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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="dtconstr.html#V1">with_terminals</a> <a href="dtconstr.html#V2">with_nonterminals</a> <a href="bintree1.html#V3" target="_self">binary</a>  <a href="lang1.html#L1">DTConstrStr</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="trees_4.html#M1">FinSequence</a> of  <a href="dtconstr.html#K4">TS</a> <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="lang1.html#NM1">Symbol</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>





<b>assume </b><a NAME="E1:19"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="lang1.html#R1">==&gt;</a>  <a href="dtconstr.html#K1">roots</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:19"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="lang1.html#NM1">Symbol</a> of <font color="Maroon">c<sub>1</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being  <a href="finseq_1.html#NM1">FinSequence</a> st <font color="Olive">b<sub>1</sub></font> <a href="lang1.html#R1">==&gt;</a> <font color="Olive">b<sub>2</sub></font> </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E3:19"/>
<font color="Maroon">c<sub>3</sub></font> is   <a href="dtconstr.html#NM3">NonTerminal</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="lang1.html#D3">LANG1:def 3</a></i>;<br/>

<b>consider </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> being   <a href="lang1.html#NM1">Symbol</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E5:19"/><i><font color="Green">E11</font></i>: 
 <a href="dtconstr.html#K1">roots</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="lang1.html#K4">&lt;*</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="lang1.html#K4">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="bintree1.html#E1:19"><i><font color="Green">E10</font></i></a>, <a class="ref" href="bintree1.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E6:19"/><i><font color="Green">E12</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="lang1.html#K4">&lt;*</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="lang1.html#K4">*&gt;</a></span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being  <a href="trees_2.html#NM4">DecoratedTree</a> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <span class="p1"><a href="lang1.html#K4">&lt;*</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="lang1.html#K4">*&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  ) ) )
 
<b>by </b><i><a class="txt" href="bintree1.html#E5:19"><i><font color="Green">E11</font></i></a>, <a class="ref" href="trees_3.html#D18">TREES_3:def 18</a></i>;<br/>
<a NAME="E7:19"/><i><font color="Green">E13</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="lang1.html#K4">&lt;*</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="lang1.html#K4">*&gt;</a></span> <a href="hidden.html#R1">=</a> 2
 
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/>
<b>hence </b><a NAME="E8:19"/>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">1,2</span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="txt" href="bintree1.html#E6:19"><i><font color="Green">E12</font></i></a>, <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>

<b>hence </b><a NAME="E9:19"/><i><font color="Green">E14</font></i>: 
( 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font> &amp; 2 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>

<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="trees_2.html#NM4">DecoratedTree</a><b> such that </b><br/><a NAME="E11:19"/><i><font color="Green">E15</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> 1 &amp; <span class="p1"><a href="lang1.html#K4">&lt;*</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="lang1.html#K4">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="txt" href="bintree1.html#E5:19"><i><font color="Green">E11</font></i></a>, <a class="ref" href="trees_3.html#D18">TREES_3:def 18</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="trees_2.html#NM4">DecoratedTree</a><b> such that </b><br/><a NAME="E13:19"/><i><font color="Green">E16</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> 2 &amp; <span class="p1"><a href="lang1.html#K4">&lt;*</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="lang1.html#K4">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="txt" href="bintree1.html#E5:19"><i><font color="Green">E11</font></i></a>, <a class="txt" href="bintree1.html#E9:19"><i><font color="Green">E14</font></i></a>, <a class="ref" href="trees_3.html#D18">TREES_3:def 18</a></i>;<br/>
<a NAME="E14:19"/>
(  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="dtconstr.html#K4">TS</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="bintree1.html#E9:19"><i><font color="Green">E14</font></i></a>, <a class="txt" href="bintree1.html#E11:19"><i><font color="Green">E15</font></i></a>, <a class="txt" href="bintree1.html#E13:19"><i><font color="Green">E16</font></i></a>, <a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>7</sub></font> as    <a href="dtconstr.html#M1">Element</a> of  <a href="dtconstr.html#K4">TS</a> <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E16:19"/>
( <span class="p1"><a href="lang1.html#K4">&lt;*</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="lang1.html#K4">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <span class="p1"><a href="lang1.html#K4">&lt;*</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="lang1.html#K4">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/>
<a NAME="E17:19"/><b>then </b><i><font color="Green">E17</font></i>: 
(  <a href="bintree1.html#K1" target="_self">root-label</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="bintree1.html#K1" target="_self">root-label</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="txt" href="bintree1.html#E11:19"><i><font color="Green">E15</font></i></a>, <a class="txt" href="bintree1.html#E13:19"><i><font color="Green">E16</font></i></a></i>;<br/>
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="lang1.html#K4">&lt;*</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="lang1.html#K4">*&gt;</a></span></span>)</span> = 
 <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="lang1.html#K4">&lt;*</a><span class="default"><font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font></span><a href="lang1.html#K4">*&gt;</a></span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
<br/>.= 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="bintree1.html#E6:19"><i><font color="Green">E12</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/>
<a NAME="E19:19"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> 2
 
<b>by </b><i><a class="txt" href="bintree1.html#E7:19"><i><font color="Green">E13</font></i></a>, <a class="ref" href="finseq_1.html#T8">FINSEQ_1:8</a></i>;<br/>
<a NAME="E20:19"/><b>then </b><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="lang1.html#K4">&lt;*</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="lang1.html#K4">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="bintree1.html#E11:19"><i><font color="Green">E15</font></i></a>, <a class="txt" href="bintree1.html#E13:19"><i><font color="Green">E16</font></i></a>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>8</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>9</sub></font>
;<br/>

<b>thus </b><a NAME="E21:19"/>
 <a href="dtconstr.html#K1">roots</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="lang1.html#K4">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="bintree1.html#K1" target="_self">root-label</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="bintree1.html#K1" target="_self">root-label</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span><a href="lang1.html#K4">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="bintree1.html#E5:19"><i><font color="Green">E11</font></i></a>, <a class="txt" href="bintree1.html#E17:19"><i><font color="Green">E17</font></i></a></i>;<br/>

<b>thus </b><a NAME="E22:19"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> 1 &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> 2 &amp; <font color="Maroon">c<sub>3</sub></font> <a href="trees_4.html#K12">-tree</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="trees_4.html#K10">-tree</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="txt" href="bintree1.html#E11:19"><i><font color="Green">E15</font></i></a>, <a class="txt" href="bintree1.html#E13:19"><i><font color="Green">E16</font></i></a>, <a class="txt" href="bintree1.html#E20:19"><i><font color="Green">E18</font></i></a>, <a class="ref" href="trees_4.html#D6">TREES_4:def 6</a></i>;<br/>

<b>thus </b><a NAME="E23:19"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="bintree1.html#E9:19"><i><font color="Green">E14</font></i></a>, <a class="txt" href="bintree1.html#E11:19"><i><font color="Green">E15</font></i></a>, <a class="txt" href="bintree1.html#E13:19"><i><font color="Green">E16</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


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