<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="dtconstr.html#V1" target="_self">with_terminals</a> <a href="dtconstr.html#V2" target="_self">with_nonterminals</a>  <a href="lang1.html#L1">DTConstrStr</a> ;<br/>

<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="trees_2.html#NM6">DecoratedTree</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font>] means for <font color="Olive">b<sub>1</sub></font> being  <a href="dtconstr.html#NM3">NonTerminal</a> of <font color="Maroon">c<sub>5</sub></font> st <font color="Maroon">a<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="Olive">b<sub>1</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font> being   <a href="trees_4.html#M1">FinSequence</a> of  <a href="dtconstr.html#K4" target="_self">TS</a> <font color="Maroon">c<sub>5</sub></font> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="trees_4.html#K12">-tree</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="lang1.html#R1">==&gt;</a>  <a href="dtconstr.html#K1" target="_self">roots</a> <font color="Olive">b<sub>2</sub></font> );<br/>
<div><i><font color="Green">E30</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="lang1.html#NM1">Symbol</a> of <font color="Maroon">c<sub>5</sub></font>;<br/><b>assume </b><a NAME="E1:42_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a>  <a href="dtconstr.html#K6" target="_self">Terminals</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><b>thus </b><a NAME="E2:42_1"/>
<font color="Maroon">S<sub>2</sub></font>[ <a href="trees_4.html#K2">root-tree</a> <font color="Maroon">c<sub>6</sub></font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="dtconstr.html#NM3">NonTerminal</a> of <font color="Maroon">c<sub>5</sub></font>;<br/>

<b>assume </b><a NAME="E1:42_1_1"/>
<span class="p1">(<span class="default"><a href="trees_4.html#K2">root-tree</a> <font color="Maroon">c<sub>6</sub></font></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>
 ;<br/>

<a NAME="E2:42_1_1"/><b>then </b><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="trees_4.html#T3">TREES_4:3</a></i>;<br/>
<a NAME="E3:42_1_1"/>
 <a href="dtconstr.html#K6" target="_self">Terminals</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</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>5</sub></font> : for <font color="Olive">b<sub>1</sub></font> being  <a href="finseq_1.html#NM1">FinSequence</a> holds not <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> 
 
<b>by </b><i><a class="ref" href="lang1.html#D2">LANG1:def 2</a></i>;<br/>
<a NAME="E4:42_1_1"/><b>then </b><i><font color="Green">E33</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="lang1.html#NM1">Symbol</a> of <font color="Maroon">c<sub>5</sub></font> st <br/>( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="finseq_1.html#NM1">FinSequence</a> holds not <font color="Olive">b<sub>1</sub></font> <a href="lang1.html#R1">==&gt;</a> <font color="Olive">b<sub>2</sub></font> ) )
 
<b>by </b><i><a class="txt" href="dtconstr.html#E1:42_1"><i><font color="Green">E31</font></i></a></i>;<br/>
<a NAME="E5:42_1_1"/>
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="dtconstr.html#K7" target="_self">NonTerminals</a> <font color="Maroon">c<sub>5</sub></font> &amp;  <a href="dtconstr.html#K7" target="_self">NonTerminals</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</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>5</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>  )
 
<b>by </b><i><a class="ref" href="lang1.html#D3">LANG1:def 3</a></i>;<br/>
<a NAME="E6:42_1_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="lang1.html#NM1">Symbol</a> of <font color="Maroon">c<sub>5</sub></font> st <br/>( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; ex <font color="Olive">b<sub>2</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> )
 
;<br/>
<b>hence </b><a NAME="E7:42_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="trees_4.html#M1">FinSequence</a> of  <a href="dtconstr.html#K4" target="_self">TS</a> <font color="Maroon">c<sub>5</sub></font> st <br/>(  <a href="trees_4.html#K2">root-tree</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="trees_4.html#K12">-tree</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="lang1.html#R1">==&gt;</a>  <a href="dtconstr.html#K1" target="_self">roots</a> <font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="dtconstr.html#E2:42_1_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="dtconstr.html#E4:42_1_1"><i><font color="Green">E33</font></i></a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<div><i><font color="Green">E31</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="42_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="lang1.html#NM1">Symbol</a> of <font color="Maroon">c<sub>5</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="trees_4.html#M1">FinSequence</a> of  <a href="dtconstr.html#K4" target="_self">TS</a> <font color="Maroon">c<sub>5</sub></font>;<br/><b>assume </b><a NAME="E1:42_2"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">c<sub>6</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> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="trees_2.html#NM6">DecoratedTree</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>7</sub></font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>] ) )
 ;<br/><b>thus </b><a NAME="E2:42_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>6</sub></font> <a href="trees_4.html#K12">-tree</a> <font color="Maroon">c<sub>7</sub></font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="dtconstr.html#NM3">NonTerminal</a> of <font color="Maroon">c<sub>5</sub></font>;<br/>

<b>assume </b><a NAME="E1:42_2_1"/><i><font color="Green">E33</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="trees_4.html#K12">-tree</a> <font color="Maroon">c<sub>7</sub></font></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>8</sub></font>
 ;<br/>

<b>take </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>7</sub></font>;<br/>

<b>thus </b><a NAME="E2:42_2_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="trees_4.html#K12">-tree</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="trees_4.html#K12">-tree</a> <font color="Maroon">c<sub>9</sub></font> &amp; <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>9</sub></font> )
 <b>by </b><i><a class="txt" href="dtconstr.html#E1:42_2"><i><font color="Green">E32</font></i></a>, <a class="txt" href="dtconstr.html#E1:42_2_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="trees_4.html#D4">TREES_4:def 4</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E3:42"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="trees_2.html#NM6">DecoratedTree</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>5</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="dtconstr.html#K4" target="_self">TS</a> <font color="Maroon">c<sub>5</sub></font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="dtconstr.html#S7" target="_self">DTCONSTR:sch 7</a>(<a class="txt" href="dtconstr.html#E1:42"><i><font color="Green">E30</font></i></a>, <a class="txt" href="dtconstr.html#E2:42"><i><font color="Green">E31</font></i></a>);<br/></i>
<b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="dtconstr.html#M1" target="_self">Element</a> of  <a href="dtconstr.html#K4" target="_self">TS</a> <font color="Maroon">c<sub>5</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="dtconstr.html#NM3">NonTerminal</a> of <font color="Maroon">c<sub>5</sub></font>;<br/>



<b>assume </b><a NAME="E4:42"/>
<font color="Maroon">c<sub>6</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>7</sub></font>
 ;<br/>

<b>hence </b><a NAME="E5:42"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="trees_4.html#M1">FinSequence</a> of  <a href="dtconstr.html#K4" target="_self">TS</a> <font color="Maroon">c<sub>5</sub></font> st <br/>( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="trees_4.html#K12">-tree</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="lang1.html#R1">==&gt;</a>  <a href="dtconstr.html#K1" target="_self">roots</a> <font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="dtconstr.html#E3:42"><i><font color="Green">E32</font></i></a></i>;<br/>


</div>
