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

<b>set </b><font color="Maroon">c<sub>2</sub></font> =  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="trees_3.html#K3" target="_self">Funcs</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> where B is    <a href="trees_3.html#M1" target="_self">Element</a> of  <a href="trees_3.html#K1" target="_self">Trees</a>  : verum </span>}</span> ;<br/>
<a NAME="E1:45_1_1"/>
 <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="trees_3.html#K3" target="_self">Funcs</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> where B is    <a href="trees_3.html#M1" target="_self">Element</a> of  <a href="trees_3.html#K1" target="_self">Trees</a>  : verum </span>}</span>  is <a href="trees_3.html#V3" target="_self">constituted-DTrees</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="trees_3.html#D5" target="_self">TREES_3:def 5</a><br/>

<b>assume </b><a NAME="E1:45_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="trees_3.html#K3" target="_self">Funcs</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> where B is    <a href="trees_3.html#M1" target="_self">Element</a> of  <a href="trees_3.html#K1" target="_self">Trees</a>  : verum </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:45_1_1_1"/><i><font color="Green">E18</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="trees_3.html#K3" target="_self">Funcs</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> where B is    <a href="trees_3.html#M1" target="_self">Element</a> of  <a href="trees_3.html#K1" target="_self">Trees</a>  : verum </span>}</span>  )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="trees_3.html#M1" target="_self">Element</a> of  <a href="trees_3.html#K1" target="_self">Trees</a> <b> such that </b><br/><a NAME="E5:45_1_1_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="trees_3.html#K3" target="_self">Funcs</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="trees_3.html#E3:45_1_1_1"><i><font color="Green">E18</font></i></a></i>;<br/>
<b>thus </b><a NAME="E6:45_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> is   <a href="trees_2.html#NM4">DecoratedTree</a>
 <b>by </b><i><a class="txt" href="trees_3.html#E3:45_1_1_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="trees_3.html#E5:45_1_1_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="trees_3.html#D6" target="_self">Def6</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> =  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="trees_3.html#K3" target="_self">Funcs</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> where B is    <a href="trees_3.html#M1" target="_self">Element</a> of  <a href="trees_3.html#K1" target="_self">Trees</a>  : verum </span>}</span>  as  <a href="trees_3.html#V3" target="_self">constituted-DTrees</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<a NAME="E3:45_1_1"/>
<font color="Maroon">c<sub>3</sub></font> is    <a href="trees_3.html#M4" target="_self">DTree-set</a> of <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="trees_3.html#D6" target="_self">TREES_3:def 6</a><br/>

<b>assume </b><a NAME="E1:45_1_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:45_1_1_2"/><i><font color="Green">E18</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="trees_3.html#K3" target="_self">Funcs</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> where B is    <a href="trees_3.html#M1" target="_self">Element</a> of  <a href="trees_3.html#K1" target="_self">Trees</a>  : verum </span>}</span>  )
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="trees_3.html#M1" target="_self">Element</a> of  <a href="trees_3.html#K1" target="_self">Trees</a> <b> such that </b><br/><a NAME="E5:45_1_1_2"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="trees_3.html#K3" target="_self">Funcs</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="trees_3.html#E3:45_1_1_2"><i><font color="Green">E18</font></i></a></i>;<br/>
<b>thus </b><a NAME="E6:45_1_1_2"/>
<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="txt" href="trees_3.html#E3:45_1_1_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="trees_3.html#E5:45_1_1_2"><i><font color="Green">E19</font></i></a>, <a class="ref" href="trees_3.html#D6" target="_self">Def6</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as    <a href="trees_3.html#M4" target="_self">DTree-set</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>take </b>
<font color="Maroon">c<sub>4</sub></font>
;<br/>

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="trees_2.html#NM6">DecoratedTree</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> =  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font> as    <a href="trees_3.html#M1" target="_self">Element</a> of  <a href="trees_3.html#K1" target="_self">Trees</a>  <b>by </b><i><a class="ref" href="trees_3.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E6:45_1_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <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/>
<a NAME="E7:45_1_1"/><b>then </b>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="trees_3.html#K3" target="_self">Funcs</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>1</sub></font> &amp;  <a href="trees_3.html#K3" target="_self">Funcs</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="trees_3.html#K3" target="_self">Funcs</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> where B is    <a href="trees_3.html#M1" target="_self">Element</a> of  <a href="trees_3.html#K1" target="_self">Trees</a>  : verum </span>}</span>  )
 
<b>by </b><i><a class="ref" href="funct_2.html#D2">FUNCT_2:def 2</a></i>;<br/>
<b>hence </b><a NAME="E8:45_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div>
