <font color="Olive" title="b2">q</font> being  <a href="trees_4.html#NM1" title="TREES_4:NM.1">Node</a> of <font color="Maroon" title="c2">T'</font>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_1.html#K3" title="TREES_1:func.3">Leaves</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c1">T</font></span>)</span> &amp; <font color="Maroon" title="c1">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" title="c3">x</font> holds <br/><font color="Maroon" title="c5">T2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b2">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">T'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">q</font>
 ;<br/>

<a NAME="E7:43_1_2"/><i><font color="Green" title="E21">A22</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c4">T1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c5">T2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">p</font> be    <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> ; <i><font color="Red">:: according to </font></i><a class="ref" href="trees_2.html#D1" title="TREES_2:def.1">TREES_2:def 1</a><br/>

<a NAME="E1:43_1_2_1"/>
( (  not <font color="Maroon" title="c6">p</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">T1</font> or <font color="Maroon" title="c6">p</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="c1">T</font> or  ex <font color="Olive" title="b1">q</font> being  <a href="trees_4.html#NM1" title="TREES_4:NM.1">Node</a> of <font color="Maroon" title="c1">T</font> ex <font color="Olive" title="b2">r</font> being  <a href="trees_4.html#NM1" title="TREES_4:NM.1">Node</a> of <font color="Maroon" title="c2">T'</font> st <br/>( <font color="Olive" title="b1">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_1.html#K3" title="TREES_1:func.3">Leaves</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c1">T</font></span>)</span> &amp; <font color="Maroon" title="c1">T</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Maroon" title="c6">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">q</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b2">r</font> ) ) &amp; ( ( <font color="Maroon" title="c6">p</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="c1">T</font> or  ex <font color="Olive" title="b1">q</font> being  <a href="trees_4.html#NM1" title="TREES_4:NM.1">Node</a> of <font color="Maroon" title="c1">T</font> ex <font color="Olive" title="b2">r</font> being  <a href="trees_4.html#NM1" title="TREES_4:NM.1">Node</a> of <font color="Maroon" title="c2">T'</font> st <br/>( <font color="Olive" title="b1">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_1.html#K3" title="TREES_1:func.3">Leaves</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c1">T</font></span>)</span> &amp; <font color="Maroon" title="c1">T</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Maroon" title="c6">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">q</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b2">r</font> ) ) implies <font color="Maroon" title="c6">p</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">T1</font> ) &amp; (  not <font color="Maroon" title="c6">p</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="c5">T2</font> or <font color="Maroon" title="c6">p</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="c1">T</font> or  ex <font color="Olive" title="b1">q</font> being  <a href="trees_4.html#NM1" title="TREES_4:NM.1">Node</a> of <font color="Maroon" title="c1">T</font> ex <font color="Olive" title="b2">r</font> being  <a href="trees_4.html#NM1" title="TREES_4:NM.1">Node</a> of <font color="Maroon" title="c2">T'</font> st <br/>( <font color="Olive" title="b1">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_1.html#K3" title="TREES_1:func.3">Leaves</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c1">T</font></span>)</span> &amp; <font color="Maroon" title="c1">T</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Maroon" title="c6">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">q</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b2">r</font> ) ) &amp; ( ( <font color="Maroon" title="c6">p</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="c1">T</font> or  ex <font color="Olive" title="b1">q</font> being  <a href="trees_4.html#NM1" title="TREES_4:NM.1">Node</a> of <font color="Maroon" title="c1">T</font> ex <font color="Olive" title="b2">r</font> being  <a href="trees_4.html#NM1" title="TREES_4:NM.1">Node</a> of <font color="Maroon" title="c2">T'</font> st <br/>( <font color="Olive" title="b1">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_1.html#K3" title="TREES_1:func.3">Leaves</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c1">T</font></span>)</span> &amp; <font color="Maroon" title="c1">T</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Maroon" title="c6">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">q</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b2">r</font> ) ) implies <font color="Maroon" title="c6">p</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="c5">T2</font> ) )
 
<b>by </b><i><a class="txt" href="trees_4.html#E1:43_1_2"><i><font color="Green" title="E15">A16</font></i></a>, <a class="txt" href="trees_4.html#E4:43_1_2"><i><font color="Green" title="E18">A19</font></i></a></i>;<br/>
<b>hence </b><a NAME="E2:43_1_2_1"/>
( (  not <font color="Maroon" title="c6">p</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">T1</font> or <font color="Maroon" title="c6">p</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="c5">T2</font> ) &amp; (  not <font color="Maroon" title="c6">p</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="c5">T2</font> or <font color="Maroon" title="c6">p</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">T1</font> ) )
 ;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon" title="c6">p'</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  as   <a href="trees_4.html#NM1" title="TREES_4:NM.1">Node</a> of <font color="Maroon" title="c2">T'</font> <b>by </b><i><a class="ref" href="trees_1.html#T47" title="TREES_1:th.47">TREES_1:47</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:43_1_2_2"/>
<font color="Maroon" title="c7">y</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">T1</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c8">p</font> = <font color="Maroon" title="c7">y</font> as   <a href="trees_4.html#NM1" title="TREES_4:NM.1">Node</a> of <font color="Maroon" title="c4">T1</font> ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:43_1_2_2_1"/>
( <font color="Maroon" title="c8">p</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="c1">T</font> or  ex <font color="Olive" title="b1">q</font> being  <a href="trees_4.html#NM1" title="TREES_4:NM.1">Node</a> of <font color="Maroon" title="c1">T</font> ex <font color="Olive" title="b2">r</font> being  <a href="trees_4.html#NM1" title="TREES_4:NM.1">Node</a> of <font color="Maroon" title="c2">T'</font> st <br/>( <font color="Olive" title="b1">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_1.html#K3" title="TREES_1:func.3">Leaves</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c1">T</font></span>)</span> &amp; <font color="Maroon" title="c1">T</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Maroon" title="c8">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">q</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b2">r</font> ) )
 </span><b>by </b><i><a class="txt" href="trees_4.html#E1:43_1_2"><i><font color="Green" title="E15">A16</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:43_1_2_2_1_1"/>
<font color="Maroon" title="c8">p</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="c1">T</font>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c9">p</font> = <font color="Maroon" title="c8">p</font> as   <a href="trees_4.html#NM1" title="TREES_4:NM.1">Node</a> of <font color="Maroon" title="c1">T</font> ;<br/><div><b>hereby </b>
<div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_2_2_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:43_1_2_2_1_1_1_1"/>
( ( <font color="Maroon" title="c9">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_1.html#K3" title="TREES_1:func.3">Leaves</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c1">T</font></span>)</span> &amp; <font color="Maroon" title="c1">T</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> ) or  not <font color="Maroon" title="c9">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_1.html#K3" title="TREES_1:func.3">Leaves</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c1">T</font></span>)</span> or <font color="Maroon" title="c1">T</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">x</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_2_2_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:43_1_2_2_1_1_1_1_1"/><i><font color="Green" title="E22">A23</font></i>: 
( <font color="Maroon" title="c9">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_1.html#K3" title="TREES_1:func.3">Leaves</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c1">T</font></span>)</span> &amp; <font color="Maroon" title="c1">T</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> )
 ;<br/><div class="add"><a NAME="E2:43_1_2_2_1_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c4">T1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c6">p'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">T'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">p'</font> &amp; <font color="Maroon" title="c9">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c6">p'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p</font> )
 <b>by </b><i><a class="txt" href="trees_4.html#E3:43_1_2"><i><font color="Green" title="E17">A18</font></i></a>, <a class="ref" href="finseq_1.html#T47" title="FINSEQ_1:th.47">FINSEQ_1:47</a></i>;<br/><b>hence </b><a NAME="E3:43_1_2_2_1_1_1_1_1"/>
<font color="Maroon" title="c4">T1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">T2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">y</font>
 <b>by </b><i><a class="txt" href="trees_4.html#E6:43_1_2"><i><font color="Green" title="E20">A21</font></i></a>, <a class="txt" href="trees_4.html#E1:43_1_2_2_1_1_1_1_1"><i><font color="Green" title="E22">A23</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_2_2_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:43_1_2_2_1_1_1_1_2"/><i><font color="Green" title="E22">A24</font></i>: 
(  not <font color="Maroon" title="c9">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_1.html#K3" title="TREES_1:func.3">Leaves</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c1">T</font></span>)</span> or <font color="Maroon" title="c1">T</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">x</font> )
 ;<br/><div class="add"><a NAME="E2:43_1_2_2_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c4">T1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">T</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">p</font>
 <b>by </b><i><a class="txt" href="trees_4.html#E2:43_1_2"><i><font color="Green" title="E16">A17</font></i></a></i>;<br/><b>hence </b><a NAME="E3:43_1_2_2_1_1_1_1_2"/>
<font color="Maroon" title="c4">T1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">T2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">y</font>
 <b>by </b><i><a class="txt" href="trees_4.html#E5:43_1_2"><i><font color="Green" title="E19">A20</font></i></a>, <a class="txt" href="trees_4.html#E1:43_1_2_2_1_1_1_1_2"><i><font color="Green" title="E22">A24</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div>
<b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:43_1_2_2_1_2"/>
 ex <font color="Olive" title="b1">q</font> being  <a href="trees_4.html#NM1" title="TREES_4:NM.1">Node</a> of <font color="Maroon" title="c1">T</font> ex <font color="Olive" title="b2">r</font> being  <a href="trees_4.html#NM1" title="TREES_4:NM.1">Node</a> of <font color="Maroon" title="c2">T'</font> st <br/>( <font color="Olive" title="b1">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_1.html#K3" title="TREES_1:func.3">Leaves</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c1">T</font></span>)</span> &amp; <font color="Maroon" title="c1">T</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Maroon" title="c8">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">q</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b2">r</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c9">q</font> being   <a href="trees_4.html#NM1" title="TREES_4:NM.1">Node</a> of <font color="Maroon" title="c1">T</font>, <font color="Maroon" title="c10">r</font> being   <a href="trees_4.html#NM1" title="TREES_4:NM.1">Node</a> of <font color="Maroon" title="c2">T'</font><b> such that </b><br/><a NAME="E3:43_1_2_2_1_2"/><i><font color="Green" title="E22">A25</font></i>: 
( <font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_1.html#K3" title="TREES_1:func.3">Leaves</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c1">T</font></span>)</span> &amp; <font color="Maroon" title="c1">T</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Maroon" title="c8">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c10">r</font> )
 ;<br/><b>thus </b><font color="Maroon" title="c4">T1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">y</font> = 
<font color="Maroon" title="c2">T'</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">r</font>
<b>by </b><i><a class="txt" href="trees_4.html#E3:43_1_2"><i><font color="Green" title="E17">A18</font></i></a>, <a class="txt" href="trees_4.html#E3:43_1_2_2_1_2"><i><font color="Green" title="E22">A25</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c5">T2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">y</font>
<b>by </b><i><a class="txt" href="trees_4.html#E6:43_1_2"><i><font color="Green" title="E20">A21</font></i></a>, <a class="txt" href="trees_4.html#E3:43_1_2_2_1_2"><i><font color="Green" title="E22">A25</font></i></a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E10:43_1_2"/>
<font color="Maroon" title="c4">T1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">T2</font>
 <b>by </b><i><a class="txt" href="trees_4.html#E7:43_1_2"><i><font color="Green" title="E21">A22</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>


</div>
