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<b>let </b><font color="Maroon">T1</font> be   <a href="trees_2.html#NM4">DecoratedTree</a>, <font color="Maroon">T2</font> be   <a href="trees_2.html#NM4">DecoratedTree</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:14_1_2"/><i><font color="Green">E61</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T1</font> <a href="hidden.html#R1">=</a>  <a href="trees_1.html#K2">elementary_tree</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> &amp; <font color="Maroon">T1</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> holds <br/><font color="Maroon">T1</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Olive">n</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) )
 <b>and </b><br/><a NAME="E2:14_1_2"/><i><font color="Green">E62</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T2</font> <a href="hidden.html#R1">=</a>  <a href="trees_1.html#K2">elementary_tree</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> &amp; <font color="Maroon">T2</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> holds <br/><font color="Maroon">T2</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Olive">n</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:14_1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="trees_1.html#K2">elementary_tree</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span>
 ;<br/><b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as    <a href="trees_1.html#M1">Element</a> of  <a href="trees_1.html#K2">elementary_tree</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> ;<br/><a NAME="E3:14_1_2_1"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or  ex <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> &amp; <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Olive">n</font></span><a href="finseq_1.html#K12">*&gt;</a></span> ) )
 <b>by </b><i><a class="ref" href="trees_1.html#T57">TREES_1:57</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_2_1_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">n</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><a NAME="E1:14_1_2_1_1"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">n</font></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 ;<br/><b>thus </b><font color="Maroon">T1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">T2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, </i>
;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:14_1_2_1"/>
<font color="Maroon">T1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:14_1_2"/>
<font color="Maroon">T1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">T2</font>
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>


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