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<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> ;<br/>

<a NAME="E1:58"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> or <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> )
 
<b>by </b><i><a class="txt" href="dtconstr.html#E15"><i><font color="Green">Lemma10</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a>
<div class="add"><b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:58_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="dtconstr.html#K19" target="_self">TerminalLanguage</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><a NAME="E2:58_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="trees_3.html#M5">Element</a> of  <a href="trees_3.html#K5">FinTrees</a> the <a href="struct_0.html#U1">carrier</a> of <a href="dtconstr.html#K5" target="_self">PeanoNat</a>  st <br/>( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="dtconstr.html#K16" target="_self">TerminalString</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="dtconstr.html#K4" target="_self">TS</a> <a href="dtconstr.html#K5" target="_self">PeanoNat</a>  &amp; <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> )
 ;<br/><a NAME="E3:58_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="lang1.html#K3">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="dtconstr.html#T14" target="_self">Th14</a></i>;<br/><b>hence </b><a NAME="E4:58_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2"><a href="lang1.html#K3">&lt;*</a><span class="default">0</span><a href="lang1.html#K3">*&gt;</a></span></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E3:58"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2"><a href="lang1.html#K3">&lt;*</a><span class="default">0</span><a href="lang1.html#K3">*&gt;</a></span></span><a href="domain_1.html#K6">}</a></span>
 ;<br/>

<a NAME="E4:58"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="lang1.html#K3">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> =  <a href="dtconstr.html#K8" target="_self">root-tree</a> <font color="Maroon">c<sub>6</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="dtconstr.html#K4" target="_self">TS</a> <a href="dtconstr.html#K5" target="_self">PeanoNat</a>  ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> =  <a href="dtconstr.html#K8" target="_self">root-tree</a> <font color="Maroon">c<sub>6</sub></font> as    <a href="dtconstr.html#M1" target="_self">Element</a> of  <a href="dtconstr.html#K4" target="_self">TS</a> <a href="dtconstr.html#K5" target="_self">PeanoNat</a>  ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="lang1.html#K3">*&gt;</a></span> 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/>
<a NAME="E8:58"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 
<b>by </b><i><a class="ref" href="trees_4.html#T3">TREES_4:3</a></i>;<br/>
<a NAME="E9:58"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="lang1.html#R1">==&gt;</a>  <a href="trees_3.html#K16">roots</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="lang1.html#K3">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="dtconstr.html#E18"><i><font color="Green">Lemma12</font></i></a>, <a class="ref" href="dtconstr.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E10:58"/><b>then </b><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="trees_4.html#K4">-tree</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="lang1.html#K3">*&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="dtconstr.html#K4" target="_self">TS</a> <a href="dtconstr.html#K5" target="_self">PeanoNat</a> 
 
<b>by </b><i><a class="ref" href="dtconstr.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E11:58"/><b>then </b>
(  <a href="dtconstr.html#K16" target="_self">TerminalString</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="trees_4.html#K12">-tree</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="trees_4.html#K4">-tree</a> <span class="p2"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="lang1.html#K3">*&gt;</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> &amp;  <a href="dtconstr.html#K16" target="_self">TerminalString</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> )
 
<b>by </b><i><a class="txt" href="dtconstr.html#E4:58"><i><font color="Green">E42</font></i></a>, <a class="ref" href="dtconstr.html#T14" target="_self">Th14</a>, <a class="ref" href="trees_4.html#D4">TREES_4:def 4</a></i>;<br/>
<b>hence </b><a NAME="E12:58"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="dtconstr.html#K19" target="_self">TerminalLanguage</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="dtconstr.html#E1:58"><i><font color="Green">E41</font></i></a>, <a class="txt" href="dtconstr.html#E8:58"><i><font color="Green">E43</font></i></a>, <a class="txt" href="dtconstr.html#E10:58"><i><font color="Green">E44</font></i></a></i>;<br/>


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