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

<a NAME="E1:44_1_1"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="trees_4.html#K13" target="_self">&lt;-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:44_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="trees_4.html#K13" target="_self">&lt;-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:44_1_1_1"/><i><font color="Green">E16</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="trees_4.html#K13" target="_self">&lt;-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="trees_4.html#K13" target="_self">&lt;-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="trees_4.html#NM1">Node</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="trees_4.html#K13" target="_self">&lt;-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <b>by </b><i><a class="txt" href="trees_4.html#E3:44_1_1_1"><i><font color="Green">E16</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> =  <a href="xboole_0.html#K1">{}</a>  as   <a href="trees_4.html#NM1">Node</a> of <font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="ref" href="trees_1.html#T47">TREES_1:47</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_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="E1:44_1_1_1_1"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> or ex <font color="Olive">b<sub>1</sub></font> being  <a href="trees_4.html#NM1">Node</a> of <font color="Maroon">c<sub>2</sub></font>ex <font color="Olive">b<sub>2</sub></font> being  <a href="trees_4.html#NM1">Node</a> of <font color="Maroon">c<sub>3</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K3">Leaves</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="trees_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Olive">b<sub>2</sub></font> ) )
 </span><b>by </b><i><a class="ref" href="trees_4.html#D7" target="_self">Def7</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:44_1_1_1_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="trees_4.html#NM1">Node</a> of <font color="Maroon">c<sub>2</sub></font> ;<br/><div><b>hereby </b>
<div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_1_1_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="E1:44_1_1_1_1_1_1_1"/>
( ( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K3">Leaves</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="trees_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> ) or  not <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K3">Leaves</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> or <font color="Maroon">c<sub>2</sub></font> <a href="trees_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:44_1_1_1_1_1_1_1_1"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K3">Leaves</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="trees_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/><div class="add"><a NAME="E2:44_1_1_1_1_1_1_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="trees_4.html#K13" target="_self">&lt;-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="trees_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="ref" href="trees_4.html#D7" target="_self">Def7</a>, <a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>;<br/><b>hence </b><a NAME="E3:44_1_1_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="trees_4.html#E3:44_1_1_1"><i><font color="Green">E16</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="44_1_1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:44_1_1_1_1_1_1_1_2"/>
(  not <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K3">Leaves</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> or <font color="Maroon">c<sub>2</sub></font> <a href="trees_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/><div class="add"><a NAME="E2:44_1_1_1_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="trees_4.html#K13" target="_self">&lt;-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="trees_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="trees_4.html#D7" target="_self">Def7</a></i>;<br/><b>hence </b><a NAME="E3:44_1_1_1_1_1_1_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="trees_4.html#E3:44_1_1_1"><i><font color="Green">E16</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="44_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:44_1_1_1_1_2"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="trees_4.html#NM1">Node</a> of <font color="Maroon">c<sub>2</sub></font>ex <font color="Olive">b<sub>2</sub></font> being  <a href="trees_4.html#NM1">Node</a> of <font color="Maroon">c<sub>3</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K3">Leaves</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="trees_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Olive">b<sub>2</sub></font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="trees_4.html#NM1">Node</a> of <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>10</sub></font> being   <a href="trees_4.html#NM1">Node</a> of <font color="Maroon">c<sub>3</sub></font><b> such that </b><br/><a NAME="E3:44_1_1_1_1_2"/><i><font color="Green">E17</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K3">Leaves</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="trees_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>10</sub></font> )
 ;<br/><a NAME="E4:44_1_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="trees_4.html#K13" target="_self">&lt;-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="trees_2.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="trees_4.html#E3:44_1_1_1_1_2"><i><font color="Green">E17</font></i></a>, <a class="ref" href="trees_4.html#D7" target="_self">Def7</a></i>;<br/><b>hence </b><a NAME="E5:44_1_1_1_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="trees_4.html#E3:44_1_1_1"><i><font color="Green">E16</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="E2:44_1_1"/>
<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="trees_4.html#K13" target="_self">&lt;-</a> <font color="Maroon">c<sub>3</sub></font> is   <a href="trees_2.html#NM6">DecoratedTree</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="trees_2.html#D9">TREES_2:def 9</a></i>;<br/>


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