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<b>let </b><font color="Maroon" title="c5">G</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="dtconstr.html#V1" title="DTCONSTR:attr.1">with_terminals</a>  <a href="lang1.html#L1" title="LANG1:struct.1">DTConstrStr</a> ;<br/><b>let </b><font color="Maroon" title="c6">tsg</font> be    <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <font color="Maroon" title="c5">G</font>;<br/><b>let </b><font color="Maroon" title="c7">s</font> be   <a href="dtconstr.html#NM2" title="DTCONSTR:NM.2">Terminal</a> of <font color="Maroon" title="c5">G</font>;<br/>





<b>assume </b><a NAME="E1:41"/><i><font color="Green" title="E29">A1</font></i>: 
<font color="Maroon" title="c6">tsg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">s</font>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">G</font>] <b>means </b>for <font color="Olive" title="b1">s</font> being  <a href="dtconstr.html#NM2" title="DTCONSTR:NM.2">Terminal</a> of <font color="Maroon" title="c5">G</font>  st $1 <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">s</font> holds <br/>$1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dtconstr.html#K8" title="DTCONSTR:func.8">root-tree</a> <font color="Olive" title="b1">s</font>;<br/>
<a NAME="E2:41"/><i><font color="Green" title="E30">A2</font></i>: 
for <font color="Olive" title="b1">s</font> being  <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon" title="c5">G</font>  st <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dtconstr.html#K6" title="DTCONSTR:func.6">Terminals</a> <font color="Maroon" title="c5">G</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[ <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Olive" title="b1">s</font>]
 
<b>by </b><i><a class="ref" href="trees_4.html#T3" title="TREES_4:th.3">TREES_4:3</a></i>;<br/>
<div><i><font color="Green" title="E31">A3</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" title="c8">nt</font> be   <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon" title="c5">G</font>;<br/><b>let </b><font color="Maroon" title="c9">ts</font> be    <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <font color="Maroon" title="c5">G</font>;<br/><b>assume </b><a NAME="E1:41_1"/><i><font color="Green" title="E32">A4</font></i>: 
( <font color="Maroon" title="c8">nt</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c9">ts</font> &amp; ( for <font color="Olive" title="b1">t</font> being  <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">G</font>  st <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c9">ts</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">t</font>] ) )
 ;<br/><b>thus </b><a NAME="E2:41_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c8">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c9">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" title="c10">s</font> be   <a href="dtconstr.html#NM2" title="DTCONSTR:NM.2">Terminal</a> of <font color="Maroon" title="c5">G</font>;<br/>

<b>assume </b><a NAME="E1:41_1_1"/><i><font color="Green" title="E33">A5</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c9">ts</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">s</font>
 ;<br/>

<a NAME="E2:41_1_1"/><i><font color="Green" title="E34">A6</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c9">ts</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">nt</font>
 
<b>by </b><i><a class="ref" href="trees_4.html#D4" title="TREES_4:def.4">TREES_4:def 4</a></i>;<br/>
<a NAME="E3:41_1_1"/>
( <font color="Maroon" title="c10">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dtconstr.html#K6" title="DTCONSTR:func.6">Terminals</a> <font color="Maroon" title="c5">G</font> &amp;  <a href="dtconstr.html#K6" title="DTCONSTR:func.6">Terminals</a> <font color="Maroon" title="c5">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">t</font> where <font color="Olive" title="b1">t</font> is   <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon" title="c5">G</font> : for <font color="Olive" title="b2">tnt</font> being  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>  holds not <font color="Olive" title="b1">t</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a> <font color="Olive" title="b2">tnt</font> </span>}</span>  )
 
<b>by </b><i><a class="ref" href="lang1.html#D2" title="LANG1:def.2">LANG1:def 2</a></i>;<br/>
<a NAME="E4:41_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">t</font> being  <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon" title="c5">G</font> st <br/>( <font color="Maroon" title="c10">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">t</font> &amp; ( for <font color="Olive" title="b2">tnt</font> being  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>  holds not <font color="Olive" title="b1">t</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a> <font color="Olive" title="b2">tnt</font> ) )
 
;<br/>
<b>hence </b><a NAME="E5:41_1_1"/>
<font color="Maroon" title="c8">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c9">ts</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dtconstr.html#K8" title="DTCONSTR:func.8">root-tree</a> <font color="Maroon" title="c10">s</font>
 <b>by </b><i><a class="txt" href="dtconstr.html#E1:41_1"><i><font color="Green" title="E32">A4</font></i></a>, <a class="txt" href="dtconstr.html#E1:41_1_1"><i><font color="Green" title="E33">A5</font></i></a>, <a class="txt" href="dtconstr.html#E2:41_1_1"><i><font color="Green" title="E34">A6</font></i></a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E4:41"/>
for <font color="Olive" title="b1">t</font> being  <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c5">G</font>  st <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <font color="Maroon" title="c5">G</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">t</font>]
 
<b>from </b><i><a class="ref" href="dtconstr.html#S7" target="_self" title="DTCONSTR:sch.7">DTCONSTR:sch 7</a>(<a class="txt" href="dtconstr.html#E2:41"><i><font color="Green" title="E30">A2</font></i></a>, <a class="txt" href="dtconstr.html#E3:41"><i><font color="Green" title="E31">A3</font></i></a>);<br/></i>
<b>hence </b><a NAME="E5:41"/>
<font color="Maroon" title="c6">tsg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dtconstr.html#K8" title="DTCONSTR:func.8">root-tree</a> <font color="Maroon" title="c7">s</font>
 <b>by </b><i><a class="txt" href="dtconstr.html#E1:41"><i><font color="Green" title="E29">A1</font></i></a></i>;<br/>


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