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<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a>,<a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a><br/>

<b>assume </b><a NAME="E1:6_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="trees_3.html#K16" title="TREES_3:func.16">roots</a> <font color="Maroon" title="c3">p</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">k</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:6_1_1"/><i><font color="Green" title="E3">A1</font></i>: 
( <font color="Maroon" title="c5">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="trees_3.html#K16" title="TREES_3:func.16">roots</a> <font color="Maroon" title="c3">p</font></span>)</span> &amp; <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="trees_3.html#K16" title="TREES_3:func.16">roots</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">k</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">k</font> = <font color="Maroon" title="c5">k</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="dtconstr.html#E3:6_1_1"><i><font color="Green" title="E3">A1</font></i></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></i>;<br/>
<a NAME="E5:6_1_1"/><i><font color="Green" title="E4">A2</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="trees_3.html#K16" title="TREES_3:func.16">roots</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c3">p</font>
 
<b>by </b><i><a class="ref" href="trees_3.html#D18" title="TREES_3:def.18">TREES_3:def 18</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">t</font> being   <a href="trees_2.html#NM4" title="TREES_2:NM.4">DecoratedTree</a><b> such that </b><br/><a NAME="E7:6_1_1"/><i><font color="Green" title="E5">A3</font></i>: 
( <font color="Maroon" title="c7">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font> &amp; <span class="p1">(<span class="default"><a href="trees_3.html#K16" title="TREES_3:func.16">roots</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">t</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 <b>by </b><i><a class="txt" href="dtconstr.html#E3:6_1_1"><i><font color="Green" title="E3">A1</font></i></a>, <a class="ref" href="trees_3.html#D18" title="TREES_3:def.18">TREES_3:def 18</a></i>;<br/>
<a NAME="E8:6_1_1"/>
<font color="Maroon" title="c7">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">T</font>
 
<b>by </b><i><a class="txt" href="dtconstr.html#E3:6_1_1"><i><font color="Green" title="E3">A1</font></i></a>, <a class="txt" href="dtconstr.html#E5:6_1_1"><i><font color="Green" title="E4">A2</font></i></a>, <a class="txt" href="dtconstr.html#E7:6_1_1"><i><font color="Green" title="E5">A3</font></i></a>, <a class="ref" href="dtconstr.html#T2" target="_self" title="DTCONSTR:th.2">Th2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c8">t</font> = <font color="Maroon" title="c7">t</font> as   <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</a> of <font color="Maroon" title="c1">D</font> <b>by </b><i><a class="ref" href="trees_3.html#D6" title="TREES_3:def.6">TREES_3:def 6</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">r</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  as   <a href="trees_4.html#NM1" title="TREES_4:NM.1">Node</a> of <font color="Maroon" title="c8">t</font> <b>by </b><i><a class="ref" href="trees_1.html#T47" title="TREES_1:th.47">TREES_1:47</a></i>;<br/>
<a NAME="E11:6_1_1"/>
<font color="Maroon" title="c8">t</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c9">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">D</font>
 
;<br/>
<b>hence </b><a NAME="E12:6_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">D</font>
 <b>by </b><i><a class="txt" href="dtconstr.html#E3:6_1_1"><i><font color="Green" title="E3">A1</font></i></a>, <a class="txt" href="dtconstr.html#E7:6_1_1"><i><font color="Green" title="E5">A3</font></i></a></i>;<br/>


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