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<b>let </b><font color="Maroon">T</font> be  <a href="trees_9.html#V4">finite-branching</a> <a href="trees_2.html#NM4">DecoratedTree</a>;<br/><b>let </b><font color="Maroon">x'</font> be    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">x</font> be    <a href="trees_1.html#M1">Element</a> of  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T</font>;<br/>





<b>assume </b><a NAME="E1:11"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">x'</font> <a href="hidden.html#R2">in</a>  <a href="trees_2.html#K1">succ</a> <font color="Maroon">x</font>
 ;<br/>

<b>consider </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><br/><a NAME="E3:11"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="finseq_1.html#K8">^</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> &amp; <font color="Maroon">x</font> <a href="finseq_1.html#K8">^</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> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">T</font> )
 <b>by </b><i/>;<br/>
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="trees_9.html#K2">succ</a> <font color="Maroon">T</font>,<font color="Maroon">x</font></span>)</span> = 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">T</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p2">(<span class="default"><font color="Maroon">x</font> <a href="trees_9.html#K1">succ</a> </span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="trees_9.html#K1">succ</a> </span>)</span>
<b>by </b><i/>
;<br/>
<a NAME="E5:11"/><b>then </b><i><font color="Green">E30</font></i>: 
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="trees_9.html#K2">succ</a> <font color="Maroon">T</font>,<font color="Maroon">x</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:11"/>
<font color="Maroon">T</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="trees_9.html#K2">succ</a> <font color="Maroon">T</font>,<font color="Maroon">x</font></span>)</span> <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/>
<b>hence </b><a NAME="E7:11"/>
<font color="Maroon">T</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x'</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="trees_9.html#K2">succ</a> <font color="Maroon">T</font>,<font color="Maroon">x</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


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