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<b>set </b><font color="Maroon" title="c1">G</font> = <font color="Maroon">F<sub>1</sub></font>();<br/>
<b>deffunc </b><font color="Maroon">H<sub>3</sub></font>(  <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon">F<sub>1</sub></font>(),  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() = <font color="Maroon">F<sub>4</sub></font>($1,<span class="p1">(<span class="default"><a href="mcart_1.html#K15" title="MCART_1:func.15">pr1</a> <span class="p2">(<span class="default"><a href="trees_3.html#K16" title="TREES_3:func.16">roots</a> $2</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="mcart_1.html#K16" title="MCART_1:func.16">pr2</a> <span class="p2">(<span class="default"><a href="trees_3.html#K16" title="TREES_3:func.16">roots</a> $2</span>)</span></span>)</span>);<br/>
<b>deffunc </b><font color="Maroon">H<sub>4</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = $2 <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">o</font>,<font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b1">o</font>,<span class="p1">(<span class="default"><a href="mcart_1.html#K15" title="MCART_1:func.15">pr1</a> <span class="p2">(<span class="default"><a href="trees_3.html#K16" title="TREES_3:func.16">roots</a> <font color="Olive" title="b2">p</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="mcart_1.html#K16" title="MCART_1:func.16">pr2</a> <span class="p2">(<span class="default"><a href="trees_3.html#K16" title="TREES_3:func.16">roots</a> <font color="Olive" title="b2">p</font></span>)</span></span>)</span>)</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Olive" title="b2">p</font></span>)</span> where <font color="Olive" title="b1">o</font> is   <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Olive" title="b2">p</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of $2 <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  :  ex <font color="Olive" title="b3">q</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="trees_3.html#K5" title="TREES_3:func.5">FinTrees</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> st <br/>( <font color="Olive" title="b2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">q</font> &amp; <font color="Olive" title="b1">o</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="dtconstr.html#K2" title="DTCONSTR:func.2">pr1</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Olive" title="b3">q</font></span>)</span> ) </span>}</span> ;<br/>
<b>consider </b><font color="Maroon" title="c2">f</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E2:23_1"/><i><font color="Green" title="E9">A1</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>and </b><br/><a NAME="E3:23_1"/><i><font color="Green" title="E10">A2</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">t</font>,<font color="Olive" title="b2">d</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> where <font color="Olive" title="b1">t</font> is   <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Olive" title="b2">d</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon">F<sub>2</sub></font>() : ( ( <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lang1.html#K5" title="LANG1:func.5">Terminals</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Olive" title="b2">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive" title="b1">t</font>) ) or ( <font color="Olive" title="b1">t</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <font color="Olive" title="b2">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b1">t</font>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ) ) ) </span>}</span> 
 <b>and </b><br/><a NAME="E4:23_1"/><i><font color="Green" title="E11">A3</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive" title="b1">n</font>,<font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font>)
 <b>from </b><i><a class="ref" href="nat_1.html#S11" title="NAT_1:sch.11">NAT_1:sch 11</a>();<br/></i>
<a NAME="E5:23_1"/><i><font color="Green" title="E12">A4</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b2">o</font>,<font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b2">o</font>,<span class="p1">(<span class="default"><a href="mcart_1.html#K15" title="MCART_1:func.15">pr1</a> <span class="p2">(<span class="default"><a href="trees_3.html#K16" title="TREES_3:func.16">roots</a> <font color="Olive" title="b3">p</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="mcart_1.html#K16" title="MCART_1:func.16">pr2</a> <span class="p2">(<span class="default"><a href="trees_3.html#K16" title="TREES_3:func.16">roots</a> <font color="Olive" title="b3">p</font></span>)</span></span>)</span>)</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Olive" title="b3">p</font></span>)</span> where <font color="Olive" title="b2">o</font> is   <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Olive" title="b3">p</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  :  ex <font color="Olive" title="b4">q</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="trees_3.html#K5" title="TREES_3:func.5">FinTrees</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> st <br/>( <font color="Olive" title="b3">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">q</font> &amp; <font color="Olive" title="b2">o</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="dtconstr.html#K2" title="DTCONSTR:func.2">pr1</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Olive" title="b4">q</font></span>)</span> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="dtconstr.html#E4:23_1"><i><font color="Green" title="E11">A3</font></i></a></i>;<br/>
<a NAME="E6:23_1"/>
 ex <font color="Olive" title="b1">X1</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="trees_3.html#K5" title="TREES_3:func.5">FinTrees</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()</span>)</span> st <br/>( <font color="Olive" title="b1">X1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b2">t</font> <a href="trees_3.html#K10" title="TREES_3:func.10">`1</a> </span>)</span> where <font color="Olive" title="b2">t</font> is    <a href="trees_3.html#M5" title="TREES_3:mode.5">Element</a> of  <a href="trees_3.html#K5" title="TREES_3:func.5">FinTrees</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> : <font color="Olive" title="b2">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c2">f</font> </span>}</span>  &amp; ( for <font color="Olive" title="b2">d</font> being  <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon">F<sub>1</sub></font>()  st <font color="Olive" title="b2">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lang1.html#K5" title="LANG1:func.5">Terminals</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/> <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Olive" title="b2">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">X1</font> ) &amp; ( for <font color="Olive" title="b2">o</font> being  <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon">F<sub>1</sub></font>()<br/> for <font color="Olive" title="b3">p</font> being   <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of <font color="Olive" title="b1">X1</font>  st <font color="Olive" title="b2">o</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="Olive" title="b3">p</font> holds <br/><font color="Olive" title="b2">o</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Olive" title="b3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">X1</font> ) &amp; ( for <font color="Olive" title="b2">F</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="trees_3.html#K5" title="TREES_3:func.5">FinTrees</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()</span>)</span>  st ( for <font color="Olive" title="b3">d</font> being  <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon">F<sub>1</sub></font>()  st <font color="Olive" title="b3">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lang1.html#K5" title="LANG1:func.5">Terminals</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/> <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Olive" title="b3">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">F</font> ) &amp; ( for <font color="Olive" title="b3">o</font> being  <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon">F<sub>1</sub></font>()<br/> for <font color="Olive" title="b4">p</font> being   <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of <font color="Olive" title="b2">F</font>  st <font color="Olive" title="b3">o</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="Olive" title="b4">p</font> holds <br/><font color="Olive" title="b3">o</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Olive" title="b4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">F</font> ) holds <br/><font color="Olive" title="b1">X1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">F</font> ) )
 
<b>from </b><i><a class="ref" href="dtconstr.html#S4" target="_self" title="DTCONSTR:sch.4">DTCONSTR:sch 4</a>(<a class="txt" href="dtconstr.html#E2:23_1"><i><font color="Green" title="E9">A1</font></i></a>, <a class="txt" href="dtconstr.html#E3:23_1"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="dtconstr.html#E5:23_1"><i><font color="Green" title="E12">A4</font></i></a>);<br/></i>
<a NAME="E7:23_1"/><b>then </b><i><font color="Green" title="E13">A5</font></i>: 
 <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <font color="Maroon">F<sub>1</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">t</font> <a href="trees_3.html#K10" title="TREES_3:func.10">`1</a> </span>)</span> where <font color="Olive" title="b1">t</font> is    <a href="trees_3.html#M5" title="TREES_3:mode.5">Element</a> of  <a href="trees_3.html#K5" title="TREES_3:func.5">FinTrees</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> : <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c2">f</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="dtconstr.html#D4" target="_self" title="DTCONSTR:def.4">Def4</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>for <font color="Olive" title="b1">dt</font> being  <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>  st <font color="Olive" title="b1">dt</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c2">f</font> &amp; $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">dt</font> <a href="trees_3.html#K10" title="TREES_3:func.10">`1</a>  holds <br/>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">dt</font> <a href="trees_3.html#K11" title="TREES_3:func.11">`2</a> </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> ;<br/>
<a NAME="E8:23_1"/><i><font color="Green" title="E14">A6</font></i>: 
for <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">e</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">F<sub>1</sub></font>() holds <br/> ex <font color="Olive" title="b2">u</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b2">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>() &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">e</font>,<font color="Olive" title="b2">u</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E1:23_1_1"/>
<font color="Maroon" title="c3">e</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">F<sub>1</sub></font>()
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">DT</font> being    <a href="trees_3.html#M5" title="TREES_3:mode.5">Element</a> of  <a href="trees_3.html#K5" title="TREES_3:func.5">FinTrees</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span><b> such that </b><br/><a NAME="E3:23_1_1"/><i><font color="Green" title="E15">A7</font></i>: 
( <font color="Maroon" title="c3">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">DT</font> <a href="trees_3.html#K10" title="TREES_3:func.10">`1</a>  &amp; <font color="Maroon" title="c4">DT</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c2">f</font> )
 <b>by </b><i><a class="txt" href="dtconstr.html#E7:23_1"><i><font color="Green" title="E13">A5</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">r</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  as   <a href="trees_4.html#NM1" title="TREES_4:NM.1">Node</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">DT</font> <a href="trees_3.html#K11" title="TREES_3:func.11">`2</a> </span>)</span> <b>by </b><i><a class="ref" href="trees_1.html#T47" title="TREES_1:th.47">TREES_1:47</a></i>;<br/>
<b>take </b><font color="Maroon" title="c6">u</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c4">DT</font> <a href="trees_3.html#K11" title="TREES_3:func.11">`2</a> </span>)</span> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c5">r</font>;<br/>

<b>thus </b><a NAME="E5:23_1_1"/>
<font color="Maroon" title="c6">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>()
 ;<br/>

<a NAME="E6:23_1_1"/><i><font color="Green" title="E16">A8</font></i>: 
for <font color="Olive" title="b1">dt1</font>, <font color="Olive" title="b2">dt2</font> being  <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>  st <font color="Olive" title="b1">dt1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c2">f</font> &amp; <font color="Olive" title="b2">dt2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c2">f</font> &amp; <font color="Olive" title="b1">dt1</font> <a href="trees_3.html#K10" title="TREES_3:func.10">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">dt2</font> <a href="trees_3.html#K10" title="TREES_3:func.10">`1</a>  holds <br/><font color="Olive" title="b1">dt1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">dt2</font>
 
<b>from </b><i><a class="ref" href="dtconstr.html#S6" target="_self" title="DTCONSTR:sch.6">DTCONSTR:sch 6</a>(<a class="txt" href="dtconstr.html#E2:23_1"><i><font color="Green" title="E9">A1</font></i></a>, <a class="txt" href="dtconstr.html#E3:23_1"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="dtconstr.html#E5:23_1"><i><font color="Green" title="E12">A4</font></i></a>);<br/></i>
<b>let </b><font color="Maroon" title="c7">dt</font> be   <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>;<br/>

<b>assume </b><a NAME="E7:23_1_1"/>
( <font color="Maroon" title="c7">dt</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c2">f</font> &amp; <font color="Maroon" title="c3">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">dt</font> <a href="trees_3.html#K10" title="TREES_3:func.10">`1</a>  )
 ;<br/>

<b>hence </b><a NAME="E8:23_1_1"/>
<font color="Maroon" title="c6">u</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">dt</font> <a href="trees_3.html#K11" title="TREES_3:func.11">`2</a> </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> 
 <b>by </b><i><a class="txt" href="dtconstr.html#E3:23_1_1"><i><font color="Green" title="E15">A7</font></i></a>, <a class="txt" href="dtconstr.html#E6:23_1_1"><i><font color="Green" title="E16">A8</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c3">ff</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><b> such that </b><br/><a NAME="E10:23_1"/><i><font color="Green" title="E15">A9</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c3">ff</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <font color="Maroon">F<sub>1</sub></font>() &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">ff</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon">F<sub>2</sub></font>() )
 <b>and </b><br/><a NAME="E11:23_1"/><i><font color="Green" title="E16">A10</font></i>: 
for <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">e</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">F<sub>1</sub></font>() holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">e</font>,<font color="Maroon" title="c3">ff</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">e</font>]
 <b>from </b><i><a class="ref" href="wellord2.html#S1" title="WELLORD2:sch.1">WELLORD2:sch 1</a>(<a class="txt" href="dtconstr.html#E8:23_1"><i><font color="Green" title="E14">A6</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon" title="c4">ff</font> = <font color="Maroon" title="c3">ff</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>() <b>by </b><i><a class="txt" href="dtconstr.html#E10:23_1"><i><font color="Green" title="E15">A9</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c4">ff</font>
;<br/>

<b>consider </b><font color="Maroon" title="c5">X</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="trees_3.html#K5" title="TREES_3:func.5">FinTrees</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span><b> such that </b><br/><a NAME="E14:23_1"/><i><font color="Green" title="E17">A11</font></i>: 
<font color="Maroon" title="c5">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c2">f</font>
 <b>and </b><br/><a NAME="E15:23_1"/><i><font color="Green" title="E18">A12</font></i>: 
for <font color="Olive" title="b1">d</font> being  <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon">F<sub>1</sub></font>()  st <font color="Olive" title="b1">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lang1.html#K5" title="LANG1:func.5">Terminals</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/> <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">d</font>,<font color="Maroon">F<sub>3</sub></font>(<font color="Olive" title="b1">d</font>)</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X</font>
 <b>and </b><br/><a NAME="E16:23_1"/><i><font color="Green" title="E19">A13</font></i>: 
for <font color="Olive" title="b1">o</font> being  <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon">F<sub>1</sub></font>()<br/> for <font color="Olive" title="b2">p</font> being   <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of <font color="Maroon" title="c5">X</font>  st <font color="Olive" title="b1">o</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="dtconstr.html#K2" title="DTCONSTR:func.2">pr1</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Olive" title="b2">p</font></span>)</span> holds <br/><span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">o</font>,<font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b1">o</font>,<span class="p1">(<span class="default"><a href="dtconstr.html#K2" title="DTCONSTR:func.2">pr1</a> <span class="p2">(<span class="default"><a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Olive" title="b2">p</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="dtconstr.html#K3" title="DTCONSTR:func.3">pr2</a> <span class="p2">(<span class="default"><a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Olive" title="b2">p</font></span>)</span></span>)</span>)</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Olive" title="b2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X</font>
 <b>and </b><br/><a NAME="E17:23_1"/>
for <font color="Olive" title="b1">F</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="trees_3.html#K5" title="TREES_3:func.5">FinTrees</a> <span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span></span>)</span>  st ( for <font color="Olive" title="b2">d</font> being  <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon">F<sub>1</sub></font>()  st <font color="Olive" title="b2">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lang1.html#K5" title="LANG1:func.5">Terminals</a> <font color="Maroon">F<sub>1</sub></font>() holds <br/> <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b2">d</font>,<font color="Maroon">F<sub>3</sub></font>(<font color="Olive" title="b2">d</font>)</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">F</font> ) &amp; ( for <font color="Olive" title="b2">o</font> being  <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon">F<sub>1</sub></font>()<br/> for <font color="Olive" title="b3">p</font> being   <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of <font color="Olive" title="b1">F</font>  st <font color="Olive" title="b2">o</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="dtconstr.html#K2" title="DTCONSTR:func.2">pr1</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Olive" title="b3">p</font></span>)</span> holds <br/><span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b2">o</font>,<font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b2">o</font>,<span class="p1">(<span class="default"><a href="dtconstr.html#K2" title="DTCONSTR:func.2">pr1</a> <span class="p2">(<span class="default"><a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Olive" title="b3">p</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="dtconstr.html#K3" title="DTCONSTR:func.3">pr2</a> <span class="p2">(<span class="default"><a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Olive" title="b3">p</font></span>)</span></span>)</span>)</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Olive" title="b3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">F</font> ) holds <br/><font color="Maroon" title="c5">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">F</font>
 <b>from </b><i><a class="ref" href="dtconstr.html#S3" target="_self" title="DTCONSTR:sch.3">DTCONSTR:sch 3</a>(<a class="txt" href="dtconstr.html#E2:23_1"><i><font color="Green" title="E9">A1</font></i></a>, <a class="txt" href="dtconstr.html#E3:23_1"><i><font color="Green" title="E10">A2</font></i></a>, <a class="txt" href="dtconstr.html#E5:23_1"><i><font color="Green" title="E12">A4</font></i></a>);<br/></i>
<div><b>hereby </b>
<div class="add"><b>let </b><font color="Maroon" title="c6">t</font> be   <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon">F<sub>1</sub></font>();<br/><b>assume </b><a NAME="E1:23_1_2"/><i><font color="Green" title="E20">A14</font></i>: 
<font color="Maroon" title="c6">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lang1.html#K5" title="LANG1:func.5">Terminals</a> <font color="Maroon">F<sub>1</sub></font>()
 ;<br/><a NAME="E2:23_1_2"/><i><font color="Green" title="E21">A15</font></i>: 
( <span class="p1">(<span class="default"><a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">t</font>,<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c6">t</font>)</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="trees_3.html#K10" title="TREES_3:func.10">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Maroon" title="c6">t</font> &amp; <span class="p1">(<span class="default"><a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">t</font>,<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c6">t</font>)</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span>)</span> <a href="trees_3.html#K11" title="TREES_3:func.11">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c6">t</font>) )
 <b>by </b><i><a class="ref" href="trees_4.html#T25" title="TREES_4:th.25">TREES_4:25</a></i>;<br/><a NAME="E3:23_1_2"/>
 <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">t</font>,<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c6">t</font>)</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c2">f</font>
 <b>by </b><i><a class="txt" href="dtconstr.html#E14:23_1"><i><font color="Green" title="E17">A11</font></i></a>, <a class="txt" href="dtconstr.html#E15:23_1"><i><font color="Green" title="E18">A12</font></i></a>, <a class="txt" href="dtconstr.html#E1:23_1_2"><i><font color="Green" title="E20">A14</font></i></a></i>;<br/><a NAME="E4:23_1_2"/><b>then </b>
 <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Maroon" title="c6">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">F<sub>1</sub></font>()
 <b>by </b><i><a class="txt" href="dtconstr.html#E7:23_1"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="dtconstr.html#E2:23_1_2"><i><font color="Green" title="E21">A15</font></i></a></i>;<br/><b>hence </b><font color="Maroon" title="c4">ff</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Maroon" title="c6">t</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c6">t</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> 
<b>by </b><i><a class="txt" href="dtconstr.html#E11:23_1"><i><font color="Green" title="E16">A10</font></i></a>, <a class="txt" href="dtconstr.html#E14:23_1"><i><font color="Green" title="E17">A11</font></i></a>, <a class="txt" href="dtconstr.html#E15:23_1"><i><font color="Green" title="E18">A12</font></i></a>, <a class="txt" href="dtconstr.html#E1:23_1_2"><i><font color="Green" title="E20">A14</font></i></a>, <a class="txt" href="dtconstr.html#E2:23_1_2"><i><font color="Green" title="E21">A15</font></i></a></i>
<br/>.= 
<font color="Maroon">F<sub>3</sub></font>(<font color="Maroon" title="c6">t</font>)
<b>by </b><i><a class="ref" href="trees_4.html#T3" title="TREES_4:th.3">TREES_4:3</a></i>
;<br/><br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon" title="c6">nt</font> be   <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon">F<sub>1</sub></font>();<br/><b>let </b><font color="Maroon" title="c7">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">F<sub>1</sub></font>();<br/>



<b>set </b><font color="Maroon" title="c8">rts</font> =  <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c7">ts</font>;<br/>
<b>assume </b><a NAME="E19:23_1"/><i><font color="Green" title="E20">A16</font></i>: 
<font color="Maroon" title="c6">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="c7">ts</font>
 ;<br/>

<b>set </b><font color="Maroon" title="c9">x</font> = <font color="Maroon" title="c4">ff</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c7">ts</font>;<br/>
<a NAME="E20:23_1"/><i><font color="Green" title="E21">A17</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">ts</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">ts</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">dt</font> being  <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> st <br/>( <font color="Olive" title="b1">dt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $2 &amp; <font color="Olive" title="b1">dt</font> <a href="trees_3.html#K10" title="TREES_3:func.10">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">ts</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 &amp; <font color="Olive" title="b1">dt</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c2">f</font> );<br/>
<a NAME="E21:23_1"/><i><font color="Green" title="E22">A18</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">ts</font></span>)</span> holds <br/> ex <font color="Olive" title="b2">x</font> being   <a href="trees_3.html#M5" title="TREES_3:mode.5">Element</a> of  <a href="trees_3.html#K5" title="TREES_3:func.5">FinTrees</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font>,<font color="Olive" title="b2">x</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:23_1_3"/>
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">ts</font></span>)</span>
 ;<br/>

<a NAME="E2:23_1_3"/><b>then </b>
( <font color="Maroon" title="c7">ts</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</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="c7">ts</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">ts</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <font color="Maroon">F<sub>1</sub></font>() )
 
<b>by </b><i><a class="txt" href="dtconstr.html#E20:23_1"><i><font color="Green" title="E21">A17</font></i></a>, <a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E3:23_1_3"/><b>then </b>
<font color="Maroon" title="c7">ts</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</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">F<sub>1</sub></font>()
 
;<br/>
<a NAME="E4:23_1_3"/><b>then </b>
 ex <font color="Olive" title="b1">x</font> being   <a href="trees_3.html#M5" title="TREES_3:mode.5">Element</a> of  <a href="trees_3.html#K5" title="TREES_3:func.5">FinTrees</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> st <br/>( <font color="Maroon" title="c7">ts</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="trees_3.html#K10" title="TREES_3:func.10">`1</a>  &amp; <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c2">f</font> )
 
<b>by </b><i><a class="txt" href="dtconstr.html#E7:23_1"><i><font color="Green" title="E13">A5</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:23_1_3"/>
 ex <font color="Olive" title="b1">x</font> being   <a href="trees_3.html#M5" title="TREES_3:mode.5">Element</a> of  <a href="trees_3.html#K5" title="TREES_3:func.5">FinTrees</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c10">k</font>,<font color="Olive" title="b1">x</font>]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c10">dts</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="trees_3.html#K5" title="TREES_3:func.5">FinTrees</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span><b> such that </b><br/><a NAME="E23:23_1"/><i><font color="Green" title="E23">A19</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c10">dts</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">ts</font></span>)</span>
 <b>and </b><br/><a NAME="E24:23_1"/><i><font color="Green" title="E24">A20</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">ts</font></span>)</span> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font>,<font color="Maroon" title="c10">dts</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font>]
 <b>from </b><i><a class="ref" href="finseq_1.html#S5" title="FINSEQ_1:sch.5">FINSEQ_1:sch 5</a>(<a class="txt" href="dtconstr.html#E21:23_1"><i><font color="Green" title="E22">A18</font></i></a>);<br/></i>
<a NAME="E25:23_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c10">dts</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c5">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:23_1_4"/>
<font color="Maroon" title="c11">x</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="c10">dts</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c12">k</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:23_1_4"/><i><font color="Green" title="E25">A21</font></i>: 
( <font color="Maroon" title="c12">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">ts</font> &amp; <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">dts</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font> )
 <b>by </b><i><a class="txt" href="dtconstr.html#E20:23_1"><i><font color="Green" title="E21">A17</font></i></a>, <a class="txt" href="dtconstr.html#E23:23_1"><i><font color="Green" title="E23">A19</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">k</font> = <font color="Maroon" title="c12">k</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="dtconstr.html#E3:23_1_4"><i><font color="Green" title="E25">A21</font></i></a></i>;<br/>
<a NAME="E5:23_1_4"/>
 ex <font color="Olive" title="b1">dt</font> being  <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> st <br/>( <font color="Olive" title="b1">dt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x</font> &amp; <font color="Olive" title="b1">dt</font> <a href="trees_3.html#K10" title="TREES_3:func.10">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">ts</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">k</font> &amp; <font color="Olive" title="b1">dt</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c2">f</font> )
 
<b>by </b><i><a class="txt" href="dtconstr.html#E20:23_1"><i><font color="Green" title="E21">A17</font></i></a>, <a class="txt" href="dtconstr.html#E24:23_1"><i><font color="Green" title="E24">A20</font></i></a>, <a class="txt" href="dtconstr.html#E3:23_1_4"><i><font color="Green" title="E25">A21</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:23_1_4"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X</font>
 <b>by </b><i><a class="txt" href="dtconstr.html#E14:23_1"><i><font color="Green" title="E17">A11</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c11">dts</font> = <font color="Maroon" title="c10">dts</font> as    <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of <font color="Maroon" title="c5">X</font> <b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E27:23_1"/><i><font color="Green" title="E25">A22</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c7">ts</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">ts</font>
 
<b>by </b><i><a class="ref" href="trees_3.html#D18" title="TREES_3:def.18">TREES_3:def 18</a></i>;<br/>
<a NAME="E28:23_1"/><i><font color="Green" title="E26">A23</font></i>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K2" title="DTCONSTR:func.2">pr1</a> <span class="p2">(<span class="default"><a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c11">dts</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c11">dts</font></span>)</span> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K3" title="DTCONSTR:func.3">pr2</a> <span class="p2">(<span class="default"><a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c11">dts</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c11">dts</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="mcart_1.html#D12" title="MCART_1:def.12">MCART_1:def 12</a>, <a class="ref" href="mcart_1.html#D13" title="MCART_1:def.13">MCART_1:def 13</a></i>;<br/>
<a NAME="E29:23_1"/><b>then </b><i><font color="Green" title="E27">A24</font></i>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K2" title="DTCONSTR:func.2">pr1</a> <span class="p2">(<span class="default"><a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c11">dts</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">ts</font> &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K3" title="DTCONSTR:func.3">pr2</a> <span class="p2">(<span class="default"><a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c11">dts</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">ts</font> )
 
<b>by </b><i><a class="txt" href="dtconstr.html#E20:23_1"><i><font color="Green" title="E21">A17</font></i></a>, <a class="txt" href="dtconstr.html#E23:23_1"><i><font color="Green" title="E23">A19</font></i></a>, <a class="ref" href="trees_3.html#D18" title="TREES_3:def.18">TREES_3:def 18</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c12">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E1:23_1_5"/><i><font color="Green" title="E28">A25</font></i>: 
<font color="Maroon" title="c12">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">ts</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c13">dt</font> being   <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span><b> such that </b><br/><a NAME="E3:23_1_5"/><i><font color="Green" title="E29">A26</font></i>: 
( <font color="Maroon" title="c13">dt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">dts</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font> &amp; <font color="Maroon" title="c13">dt</font> <a href="trees_3.html#K10" title="TREES_3:func.10">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">ts</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font> &amp; <font color="Maroon" title="c13">dt</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c2">f</font> )
 <b>by </b><i><a class="txt" href="dtconstr.html#E20:23_1"><i><font color="Green" title="E21">A17</font></i></a>, <a class="txt" href="dtconstr.html#E24:23_1"><i><font color="Green" title="E24">A20</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c14">r</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  as   <a href="trees_4.html#NM1" title="TREES_4:NM.1">Node</a> of <font color="Maroon" title="c13">dt</font> <b>by </b><i><a class="ref" href="trees_1.html#T47" title="TREES_1:th.47">TREES_1:47</a></i>;<br/><a NAME="E5:23_1_5"/>
 ex <font color="Olive" title="b1">T</font> being  <a href="trees_2.html#NM4" title="TREES_2:NM.4">DecoratedTree</a> st <br/>( <font color="Olive" title="b1">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">ts</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font> &amp; <span class="p1">(<span class="default"><a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c7">ts</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">T</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>  )
 <b>by </b><i><a class="txt" href="dtconstr.html#E1:23_1_5"><i><font color="Green" title="E28">A25</font></i></a>, <a class="ref" href="trees_3.html#D18" title="TREES_3:def.18">TREES_3:def 18</a></i>;<br/><a NAME="E6:23_1_5"/><b>then </b><i><font color="Green" title="E30">A27</font></i>: 
<span class="p1">(<span class="default"><a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c7">ts</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">dt</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c14">r</font></span>)</span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a> 
 <b>by </b><i><a class="txt" href="dtconstr.html#E3:23_1_5"><i><font color="Green" title="E29">A26</font></i></a>, <a class="ref" href="trees_3.html#T41" title="TREES_3:th.41">TREES_3:41</a></i>;<br/><a NAME="E7:23_1_5"/>
 ex <font color="Olive" title="b1">T</font> being  <a href="trees_2.html#NM4" title="TREES_2:NM.4">DecoratedTree</a> st <br/>( <font color="Olive" title="b1">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">dts</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font> &amp; <span class="p1">(<span class="default"><a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c11">dts</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">T</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>  )
 <b>by </b><i><a class="txt" href="dtconstr.html#E20:23_1"><i><font color="Green" title="E21">A17</font></i></a>, <a class="txt" href="dtconstr.html#E23:23_1"><i><font color="Green" title="E23">A19</font></i></a>, <a class="txt" href="dtconstr.html#E1:23_1_5"><i><font color="Green" title="E28">A25</font></i></a>, <a class="ref" href="trees_3.html#D18" title="TREES_3:def.18">TREES_3:def 18</a></i>;<br/><b>hence </b><a NAME="E8:23_1_5"/>
<span class="p1">(<span class="default"><a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c7">ts</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K2" title="DTCONSTR:func.2">pr1</a> <span class="p2">(<span class="default"><a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c11">dts</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">k</font>
 <b>by </b><i><a class="txt" href="dtconstr.html#E28:23_1"><i><font color="Green" title="E26">A23</font></i></a>, <a class="txt" href="dtconstr.html#E29:23_1"><i><font color="Green" title="E27">A24</font></i></a>, <a class="txt" href="dtconstr.html#E1:23_1_5"><i><font color="Green" title="E28">A25</font></i></a>, <a class="txt" href="dtconstr.html#E3:23_1_5"><i><font color="Green" title="E29">A26</font></i></a>, <a class="txt" href="dtconstr.html#E6:23_1_5"><i><font color="Green" title="E30">A27</font></i></a>, <a class="ref" href="mcart_1.html#D12" title="MCART_1:def.12">MCART_1:def 12</a></i>;<br/></div><b>end;</b></div>
<a NAME="E31:23_1"/><b>then </b><i><font color="Green" title="E28">A28</font></i>: 
 <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c7">ts</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dtconstr.html#K2" title="DTCONSTR:func.2">pr1</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c11">dts</font></span>)</span>
 
<b>by </b><i><a class="txt" href="dtconstr.html#E27:23_1"><i><font color="Green" title="E25">A22</font></i></a>, <a class="txt" href="dtconstr.html#E29:23_1"><i><font color="Green" title="E27">A24</font></i></a>, <a class="ref" href="finseq_1.html#T17" title="FINSEQ_1:th.17">FINSEQ_1:17</a></i>;<br/>
<a NAME="E32:23_1"/><b>then </b><i><font color="Green" title="E29">A29</font></i>: 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">nt</font>,<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c6">nt</font>,<font color="Maroon" title="c11">dts</font>)</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c11">dts</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">X</font>
 
<b>by </b><i><a class="txt" href="dtconstr.html#E16:23_1"><i><font color="Green" title="E19">A13</font></i></a>, <a class="txt" href="dtconstr.html#E19:23_1"><i><font color="Green" title="E20">A16</font></i></a></i>;<br/>
<a NAME="E33:23_1"/><i><font color="Green" title="E30">A30</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c11">dts</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="trees_3.html#K5" title="TREES_3:func.5">FinTrees</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E34:23_1"/>
 <a href="trees_3.html#K5" title="TREES_3:func.5">FinTrees</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="trees_3.html#K4" title="TREES_3:func.4">Trees</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="trees_3.html#T22" title="TREES_3:th.22">TREES_3:22</a></i>;<br/>
<a NAME="E35:23_1"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c11">dts</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="trees_3.html#K4" title="TREES_3:func.4">Trees</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="txt" href="dtconstr.html#E33:23_1"><i><font color="Green" title="E30">A30</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c12">dts'</font> = <font color="Maroon" title="c11">dts</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="trees_3.html#K4" title="TREES_3:func.4">Trees</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E37:23_1"/><i><font color="Green" title="E31">A31</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">ts</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="trees_3.html#K5" title="TREES_3:func.5">FinTrees</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E38:23_1"/>
 <a href="trees_3.html#K5" title="TREES_3:func.5">FinTrees</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="trees_3.html#K4" title="TREES_3:func.4">Trees</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()
 
<b>by </b><i><a class="ref" href="trees_3.html#T22" title="TREES_3:th.22">TREES_3:22</a></i>;<br/>
<a NAME="E39:23_1"/><b>then </b>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">ts</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="trees_3.html#K4" title="TREES_3:func.4">Trees</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()
 
<b>by </b><i><a class="txt" href="dtconstr.html#E37:23_1"><i><font color="Green" title="E31">A31</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c13">ts'</font> = <font color="Maroon" title="c7">ts</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="trees_3.html#K4" title="TREES_3:func.4">Trees</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>() <b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c14">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:23_1_6"/>
<font color="Maroon" title="c14">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">dts</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c15">dt</font> being   <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span><b> such that </b><br/><a NAME="E3:23_1_6"/><i><font color="Green" title="E32">A32</font></i>: 
( <font color="Maroon" title="c15">dt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">dts</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font> &amp; <font color="Maroon" title="c15">dt</font> <a href="trees_3.html#K10" title="TREES_3:func.10">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">ts</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font> &amp; <font color="Maroon" title="c15">dt</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c2">f</font> )
 <b>by </b><i><a class="txt" href="dtconstr.html#E23:23_1"><i><font color="Green" title="E23">A19</font></i></a>, <a class="txt" href="dtconstr.html#E24:23_1"><i><font color="Green" title="E24">A20</font></i></a></i>;<br/><b>let </b><font color="Maroon" title="c16">T</font> be   <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>;<br/><b>assume </b><a NAME="E4:23_1_6"/>
<font color="Maroon" title="c16">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">dts</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font>
 ;<br/><b>hence </b><a NAME="E5:23_1_6"/>
<font color="Maroon" title="c7">ts</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">T</font> <a href="trees_3.html#K10" title="TREES_3:func.10">`1</a> 
 <b>by </b><i><a class="txt" href="dtconstr.html#E3:23_1_6"><i><font color="Green" title="E32">A32</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E42:23_1"/><b>then </b><i><font color="Green" title="E32">A33</font></i>: 
<span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">nt</font>,<font color="Maroon">H<sub>3</sub></font>(<font color="Maroon" title="c6">nt</font>,<font color="Maroon" title="c11">dts</font>)</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="trees_4.html#K8" title="TREES_4:func.8">-tree</a> <font color="Maroon" title="c12">dts'</font></span>)</span> <a href="trees_3.html#K10" title="TREES_3:func.10">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">nt</font> <a href="trees_4.html#K8" title="TREES_4:func.8">-tree</a> <font color="Maroon" title="c13">ts'</font>
 
<b>by </b><i><a class="txt" href="dtconstr.html#E20:23_1"><i><font color="Green" title="E21">A17</font></i></a>, <a class="txt" href="dtconstr.html#E23:23_1"><i><font color="Green" title="E23">A19</font></i></a>, <a class="ref" href="trees_4.html#T27" title="TREES_4:th.27">TREES_4:27</a></i>;<br/>
<a NAME="E43:23_1"/><i><font color="Green" title="E33">A34</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c7">ts</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">ff</font>
 
<b>by </b><i><a class="txt" href="dtconstr.html#E10:23_1"><i><font color="Green" title="E15">A9</font></i></a>, <a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E44:23_1"/><b>then </b><i><font color="Green" title="E34">A35</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">ff</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c7">ts</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">ts</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_7"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c14">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E1:23_1_7"/><i><font color="Green" title="E35">A36</font></i>: 
<font color="Maroon" title="c14">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">ts</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c15">dt</font> being   <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>(),<font color="Maroon">F<sub>2</sub></font>()</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span><b> such that </b><br/><a NAME="E3:23_1_7"/><i><font color="Green" title="E36">A37</font></i>: 
( <font color="Maroon" title="c15">dt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">dts</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">k</font> &amp; <font color="Maroon" title="c15">dt</font> <a href="trees_3.html#K10" title="TREES_3:func.10">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">ts</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">k</font> &amp; <font color="Maroon" title="c15">dt</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon" title="c2">f</font> )
 <b>by </b><i><a class="txt" href="dtconstr.html#E20:23_1"><i><font color="Green" title="E21">A17</font></i></a>, <a class="txt" href="dtconstr.html#E24:23_1"><i><font color="Green" title="E24">A20</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon" title="c16">r</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  as   <a href="trees_4.html#NM1" title="TREES_4:NM.1">Node</a> of <font color="Maroon" title="c15">dt</font> <b>by </b><i><a class="ref" href="trees_1.html#T47" title="TREES_1:th.47">TREES_1:47</a></i>;<br/><a NAME="E5:23_1_7"/><i><font color="Green" title="E37">A38</font></i>: 
<font color="Maroon" title="c7">ts</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">k</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="c7">ts</font>
 <b>by </b><i><a class="txt" href="dtconstr.html#E1:23_1_7"><i><font color="Green" title="E35">A36</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><i><font color="Green" title="E38">A39</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c4">ff</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c7">ts</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">k</font> = 
<font color="Maroon" title="c4">ff</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">dt</font> <a href="trees_3.html#K10" title="TREES_3:func.10">`1</a> </span>)</span>
<b>by </b><i><a class="txt" href="dtconstr.html#E44:23_1"><i><font color="Green" title="E34">A35</font></i></a>, <a class="txt" href="dtconstr.html#E1:23_1_7"><i><font color="Green" title="E35">A36</font></i></a>, <a class="txt" href="dtconstr.html#E3:23_1_7"><i><font color="Green" title="E36">A37</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">dt</font> <a href="trees_3.html#K11" title="TREES_3:func.11">`2</a> </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> 
<b>by </b><i><a class="txt" href="dtconstr.html#E10:23_1"><i><font color="Green" title="E15">A9</font></i></a>, <a class="txt" href="dtconstr.html#E11:23_1"><i><font color="Green" title="E16">A10</font></i></a>, <a class="txt" href="dtconstr.html#E43:23_1"><i><font color="Green" title="E33">A34</font></i></a>, <a class="txt" href="dtconstr.html#E3:23_1_7"><i><font color="Green" title="E36">A37</font></i></a>, <a class="txt" href="dtconstr.html#E5:23_1_7"><i><font color="Green" title="E37">A38</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">dt</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c16">r</font></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> 
<b>by </b><i><a class="ref" href="trees_3.html#T41" title="TREES_3:th.41">TREES_3:41</a></i>
;<br/><a NAME="E7:23_1_7"/>
 ex <font color="Olive" title="b1">T</font> being  <a href="trees_2.html#NM4" title="TREES_2:NM.4">DecoratedTree</a> st <br/>( <font color="Olive" title="b1">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">dts</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">k</font> &amp; <span class="p1">(<span class="default"><a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c11">dts</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">T</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">r</font> )
 <b>by </b><i><a class="txt" href="dtconstr.html#E20:23_1"><i><font color="Green" title="E21">A17</font></i></a>, <a class="txt" href="dtconstr.html#E23:23_1"><i><font color="Green" title="E23">A19</font></i></a>, <a class="txt" href="dtconstr.html#E1:23_1_7"><i><font color="Green" title="E35">A36</font></i></a>, <a class="ref" href="trees_3.html#D18" title="TREES_3:def.18">TREES_3:def 18</a></i>;<br/><b>hence </b><a NAME="E8:23_1_7"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">ff</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c7">ts</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K3" title="DTCONSTR:func.3">pr2</a> <span class="p2">(<span class="default"><a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c11">dts</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">k</font>
 <b>by </b><i><a class="txt" href="dtconstr.html#E28:23_1"><i><font color="Green" title="E26">A23</font></i></a>, <a class="txt" href="dtconstr.html#E29:23_1"><i><font color="Green" title="E27">A24</font></i></a>, <a class="txt" href="dtconstr.html#E1:23_1_7"><i><font color="Green" title="E35">A36</font></i></a>, <a class="txt" href="dtconstr.html#E3:23_1_7"><i><font color="Green" title="E36">A37</font></i></a>, <a class="txt" href="dtconstr.html#E6:23_1_7"><i><font color="Green" title="E38">A39</font></i></a>, <a class="ref" href="mcart_1.html#D13" title="MCART_1:def.13">MCART_1:def 13</a></i>;<br/></div><b>end;</b></div>
<a NAME="E46:23_1"/><b>then </b><i><font color="Green" title="E35">A40</font></i>: 
<font color="Maroon" title="c4">ff</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c7">ts</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dtconstr.html#K3" title="DTCONSTR:func.3">pr2</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c11">dts</font></span>)</span>
 
<b>by </b><i><a class="txt" href="dtconstr.html#E29:23_1"><i><font color="Green" title="E27">A24</font></i></a>, <a class="txt" href="dtconstr.html#E44:23_1"><i><font color="Green" title="E34">A35</font></i></a>, <a class="ref" href="finseq_1.html#T17" title="FINSEQ_1:th.17">FINSEQ_1:17</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">r</font> =  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  as   <a href="trees_4.html#NM1" title="TREES_4:NM.1">Node</a> of <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">nt</font>,<font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c6">nt</font>,<span class="p1">(<span class="default"><a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c7">ts</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">ff</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c7">ts</font></span>)</span>)</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c11">dts</font></span>)</span> <b>by </b><i><a class="ref" href="trees_1.html#T47" title="TREES_1:th.47">TREES_1:47</a></i>;<br/>
<a NAME="E48:23_1"/>
<font color="Maroon" title="c6">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c7">ts</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">F<sub>1</sub></font>()
 
<b>by </b><i><a class="txt" href="dtconstr.html#E7:23_1"><i><font color="Green" title="E13">A5</font></i></a>, <a class="txt" href="dtconstr.html#E14:23_1"><i><font color="Green" title="E17">A11</font></i></a>, <a class="txt" href="dtconstr.html#E32:23_1"><i><font color="Green" title="E29">A29</font></i></a>, <a class="txt" href="dtconstr.html#E42:23_1"><i><font color="Green" title="E32">A33</font></i></a></i>;<br/>
<b>then </b><font color="Maroon" title="c4">ff</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c7">ts</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">nt</font>,<font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c6">nt</font>,<span class="p1">(<span class="default"><a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c7">ts</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">ff</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c7">ts</font></span>)</span>)</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c11">dts</font></span>)</span> <a href="trees_3.html#K11" title="TREES_3:func.11">`2</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">r</font>
<b>by </b><i><a class="txt" href="dtconstr.html#E11:23_1"><i><font color="Green" title="E16">A10</font></i></a>, <a class="txt" href="dtconstr.html#E14:23_1"><i><font color="Green" title="E17">A11</font></i></a>, <a class="txt" href="dtconstr.html#E16:23_1"><i><font color="Green" title="E19">A13</font></i></a>, <a class="txt" href="dtconstr.html#E19:23_1"><i><font color="Green" title="E20">A16</font></i></a>, <a class="txt" href="dtconstr.html#E31:23_1"><i><font color="Green" title="E28">A28</font></i></a>, <a class="txt" href="dtconstr.html#E42:23_1"><i><font color="Green" title="E32">A33</font></i></a>, <a class="txt" href="dtconstr.html#E46:23_1"><i><font color="Green" title="E35">A40</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">nt</font>,<font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c6">nt</font>,<span class="p1">(<span class="default"><a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c7">ts</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">ff</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c7">ts</font></span>)</span>)</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c11">dts</font></span>)</span> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c14">r</font></span>)</span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> 
<b>by </b><i><a class="ref" href="trees_3.html#T41" title="TREES_3:th.41">TREES_3:41</a></i>
<br/>.= 
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c6">nt</font>,<font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c6">nt</font>,<span class="p1">(<span class="default"><a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c7">ts</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">ff</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c7">ts</font></span>)</span>)</span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a> 
<b>by </b><i><a class="ref" href="trees_4.html#D4" title="TREES_4:def.4">TREES_4:def 4</a></i>
;<br/>
<b>hence </b><a NAME="E50:23_1"/>
<font color="Maroon" title="c4">ff</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c7">ts</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon" title="c6">nt</font>,<span class="p1">(<span class="default"><a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c7">ts</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">ff</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c7">ts</font></span>)</span>)
 <b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>;<br/>


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