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<b>consider </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="dtconstr.html#K4" target="_self">TS</a> <a href="dtconstr.html#K5" target="_self">PeanoNat</a> ,<span class="p1">(<span class="default"><a href="dtconstr.html#K6" target="_self">Terminals</a> <a href="dtconstr.html#K5" target="_self">PeanoNat</a> </span>)</span> <a href="finseq_2.html#K3">*</a> <b> such that </b><br/><a NAME="E2:57"/><i><font color="Green">E31</font></i>: 
(  <a href="dtconstr.html#K16" target="_self">TerminalString</a> <span class="p1">(<span class="default"><a href="trees_4.html#K2">root-tree</a> <font color="Maroon">O</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K8" target="_self">root-tree</a> <font color="Maroon">O</font></span>)</span> &amp; ( for <font color="Olive">t</font> being  <a href="lang1.html#NM1">Symbol</a> of <a href="dtconstr.html#K5" target="_self">PeanoNat</a>   st <font color="Olive">t</font> <a href="hidden.html#R2">in</a>  <a href="dtconstr.html#K6" target="_self">Terminals</a> <a href="dtconstr.html#K5" target="_self">PeanoNat</a>  holds <br/><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="trees_4.html#K2">root-tree</a> <font color="Olive">t</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Olive">t</font></span><a href="lang1.html#K3">*&gt;</a></span> ) &amp; ( for <font color="Olive">nt</font> being  <a href="lang1.html#NM1">Symbol</a> of <a href="dtconstr.html#K5" target="_self">PeanoNat</a> <br/> for <font color="Olive">ts</font> being   <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>   st <font color="Olive">nt</font> <a href="lang1.html#R1">==&gt;</a>  <a href="dtconstr.html#K1" target="_self">roots</a> <font color="Olive">ts</font> holds <br/><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">nt</font> <a href="trees_4.html#K12">-tree</a> <font color="Olive">ts</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="dtconstr.html#K15" target="_self">FlattenSeq</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseqop.html#K5">*</a> <font color="Olive">ts</font></span>)</span> ) )
 <b>by </b><i><a class="ref" href="dtconstr.html#T3" target="_self">Th3</a></i>;<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 <a href="dtconstr.html#K5" target="_self">PeanoNat</a> ] <b>means </b> <a href="dtconstr.html#K16" target="_self">TerminalString</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default">0</span><a href="lang1.html#K3">*&gt;</a></span>;<br/>
<div><i><font color="Green">E32</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">s</font> be   <a href="lang1.html#NM1">Symbol</a> of <a href="dtconstr.html#K5" target="_self">PeanoNat</a> ;<br/><b>assume </b><a NAME="E1:57_1"/>
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="dtconstr.html#K6" target="_self">Terminals</a> <a href="dtconstr.html#K5" target="_self">PeanoNat</a> 
 ;<br/><a NAME="E2:57_1"/><b>then </b>
<font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">O</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:57_1"/>
<font color="Maroon">S<sub>2</sub></font>[ <a href="trees_4.html#K2">root-tree</a> <font color="Maroon">s</font>]
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E34</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="57_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">nt</font> be   <a href="lang1.html#NM1">Symbol</a> of <a href="dtconstr.html#K5" target="_self">PeanoNat</a> ;<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> <a href="dtconstr.html#K5" target="_self">PeanoNat</a> ;<br/><b>assume </b><a NAME="E1:57_2"/><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 <a href="dtconstr.html#K5" target="_self">PeanoNat</a>   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/><a NAME="E2:57_2"/><b>then </b><i><font color="Green">E59</font></i>: 
<font color="Maroon">nt</font> <a href="trees_4.html#K12">-tree</a> <font color="Maroon">ts</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> 
 <b>by </b><i/>;<br/><a NAME="E3:57_2"/>
( <font color="Maroon">nt</font> <a href="hidden.html#R1">=</a> <font color="Maroon">S</font> &amp; (  <a href="dtconstr.html#K1" target="_self">roots</a> <font color="Maroon">ts</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">O</font></span><a href="lang1.html#K3">*&gt;</a></span> or  <a href="dtconstr.html#K1" target="_self">roots</a> <font color="Maroon">ts</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default">1</span><a href="lang1.html#K3">*&gt;</a></span> ) )
 <b>by </b><i>, </i>;<br/><a NAME="E4:57_2"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K1" target="_self">roots</a> <font color="Maroon">ts</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/><b>then consider </b><font color="Maroon">x</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> <b> such that </b><br/><a NAME="E6:57_2"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">ts</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">x</font></span><a href="lang1.html#K3">*&gt;</a></span> &amp; <font color="Maroon">x</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>  )
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</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>  <b>by </b><i><a class="ref" href="dtconstr.html#T7" target="_self">Th7</a></i>;<br/><a NAME="E8:57_2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">ts</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="dtconstr.html#T7" target="_self">Th7</a>, <a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/><a NAME="E9:57_2"/><b>then </b><i><font color="Green">E63</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">ts</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E10:57_2"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x'</font> is    <a href="finseq_2.html#M2">Element</a> of <span class="p1">(<span class="default"><a href="dtconstr.html#K6" target="_self">Terminals</a> <a href="dtconstr.html#K5" target="_self">PeanoNat</a> </span>)</span> <a href="finseq_2.html#K3">*</a> 
 ;<br/><b>then </b><i><font color="Green">E64</font></i>: <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x'</font> = 
 <a href="dtconstr.html#K16" target="_self">TerminalString</a> <font color="Maroon">x</font>
<b>by </b><i>, <a class="ref" href="dtconstr.html#T3" target="_self">Th3</a></i>
<br/>.= 
<span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">O</font></span><a href="lang1.html#K3">*&gt;</a></span>
<b>by </b><i>, </i>
;<br/><a NAME="E12:57_2"/>
<font color="Maroon">f</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">ts</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="dtconstr.html#T7" target="_self">Th7</a>, <a class="ref" href="finseq_2.html#T39">FINSEQ_2:39</a></i>;<br/><b>then </b><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <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="dtconstr.html#K15" target="_self">FlattenSeq</a> <span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x'</font></span>)</span></span><a href="lang1.html#K3">*&gt;</a></span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1"><a href="lang1.html#K3">&lt;*</a><span class="default"><font color="Maroon">O</font></span><a href="lang1.html#K3">*&gt;</a></span>
<b>by </b><i>, <a class="txt" href="dtconstr.html#E3"><i><font color="Green">Lemma36</font></i></a></i>
;<br/><b>hence </b><a NAME="E14:57_2"/>
<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>]
 <b>by </b><i>, <a class="txt" href="dtconstr.html#E1:57_2"><i><font color="Green">E45</font></i></a>, <a class="ref" href="dtconstr.html#T3" target="_self">Th3</a></i>;<br/></div><b>end;</b></div>
<b>thus </b><a NAME="E5:57"/>
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 <a href="dtconstr.html#K5" target="_self">PeanoNat</a>   st <font color="Olive">t</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>  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>(, );</i><br/>


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