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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">F<sub>5</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>6</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E1:2_1"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">s</font> being  <a href="lang1.html#NM1">Symbol</a> of <font color="Maroon">F<sub>1</sub></font>()  st <font color="Olive">s</font> <a href="hidden.html#R2">in</a>  <a href="lang1.html#K5">Terminals</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="trees_4.html#K2">root-tree</a> <font color="Olive">s</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">s</font> be   <a href="lang1.html#NM1">Symbol</a> of <font color="Maroon">F<sub>1</sub></font>();<br/>

<b>assume </b><a NAME="E1:2_1_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">s</font> <a href="hidden.html#R2">in</a>  <a href="lang1.html#K5">Terminals</a> <font color="Maroon">F<sub>1</sub></font>()
 ;<br/>

<b>hence </b><font color="Maroon">F<sub>5</sub></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="Maroon">s</font></span>)</span> = 
<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">s</font>)
<b>by </b><i/>
<br/>.= 
<font color="Maroon">F<sub>6</sub></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="Maroon">s</font></span>)</span>
<b>by </b><i>, </i>
;<br/><br/>


</div><b>end;</b></div>
<a NAME="E2:2_1"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">nt</font> being  <a href="lang1.html#NM1">Symbol</a> of <font color="Maroon">F<sub>1</sub></font>()<br/> for <font color="Olive">ts</font> being   <a href="trees_4.html#M1">FinSequence</a> of  <a href="dtconstr.html#K4">TS</a> <font color="Maroon">F<sub>1</sub></font>()  st <font color="Olive">nt</font> <a href="lang1.html#R1">==&gt;</a>  <a href="dtconstr.html#K1">roots</a> <font color="Olive">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">F<sub>1</sub></font>()  st <font color="Olive">t</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Olive">ts</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">t</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">nt</font> <a href="trees_4.html#K12">-tree</a> <font color="Olive">ts</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_2"><b>proof </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">F<sub>1</sub></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">TS</a> <font color="Maroon">F<sub>1</sub></font>();<br/>



<b>assume </b><a NAME="E1:2_1_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">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">F<sub>1</sub></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">F<sub>5</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Olive">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>6</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Olive">t</font> ) )
 ;<br/>

<a NAME="E2:2_1_2"/><i><font color="Green">E46</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">ts</font> <a href="tarski.html#R1">c=</a>  <a href="dtconstr.html#K4">TS</a> <font color="Maroon">F<sub>1</sub></font>()
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E3:2_1_2"/><b>then </b><i><font color="Green">E47</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">ts</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F<sub>5</sub></font>()
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<b>then </b> <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>5</sub></font>() <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">ts</font></span>)</span> = 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">ts</font>
<b>by </b><i><a class="ref" href="relat_1.html#T46">RELAT_1:46</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">ts</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>
;<br/>
<b>then reconsider </b><font color="Maroon">ntv1</font> = <font color="Maroon">F<sub>5</sub></font>() <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">ts</font> as   <a href="finseq_1.html#NM1">FinSequence</a> <b>by </b><i><a class="ref" href="finseq_1.html#D2">FINSEQ_1:def 2</a></i>;<br/>
<a NAME="E6:2_1_2"/><i><font color="Green">E49</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>5</sub></font>() <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">ts</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">ts</font>
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/>
<a NAME="E7:2_1_2"/>
(  <a href="relat_1.html#K2">rng</a> <font color="Maroon">ntv1</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F<sub>5</sub></font>() &amp;  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F<sub>5</sub></font>() <a href="tarski.html#R1">c=</a> <font color="Maroon">F<sub>2</sub></font>() )
 
<b>by </b><i><a class="ref" href="relat_1.html#T45">RELAT_1:45</a>, <a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<a NAME="E8:2_1_2"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">ntv1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F<sub>2</sub></font>()
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">ntv1</font> = <font color="Maroon">ntv1</font> as    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">F<sub>2</sub></font>() <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E10:2_1_2"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">ts</font> <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F<sub>6</sub></font>()
 
<b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E11:2_1_2"/><b>then </b><i><font color="Green">E50</font></i>: 
 <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F<sub>6</sub></font>() <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">ts</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">ts</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:2_1_2_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">ts</font>
 ;<br/><b>then reconsider </b><font color="Maroon">t</font> = <font color="Maroon">ts</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> as    <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 <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i><a class="ref" href="dtconstr.html#T2">DTCONSTR:2</a></i>;<br/><a NAME="E3:2_1_2_2"/>
<font color="Maroon">t</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="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E4:2_1_2_2"/><b>then </b><i><font color="Green">E52</font></i>: 
<font color="Maroon">F<sub>5</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>6</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font>
 <b>by </b><i/>;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon">F<sub>5</sub></font>() <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">ts</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">F<sub>5</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">t</font>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">F<sub>6</sub></font>() <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">ts</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
<b>by </b><i>, , , <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E13:2_1_2"/><b>then </b><i><font color="Green">E53</font></i>: 
<font color="Maroon">F<sub>5</sub></font>() <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">ts</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>6</sub></font>() <a href="funct_1.html#NK3" target="_self">*</a> <font color="Maroon">ts</font>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>;<br/>
<b>thus </b><font color="Maroon">F<sub>5</sub></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> = 
<font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">nt</font>,<font color="Maroon">ts</font>,<font color="Maroon">ntv1</font>)
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">F<sub>6</sub></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>
<b>by </b><i>, , </i>
;<br/>


</div><b>end;</b></div>
<a NAME="E3:2_1"/><i><font color="Green">E54</font></i>: 
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">F<sub>1</sub></font>()  st <font color="Olive">t</font> <a href="hidden.html#R2">in</a>  <a href="dtconstr.html#K4">TS</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">t</font>]
 
<b>from </b><i><a class="ref" href="dtconstr.html#S7">DTCONSTR:sch 7</a>(, );<br/></i>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:2_1_3"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="dtconstr.html#K4">TS</a> <font color="Maroon">F<sub>1</sub></font>()
 ;<br/><b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as    <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 <font color="Maroon">F<sub>1</sub></font>() ;<br/><a NAME="E3:2_1_3"/>
<font color="Maroon">x'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font>
 ;<br/><b>hence </b><a NAME="E4:2_1_3"/>
<font color="Maroon">F<sub>5</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>6</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:2_1"/>
<font color="Maroon">F<sub>5</sub></font>() <a href="funct_2.html#R2">=</a> <font color="Maroon">F<sub>6</sub></font>()
 <b>by </b><i><a class="ref" href="funct_2.html#T18">FUNCT_2:18</a></i>;<br/>


</div>
