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<b>deffunc </b><font color="Maroon">H<sub>3</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">H<sub>2</sub></font>(<font color="Olive" title="b1">o</font>, <a href="mcart_1.html#K15" title="MCART_1:func.15">pr1</a> <span class="p1">(<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>, <a href="mcart_1.html#K16" title="MCART_1:func.16">pr2</a> <span class="p1">(<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><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" title="c1">G</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" title="c1">G</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </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="E2:21_1_1"/><i><font color="Green" title="E8">A1</font></i>: 
 <a href="relat_1.html#K1" title="RELAT_1:func.1">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="E3:21_1_1"/><i><font color="Green" title="E9">A2</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0 <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" title="c1">G</font>, <font color="Olive" title="b2">d</font> is    <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>  : ( ( <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" title="c1">G</font> &amp; <font color="Olive" title="b2">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</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">H<sub>2</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="E4:21_1_1"/><i><font color="Green" title="E10">A3</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#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>3</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="recdef_1.html#S3" title="RECDEF_1:sch.3">RECDEF_1:sch 3</a>();<br/></i>
<a NAME="E5:21_1_1"/><i><font color="Green" title="E11">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#K1" title="NAT_1:func.1">+</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">H<sub>2</sub></font>(<font color="Olive" title="b2">o</font>, <a href="mcart_1.html#K15" title="MCART_1:func.15">pr1</a> <span class="p1">(<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>, <a href="mcart_1.html#K16" title="MCART_1:func.16">pr2</a> <span class="p1">(<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><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" title="c1">G</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" title="c1">G</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </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#E4:21_1_1"><i><font color="Green" title="E10">A3</font></i></a></i>;<br/>
<a NAME="E6:21_1_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" title="c1">G</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" title="c1">G</font>,<a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </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" title="c1">G</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" title="c1">G</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" title="c1">G</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" title="c1">G</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" title="c1">G</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" title="c1">G</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" title="c1">G</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#E2:21_1_1"><i><font color="Green" title="E8">A1</font></i></a>, <a class="txt" href="dtconstr.html#E3:21_1_1"><i><font color="Green" title="E9">A2</font></i></a>, <a class="txt" href="dtconstr.html#E5:21_1_1"><i><font color="Green" title="E11">A4</font></i></a>);<br/></i>
<b>hence </b><a NAME="E7:21_1_1"/>
 ex <font color="Olive">b<sub>1</sub></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" title="c1">G</font></span>)</span> st <br/>( ( for <font color="Olive" title="b2">d</font> being  <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon" title="c1">G</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" title="c1">G</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">b<sub>1</sub></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" title="c1">G</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">b<sub>1</sub></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">b<sub>1</sub></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" title="c1">G</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" title="c1">G</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" title="c1">G</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" title="c1">G</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">b<sub>1</sub></font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">F</font> ) )
 ;<br/>


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