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<b>thus </b><a NAME="E1:30"/>
 <a href="lang1.html#K5">Terminals</a> <a href="dtconstr.html#K5" target="_self">PeanoNat</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="dtconstr.html#E22"><i><font color="Green">Lemma16</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="dtconstr.html#D6" target="_self">DTCONSTR:def 6</a><br/>

<b>thus </b><a NAME="E2:30"/>
 <a href="lang1.html#K6">NonTerminals</a> <a href="dtconstr.html#K5" target="_self">PeanoNat</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="dtconstr.html#E25"><i><font color="Green">Lemma19</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="dtconstr.html#D7" target="_self">DTCONSTR:def 7</a><br/>

<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> =  <a href="trees_4.html#K2">root-tree</a> <font color="Maroon">c<sub>2</sub></font> as    <a href="dtconstr.html#M1" target="_self">Element</a> of <font color="Maroon">c<sub>4</sub></font> <b>by </b><i><a class="ref" href="dtconstr.html#D4" target="_self">Def4</a>, <a class="txt" href="dtconstr.html#E22"><i><font color="Green">Lemma16</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="lang1.html#K3">*&gt;</a></span> as    <a href="trees_4.html#M1">FinSequence</a> of <font color="Maroon">c<sub>4</sub></font> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as    <a href="trees_4.html#M1">FinSequence</a> of  <a href="dtconstr.html#K4" target="_self">TS</a> <a href="dtconstr.html#K5" target="_self">PeanoNat</a>  ;<br/>
<b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="lang1.html#NM1">Symbol</a> of <a href="dtconstr.html#K5" target="_self">PeanoNat</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="dtconstr.html#D8" target="_self">DTCONSTR:def 8</a><br/>

<b>assume </b><a NAME="E6:30"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="lang1.html#K6">NonTerminals</a> <a href="dtconstr.html#K5" target="_self">PeanoNat</a> 
 ;<br/>

<a NAME="E7:30"/><b>then </b><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="dtconstr.html#E27"><i><font color="Green">Lemma21</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>7</sub></font>
;<br/>

<a NAME="E8:30"/><i><font color="Green">E28</font></i>: 
(  <a href="finsop_1.html#K5">dom</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">c<sub>5</sub></font></span><a href="lang1.html#K3">*&gt;</a></span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> 1 &amp;  <a href="finsop_1.html#K5">dom</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="lang1.html#K3">*&gt;</a></span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> 1 )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T55">FINSEQ_1:55</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:30_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>7</sub></font>
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="trees_2.html#NM4">DecoratedTree</a> ;<br/><b>take </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font>;<br/><a NAME="E3:30_1"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="lang1.html#K3">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="dtconstr.html#E8:30"><i><font color="Green">E28</font></i></a>, <a class="txt" href="dtconstr.html#E1:30_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E4:30_1"/>
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> &amp; <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="lang1.html#K3">*&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a>, <a class="ref" href="trees_4.html#T3">TREES_4:3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E10:30"/>
<font color="Maroon">c<sub>8</sub></font> <a href="lang1.html#R1">==&gt;</a>  <a href="dtconstr.html#K1" target="_self">roots</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="dtconstr.html#E7:30"><i><font color="Green">E27</font></i></a>, <a class="txt" href="dtconstr.html#E8:30"><i><font color="Green">E28</font></i></a>, <a class="txt" href="dtconstr.html#E18"><i><font color="Green">Lemma12</font></i></a>, <a class="ref" href="trees_3.html#D18">TREES_3:def 18</a></i>;<br/>


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