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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="trees_2.html#NM4">DecoratedTree</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font>;<br/>





<a NAME="E1:20"/><i><font color="Green">E17</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="trees_a.html#K3" target="_self">tree</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2"><a href="trees_a.html#K2" target="_self">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="trees_a.html#K2" target="_self">}</a></span>,<font color="Maroon">c<sub>2</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>1</sub></font> <a href="trees_2.html#K8">with-replacement</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_1"><b>proof </b></a><div class="add">

 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="trees_a.html#K3" target="_self">tree</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2"><a href="trees_a.html#K2" target="_self">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="trees_a.html#K2" target="_self">}</a></span>,<font color="Maroon">c<sub>2</sub></font></span>)</span> = 
 <a href="trees_a.html#K1" target="_self">tree</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1"><a href="trees_a.html#K2" target="_self">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="trees_a.html#K2" target="_self">}</a></span>,<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="trees_a.html#D2" target="_self">Def2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="trees_1.html#K5">with-replacement</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="trees_a.html#T10" target="_self">Th10</a></i>
<br/>.= 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="trees_2.html#K8">with-replacement</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="trees_2.html#D12">TREES_2:def 12</a></i>
;<br/>
<b>hence </b><a NAME="E2:20_1"/>
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="trees_a.html#K3" target="_self">tree</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2"><a href="trees_a.html#K2" target="_self">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="trees_a.html#K2" target="_self">}</a></span>,<font color="Maroon">c<sub>2</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>1</sub></font> <a href="trees_2.html#K8">with-replacement</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:20"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="trees_a.html#K3" target="_self">tree</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2"><a href="trees_a.html#K2" target="_self">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="trees_a.html#K2" target="_self">}</a></span>,<font color="Maroon">c<sub>2</sub></font></span>)</span> holds <br/><span class="p1">(<span class="default"><a href="trees_a.html#K3" target="_self">tree</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2"><a href="trees_a.html#K2" target="_self">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="trees_a.html#K2" target="_self">}</a></span>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="trees_2.html#K8">with-replacement</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:20_2"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="trees_a.html#K3" target="_self">tree</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2"><a href="trees_a.html#K2" target="_self">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="trees_a.html#K2" target="_self">}</a></span>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:20_2"/><b>then </b><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="trees_a.html#K1" target="_self">tree</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1"><a href="trees_a.html#K2" target="_self">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="trees_a.html#K2" target="_self">}</a></span>,<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="trees_a.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E3:20_2"/><i><font color="Green">E20</font></i>: 
 <a href="trees_a.html#K1" target="_self">tree</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1"><a href="trees_a.html#K2" target="_self">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="trees_a.html#K2" target="_self">}</a></span>,<span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font> : for <font color="Olive">b<sub>1</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="trees_a.html#K2" target="_self">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="trees_a.html#K2" target="_self">}</a></span> holds <br/> not <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Olive">b<sub>1</sub></font> </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font>, B is    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="trees_a.html#K2" target="_self">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="trees_a.html#K2" target="_self">}</a></span> </span>}</span> 
 
<b>by </b><i><a class="ref" href="trees_a.html#T7" target="_self">Th7</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:20_2_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font> : for <font color="Olive">b<sub>1</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="trees_a.html#K2" target="_self">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="trees_a.html#K2" target="_self">}</a></span> holds <br/> not <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Olive">b<sub>1</sub></font> </span>}</span>  or <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"><font color="Olive">b<sub>1</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font>, B is    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="trees_a.html#K2" target="_self">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="trees_a.html#K2" target="_self">}</a></span> </span>}</span>  )
 </span><b>by </b><i><a class="txt" href="trees_a.html#E2:20_2"><i><font color="Green">E19</font></i></a>, <a class="txt" href="trees_a.html#E3:20_2"><i><font color="Green">E20</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_2_1_1"><b>suppose </b></a><a NAME="E1:20_2_1_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font> : for <font color="Olive">b<sub>1</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="trees_a.html#K2" target="_self">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="trees_a.html#K2" target="_self">}</a></span> holds <br/> not <font color="Olive">b<sub>2</sub></font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:20_2_1_1"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="trees_a.html#K2" target="_self">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="trees_a.html#K2" target="_self">}</a></span> holds <br/> not <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Maroon">c<sub>5</sub></font> ) )
 ;<br/><b>consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E5:20_2_1_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E6:20_2_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="trees_a.html#K2" target="_self">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="trees_a.html#K2" target="_self">}</a></span>
 <b>by </b><i><a class="txt" href="trees_a.html#E5:20_2_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E7:20_2_1_1"/><b>then </b><i><font color="Green">E24</font></i>: 
not <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="trees_a.html#E3:20_2_1_1"><i><font color="Green">E22</font></i></a></i>;<br/><a NAME="E8:20_2_1_1"/>
( <font color="Maroon">c<sub>4</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>1</sub></font> <a href="trees_2.html#K8">with-replacement</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font> : not <font color="Maroon">c<sub>6</sub></font> <a href="tarski.html#R1">is_a_prefix_of</a> <font color="Olive">b<sub>1</sub></font> </span>}</span>  implies <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="trees_2.html#K8">with-replacement</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="trees_a.html#E5:20_2_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="trees_a.html#T16" target="_self">Th16</a></i>;<br/><b>hence </b><a NAME="E9:20_2_1_1"/>
<span class="p1">(<span class="default"><a href="trees_a.html#K3" target="_self">tree</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2"><a href="trees_a.html#K2" target="_self">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="trees_a.html#K2" target="_self">}</a></span>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="trees_2.html#K8">with-replacement</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="trees_a.html#E1:20"><i><font color="Green">E17</font></i></a>, <a class="txt" href="trees_a.html#E1:20_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="trees_a.html#E1:20_2_1_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="trees_a.html#E3:20_2_1_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="trees_a.html#E7:20_2_1_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="trees_a.html#T15" target="_self">Th15</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_2_1_2"><b>suppose </b></a><a NAME="E1:20_2_1_2"/><i><font color="Green">E21</font></i>: 
<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"><font color="Olive">b<sub>1</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Olive">b<sub>2</sub></font></span>)</span> where B is    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font>, B is    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="trees_a.html#K2" target="_self">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="trees_a.html#K2" target="_self">}</a></span> </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>6</sub></font> being    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E3:20_2_1_2"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="trees_a.html#K2" target="_self">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="trees_a.html#K2" target="_self">}</a></span> )
 ;<br/><a NAME="E4:20_2_1_2"/><i><font color="Green">E23</font></i>: 
<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"><font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> 
 <b>by </b><i><a class="txt" href="trees_a.html#E3:20_2_1_2"><i><font color="Green">E22</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>8</sub></font> being    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E6:20_2_1_2"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="trees_a.html#K2" target="_self">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="trees_a.html#K2" target="_self">}</a></span> &amp; <span class="p1">(<span class="default"><a href="trees_a.html#K3" target="_self">tree</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2"><a href="trees_a.html#K2" target="_self">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="trees_a.html#K2" target="_self">}</a></span>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="trees_a.html#E1:20_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="trees_a.html#E1:20_2_1_2"><i><font color="Green">E21</font></i></a>, <a class="ref" href="trees_a.html#T17" target="_self">Th17</a></i>;<br/><a NAME="E7:20_2_1_2"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="trees_a.html#E3:20_2_1_2"><i><font color="Green">E22</font></i></a>, <a class="txt" href="trees_a.html#E6:20_2_1_2"><i><font color="Green">E24</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E9:20_2_1_2"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="finseq_1.html#K8">^</a> <font color="Maroon">c<sub>9</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="trees_2.html#K8">with-replacement</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="txt" href="trees_a.html#E1:20"><i><font color="Green">E17</font></i></a>, <a class="txt" href="trees_a.html#E1:20_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="trees_a.html#E4:20_2_1_2"><i><font color="Green">E23</font></i></a>, <a class="ref" href="trees_a.html#T18" target="_self">Th18</a></i>;<br/><b>thus </b><a NAME="E10:20_2_1_2"/>
<span class="p1">(<span class="default"><a href="trees_a.html#K3" target="_self">tree</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p2"><a href="trees_a.html#K2" target="_self">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="trees_a.html#K2" target="_self">}</a></span>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="trees_2.html#K8">with-replacement</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="trees_a.html#E6:20_2_1_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="trees_a.html#E7:20_2_1_2"><i><font color="Green">E25</font></i></a>, <a class="txt" href="trees_a.html#E9:20_2_1_2"><i><font color="Green">E26</font></i></a>, <a class="ref" href="finseq_1.html#T46">FINSEQ_1:46</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E3:20"/>
 <a href="trees_a.html#K3" target="_self">tree</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1"><a href="trees_a.html#K2" target="_self">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="trees_a.html#K2" target="_self">}</a></span>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="trees_2.html#K8">with-replacement</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="trees_a.html#E1:20"><i><font color="Green">E17</font></i></a>, <a class="ref" href="trees_2.html#T33">TREES_2:33</a></i>;<br/>


</div>
