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<b>let </b><font color="Maroon">G</font> be  non <a href="struct_0.html#V3">empty</a> <a href="dtconstr.html#V1" target="_self">with_terminals</a>  <a href="lang1.html#L1">DTConstrStr</a> ;<br/><b>let </b><font color="Maroon">tsg</font> be    <a href="dtconstr.html#M1" target="_self">Element</a> of  <a href="dtconstr.html#K4" target="_self">TS</a> <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">s</font> be   <a href="dtconstr.html#NM2">Terminal</a> of <font color="Maroon">G</font>;<br/>





<b>assume </b><a NAME="E1:41"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">tsg</font> <a href="funct_1.html#K1">.</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">s</font>
 ;<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">G</font>] <b>means </b>for <font color="Olive">s</font> being  <a href="dtconstr.html#NM2">Terminal</a> of <font color="Maroon">G</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">s</font> holds <br/><font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="dtconstr.html#K8" target="_self">root-tree</a> <font color="Olive">s</font>;<br/>
<a NAME="E2:41"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">s</font> being  <a href="lang1.html#NM1">Symbol</a> of <font color="Maroon">G</font>  st <font color="Olive">s</font> <a href="hidden.html#R2">in</a>  <a href="dtconstr.html#K6" target="_self">Terminals</a> <font color="Maroon">G</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[ <a href="trees_4.html#K2">root-tree</a> <font color="Olive">s</font>]
 
<b>by </b><i><a class="ref" href="trees_4.html#T3">TREES_4:3</a></i>;<br/>
<div><i><font color="Green">E34</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">nt</font> be   <a href="lang1.html#NM1">Symbol</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">ts</font> be    <a href="trees_4.html#M1">FinSequence</a> of  <a href="dtconstr.html#K4" target="_self">TS</a> <font color="Maroon">G</font>;<br/><b>assume </b><a NAME="E1:41_1"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">nt</font> <a href="lang1.html#R1">==&gt;</a>  <a href="dtconstr.html#K1" target="_self">roots</a> <font color="Maroon">ts</font> &amp; ( for <font color="Olive">t</font> being  <a href="trees_2.html#NM6">DecoratedTree</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font>  st <font color="Olive">t</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">ts</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">t</font>] ) )
 ;<br/><b>thus </b><a NAME="E2:41_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">nt</font> <a href="trees_4.html#K12">-tree</a> <font color="Maroon">ts</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s</font> be   <a href="dtconstr.html#NM2">Terminal</a> of <font color="Maroon">G</font>;<br/>

<b>assume </b><a NAME="E1:41_1_1"/><i><font color="Green">E59</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">nt</font> <a href="trees_4.html#K12">-tree</a> <font color="Maroon">ts</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">s</font>
 ;<br/>

<a NAME="E2:41_1_1"/><i><font color="Green">E62</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">nt</font> <a href="trees_4.html#K12">-tree</a> <font color="Maroon">ts</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">nt</font>
 
<b>by </b><i><a class="ref" href="trees_4.html#D4">TREES_4:def 4</a></i>;<br/>
<a NAME="E3:41_1_1"/>
( <font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="dtconstr.html#K6" target="_self">Terminals</a> <font color="Maroon">G</font> &amp;  <a href="dtconstr.html#K6" target="_self">Terminals</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">t</font> where <font color="Olive">t</font> is   <a href="lang1.html#NM1">Symbol</a> of <font color="Maroon">G</font> : for <font color="Olive">tnt</font> being  <a href="finseq_1.html#NM1">FinSequence</a>  holds not <font color="Olive">t</font> <a href="lang1.html#R1">==&gt;</a> <font color="Olive">tnt</font> </span>}</span>  )
 
<b>by </b><i><a class="ref" href="lang1.html#D2">LANG1:def 2</a></i>;<br/>
<a NAME="E4:41_1_1"/><b>then </b>
 ex <font color="Olive">t</font> being  <a href="lang1.html#NM1">Symbol</a> of <font color="Maroon">G</font> st <br/>( <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Olive">t</font> &amp; ( for <font color="Olive">tnt</font> being  <a href="finseq_1.html#NM1">FinSequence</a>  holds not <font color="Olive">t</font> <a href="lang1.html#R1">==&gt;</a> <font color="Olive">tnt</font> ) )
 
;<br/>
<b>hence </b><a NAME="E5:41_1_1"/>
<font color="Maroon">nt</font> <a href="trees_4.html#K12">-tree</a> <font color="Maroon">ts</font> <a href="hidden.html#R1">=</a>  <a href="dtconstr.html#K8" target="_self">root-tree</a> <font color="Maroon">s</font>
 <b>by </b><i><a class="txt" href="dtconstr.html#E2:41"><i><font color="Green">E32</font></i></a>, , <a class="txt" href="dtconstr.html#E3:41"><i><font color="Green">E34</font></i></a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E4:41"/>
for <font color="Olive">t</font> being  <a href="trees_2.html#NM6">DecoratedTree</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font>  st <font color="Olive">t</font> <a href="hidden.html#R2">in</a>  <a href="dtconstr.html#K4" target="_self">TS</a> <font color="Maroon">G</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">t</font>]
 
<b>from </b><i><a class="ref" href="dtconstr.html#S7" target="_self">DTCONSTR:sch 7</a>(, <a class="txt" href="dtconstr.html#E1:41"><i><font color="Green">E31</font></i></a>);<br/></i>
<b>hence </b><a NAME="E5:41"/>
<font color="Maroon">tsg</font> <a href="hidden.html#R1">=</a>  <a href="dtconstr.html#K8" target="_self">root-tree</a> <font color="Maroon">s</font>
 <b>by </b><i/>;<br/>


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