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

<b>set </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> <a href="trees_4.html#K4" target="_self">-tree</a> <font color="Maroon">c<sub>4</sub></font>;<br/>
<a NAME="E1:35_1_1"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="trees_4.html#K4" target="_self">-tree</a> <font color="Maroon">c<sub>4</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="35_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</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="E1:35_1_1_1"/>
<font color="Maroon">c<sub>6</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>3</sub></font> <a href="trees_4.html#K4" target="_self">-tree</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:35_1_1_1"/><i><font color="Green">E15</font></i>: 
( <font color="Maroon">c<sub>7</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>3</sub></font> <a href="trees_4.html#K4" target="_self">-tree</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="trees_4.html#K4" target="_self">-tree</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</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>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="trees_4.html#NM1">Node</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="trees_4.html#K4" target="_self">-tree</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <b>by </b><i><a class="txt" href="trees_4.html#E3:35_1_1_1"><i><font color="Green">E15</font></i></a></i>;<br/>
<a NAME="E5:35_1_1_1"/><i><font color="Green">E16</font></i>: 
(  <a href="trees_3.html#K13">tree</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="trees_4.html#K4" target="_self">-tree</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="trees_3.html#K13">tree</a> <span class="p2">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="trees_2.html#K2">-level</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> </span>}</span>  &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="trees_4.html#K4" target="_self">-tree</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2">doms</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="trees_4.html#K4" target="_self">-tree</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="trees_2.html#K5">|</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) )
 
<b>by </b><i><a class="ref" href="trees_4.html#D4" target="_self">Def4</a>, <a class="ref" href="trees_4.html#T10" target="_self">Th10</a>, <a class="ref" href="trees_3.html#T40">TREES_3:40</a>, <a class="ref" href="trees_3.html#T52">TREES_3:52</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:35_1_1_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>then consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">c<sub>10</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:35_1_1_1_1"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="ref" href="modal_1.html#T4">MODAL_1:4</a></i>;<br/><a NAME="E4:35_1_1_1_1"/>
( <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> <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>3</sub></font> <a href="trees_4.html#K4" target="_self">-tree</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="txt" href="trees_4.html#E3:35_1_1_1_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a>, <a class="ref" href="trees_1.html#T46">TREES_1:46</a></i>;<br/><a NAME="E5:35_1_1_1_1"/><b>then </b><i><font color="Green">E18</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="trees_4.html#K4" target="_self">-tree</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="trees_1.html#K4">|</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> &amp; <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="trees_4.html#K4" target="_self">-tree</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="trees_2.html#K2">-level</a> 1 &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="trees_4.html#K4" target="_self">-tree</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="trees_2.html#K5">|</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="trees_4.html#K4" target="_self">-tree</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="trees_1.html#K4">|</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="trees_4.html#E3:35_1_1_1_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="trees_1.html#D9">TREES_1:def 9</a>, <a class="ref" href="trees_2.html#D11">TREES_2:def 11</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E7:35_1_1_1_1"/><i><font color="Green">E19</font></i>: 
( <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="trees_4.html#E5:35_1_1_1"><i><font color="Green">E16</font></i></a></i>;<br/><a NAME="E8:35_1_1_1_1"/>
( <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/><a NAME="E9:35_1_1_1_1"/><b>then </b><i><font color="Green">E20</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="trees_4.html#K4" target="_self">-tree</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="trees_2.html#K5">|</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="trees_4.html#E7:35_1_1_1_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="trees_4.html#D4" target="_self">Def4</a>, <a class="txt" href="trees_4.html#E5"><i><font color="Green">Lemma4</font></i></a>, <a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as    <a href="trees_3.html#M5">Element</a> of <font color="Maroon">c<sub>2</sub></font> ;<br/><a NAME="E11:35_1_1_1_1"/>
( <font color="Maroon">c<sub>12</sub></font> <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>6</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>12</sub></font> &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>12</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="trees_4.html#E3:35_1_1_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="trees_4.html#E3:35_1_1_1_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="trees_4.html#E5:35_1_1_1_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="trees_4.html#E9:35_1_1_1_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a>, <a class="ref" href="trees_2.html#D9">TREES_2:def 9</a>, <a class="ref" href="trees_2.html#D11">TREES_2:def 11</a></i>;<br/><b>hence </b><a NAME="E12:35_1_1_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:35_1_1_1"/>
<font color="Maroon">c<sub>6</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#E3:35_1_1_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="trees_4.html#E5:35_1_1_1"><i><font color="Green">E16</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:35_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="trees_4.html#K4" target="_self">-tree</a> <font color="Maroon">c<sub>4</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/>


</div>
