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<span class="kw">reconsider </span><font color="Maroon" title="c1">D</font> =  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon">F<sub>3</sub></font>() as  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
 <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="msafree.html#K6" title="MSAFREE:func.6">DTConMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> = 
 <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="msafree.html#K9" title="MSAFREE:func.9">FreeSort</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="msafree.html#T12" title="MSAFREE:th.12">MSAFREE:12</a></span>
<br/>.= 
 <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="msafree.html#K9" title="MSAFREE:func.9">FreeSort</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_3.html#D4" title="CARD_3:def.4">CARD_3:def 4</a></span>
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c2">f1</font> =  <a href="msafree1.html#K1" title="MSAFREE1:func.1">Flatten</a> <font color="Maroon">F<sub>6</sub></font>(), <font color="Maroon" title="c3">f2</font> =  <a href="msafree1.html#K1" title="MSAFREE1:func.1">Flatten</a> <font color="Maroon">F<sub>7</sub></font>() as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p2">(<span class="default"><a href="msafree.html#K6" title="MSAFREE:func.6">DTConMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span></span>)</span>,<font color="Maroon" title="c1">D</font> ;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="msafree.html#K6" title="MSAFREE:func.6">DTConMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span>,   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p2">(<span class="default"><a href="msafree.html#K6" title="MSAFREE:func.6">DTConMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span></span>)</span> <a href="finseq_1.html#K13" title="FINSEQ_1:func.13">*</a> ,   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span> <a href="finseq_1.html#K13" title="FINSEQ_1:func.13">*</a> ) <span class="kw">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon">F<sub>3</sub></font>() = <font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default">$1 <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span>,$2,$3);<br/>
<span class="kw">consider </span><font color="Maroon" title="c4">H</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="msafree.html#K6" title="MSAFREE:func.6">DTConMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p4">(<span class="default"><a href="msafree.html#K6" title="MSAFREE:func.6">DTConMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span></span>)</span> <a href="finseq_1.html#K13" title="FINSEQ_1:func.13">*</a> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span> <a href="finseq_1.html#K13" title="FINSEQ_1:func.13">*</a> </span>)</span></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span>,<span class="p1">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span><span class="kw"> such that </span><br/><a NAME="E5:19_1"/><span class="lab"><font color="Green" title="E13">A5</font></span>: 
for <font color="Olive" title="b1">nt</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="msafree.html#K6" title="MSAFREE:func.6">DTConMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span><br/> for <font color="Olive" title="b2">ts</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p2">(<span class="default"><a href="msafree.html#K6" title="MSAFREE:func.6">DTConMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span></span>)</span> <a href="finseq_1.html#K13" title="FINSEQ_1:func.13">*</a> <br/> for <font color="Olive" title="b3">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span> <a href="finseq_1.html#K13" title="FINSEQ_1:func.13">*</a>  holds  <font color="Maroon" title="c4">H</font> <a href="multop_1.html#K2" title="MULTOP_1:func.2">.</a> <font color="Olive" title="b1">nt</font>,<font color="Olive" title="b2">ts</font>,<font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">nt</font>,<font color="Olive" title="b2">ts</font>,<font color="Olive" title="b3">x</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="multop_1.html#S4" title="MULTOP_1:sch.4">MULTOP_1:sch 4</a>();<br/></span>
<span class="kw">reconsider </span><font color="Maroon" title="c5">H</font> = <font color="Maroon" title="c4">H</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="msafree.html#K6" title="MSAFREE:func.6">DTConMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p4">(<span class="default"><a href="msafree.html#K6" title="MSAFREE:func.6">DTConMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span></span>)</span> <a href="finseq_1.html#K13" title="FINSEQ_1:func.13">*</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">D</font> <a href="finseq_1.html#K13" title="FINSEQ_1:func.13">*</a> </span>)</span></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span>,<font color="Maroon" title="c1">D</font> ;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>2</sub></font>(  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="msafree.html#K6" title="MSAFREE:func.6">DTConMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span>,   <a href="msafree1.html#M1" title="MSAFREE1:mode.1">FinSequence</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="msafree.html#K6" title="MSAFREE:func.6">DTConMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span>,   <a href="msafree1.html#M1" title="MSAFREE1:mode.1">FinSequence</a> of <font color="Maroon" title="c1">D</font>) <span class="kw">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font> = <font color="Maroon" title="c5">H</font> <a href="multop_1.html#K2" title="MULTOP_1:func.2">.</a> $1,$2,$3;<br/>
<a NAME="E7:19_1"/><span class="lab"><font color="Green" title="E14">A6</font></span>: 
 <a href="msafree.html#K6" title="MSAFREE:func.6">DTConMSA</a> <font color="Maroon">F<sub>2</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lang1.html#G1" title="LANG1:aggr.1">DTConstrStr</a>(# <span class="p1">(<span class="default"><span class="p2"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon">F<sub>1</sub></font>(),<span class="p3"><a href="tarski.html#K1" title="TARSKI:func.1">{</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>()</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p3">(<span class="default"><a href="msafree.html#K4" title="MSAFREE:func.4">coprod</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="msafree.html#K5" title="MSAFREE:func.5">REL</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> #)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msafree.html#D10" title="MSAFREE:def.10">MSAFREE:def 10</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E15">A7</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c6">f</font> be    <a href="pboole.html#M2" title="PBOOLE:mode.2">ManySortedFunction</a> of  <a href="msafree.html#K9" title="MSAFREE:func.9">FreeSort</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>(); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( for <font color="Olive" title="b1">o</font> being  <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon">F<sub>1</sub></font>()<br/> for <font color="Olive" title="b2">ts</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Olive" title="b1">o</font>,<span class="p1">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span><br/> for <font color="Olive" title="b3">x</font> being   <a href="msafree1.html#M1" title="MSAFREE1:mode.1">FinSequence</a> of <font color="Maroon" title="c1">D</font>  st <font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msafree1.html#K1" title="MSAFREE1:func.1">Flatten</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Olive" title="b2">ts</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Olive" title="b1">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Olive" title="b1">o</font>,<span class="p3">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">ts</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b1">o</font>,<font color="Olive" title="b2">ts</font>,<font color="Olive" title="b3">x</font>) ) implies for <font color="Olive" title="b1">nt</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="msafree.html#K6" title="MSAFREE:func.6">DTConMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span><br/> for <font color="Olive" title="b2">ts</font> being   <a href="msafree1.html#M1" title="MSAFREE1:mode.1">FinSequence</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="msafree.html#K6" title="MSAFREE:func.6">DTConMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span>  st <font color="Olive" title="b1">nt</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="trees_3.html#K16" title="TREES_3:func.16">roots</a> <font color="Olive" title="b2">ts</font> holds <br/>for <font color="Olive" title="b3">x</font> being   <a href="msafree1.html#M1" title="MSAFREE1:mode.1">FinSequence</a> of <font color="Maroon" title="c1">D</font>  st <font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msafree1.html#K1" title="MSAFREE1:func.1">Flatten</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Olive" title="b2">ts</font> holds <br/><span class="p1">(<span class="default"><a href="msafree1.html#K1" title="MSAFREE1:func.1">Flatten</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">nt</font> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Olive" title="b2">ts</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">H</font> <a href="multop_1.html#K2" title="MULTOP_1:func.2">.</a> <font color="Olive" title="b1">nt</font>,<font color="Olive" title="b2">ts</font>,<font color="Olive" title="b3">x</font> )</span><br/><span class="kw">assume </span><a NAME="E1:19_1_2"/><span class="lab"><font color="Green" title="E16">A8</font></span>: 
for <font color="Olive" title="b1">o</font> being  <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon">F<sub>1</sub></font>()<br/> for <font color="Olive" title="b2">ts</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Olive" title="b1">o</font>,<span class="p1">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span><br/> for <font color="Olive" title="b3">x</font> being   <a href="msafree1.html#M1" title="MSAFREE1:mode.1">FinSequence</a> of <font color="Maroon" title="c1">D</font>  st <font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msafree1.html#K1" title="MSAFREE1:func.1">Flatten</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Olive" title="b2">ts</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Olive" title="b1">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Olive" title="b1">o</font>,<span class="p3">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">ts</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Olive" title="b1">o</font>,<font color="Olive" title="b2">ts</font>,<font color="Olive" title="b3">x</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">nt</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="msafree.html#K6" title="MSAFREE:func.6">DTConMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span><br/> for <font color="Olive" title="b2">ts</font> being   <a href="msafree1.html#M1" title="MSAFREE1:mode.1">FinSequence</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="msafree.html#K6" title="MSAFREE:func.6">DTConMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span>  st <font color="Olive" title="b1">nt</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="trees_3.html#K16" title="TREES_3:func.16">roots</a> <font color="Olive" title="b2">ts</font> holds <br/>for <font color="Olive" title="b3">x</font> being   <a href="msafree1.html#M1" title="MSAFREE1:mode.1">FinSequence</a> of <font color="Maroon" title="c1">D</font>  st <font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msafree1.html#K1" title="MSAFREE1:func.1">Flatten</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Olive" title="b2">ts</font> holds <br/><span class="p1">(<span class="default"><a href="msafree1.html#K1" title="MSAFREE1:func.1">Flatten</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">nt</font> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Olive" title="b2">ts</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">H</font> <a href="multop_1.html#K2" title="MULTOP_1:func.2">.</a> <font color="Olive" title="b1">nt</font>,<font color="Olive" title="b2">ts</font>,<font color="Olive" title="b3">x</font></span><br/><span class="kw">let </span><font color="Maroon" title="c7">nt</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="msafree.html#K6" title="MSAFREE:func.6">DTConMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">ts</font> being   <a href="msafree1.html#M1" title="MSAFREE1:mode.1">FinSequence</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="msafree.html#K6" title="MSAFREE:func.6">DTConMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span>  st <font color="Maroon" title="c7">nt</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="trees_3.html#K16" title="TREES_3:func.16">roots</a> <font color="Olive" title="b1">ts</font> holds <br/>for <font color="Olive" title="b2">x</font> being   <a href="msafree1.html#M1" title="MSAFREE1:mode.1">FinSequence</a> of <font color="Maroon" title="c1">D</font>  st <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msafree1.html#K1" title="MSAFREE1:func.1">Flatten</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Olive" title="b1">ts</font> holds <br/><span class="p1">(<span class="default"><a href="msafree1.html#K1" title="MSAFREE1:func.1">Flatten</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">nt</font> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Olive" title="b1">ts</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">H</font> <a href="multop_1.html#K2" title="MULTOP_1:func.2">.</a> <font color="Maroon" title="c7">nt</font>,<font color="Olive" title="b1">ts</font>,<font color="Olive" title="b2">x</font></span><br/><span class="kw">let </span><font color="Maroon" title="c8">ts</font> be    <a href="msafree1.html#M1" title="MSAFREE1:mode.1">FinSequence</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="msafree.html#K6" title="MSAFREE:func.6">DTConMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">nt</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="trees_3.html#K16" title="TREES_3:func.16">roots</a> <font color="Maroon" title="c8">ts</font> implies for <font color="Olive" title="b1">x</font> being   <a href="msafree1.html#M1" title="MSAFREE1:mode.1">FinSequence</a> of <font color="Maroon" title="c1">D</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msafree1.html#K1" title="MSAFREE1:func.1">Flatten</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c8">ts</font> holds <br/><span class="p1">(<span class="default"><a href="msafree1.html#K1" title="MSAFREE1:func.1">Flatten</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">nt</font> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c8">ts</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">H</font> <a href="multop_1.html#K2" title="MULTOP_1:func.2">.</a> <font color="Maroon" title="c7">nt</font>,<font color="Maroon" title="c8">ts</font>,<font color="Olive" title="b1">x</font> )</span><br/><span class="kw">assume </span><a NAME="E2:19_1_2"/><span class="lab"><font color="Green" title="E17">A9</font></span>: 
<font color="Maroon" title="c7">nt</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="trees_3.html#K16" title="TREES_3:func.16">roots</a> <font color="Maroon" title="c8">ts</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">x</font> being   <a href="msafree1.html#M1" title="MSAFREE1:mode.1">FinSequence</a> of <font color="Maroon" title="c1">D</font>  st <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msafree1.html#K1" title="MSAFREE1:func.1">Flatten</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c8">ts</font> holds <br/><span class="p1">(<span class="default"><a href="msafree1.html#K1" title="MSAFREE1:func.1">Flatten</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">nt</font> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c8">ts</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">H</font> <a href="multop_1.html#K2" title="MULTOP_1:func.2">.</a> <font color="Maroon" title="c7">nt</font>,<font color="Maroon" title="c8">ts</font>,<font color="Olive" title="b1">x</font></span><br/><a NAME="E3:19_1_2"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">nt</font>,<span class="p2">(<span class="default"><a href="trees_3.html#K16" title="TREES_3:func.16">roots</a> <font color="Maroon" title="c8">ts</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msafree.html#K5" title="MSAFREE:func.5">REL</a> <font color="Maroon">F<sub>2</sub></font>()
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:19_1"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="ref" href="lang1.html#D1" title="LANG1:def.1">LANG1:def 1</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">o</font> being   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon">F<sub>1</sub></font>(), <font color="Maroon" title="c10">x2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</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>()</span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span><span class="kw"> such that </span><br/><a NAME="E5:19_1_2"/><span class="lab"><font color="Green" title="E18">A10</font></span>: 
<font color="Maroon" title="c7">nt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">o</font>,<font color="Maroon" title="c10">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="msafree1.html#T2" target="_self" title="MSAFREE1:th.2">Th2</a>, <a class="ref" href="domain_1.html#T9" title="DOMAIN_1:th.9">DOMAIN_1:9</a></span>;<br/><span class="kw">let </span><font color="Maroon" title="c11">x</font> be    <a href="msafree1.html#M1" title="MSAFREE1:mode.1">FinSequence</a> of <font color="Maroon" title="c1">D</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msafree1.html#K1" title="MSAFREE1:func.1">Flatten</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c8">ts</font> implies <span class="p1">(<span class="default"><a href="msafree1.html#K1" title="MSAFREE1:func.1">Flatten</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">nt</font> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c8">ts</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">H</font> <a href="multop_1.html#K2" title="MULTOP_1:func.2">.</a> <font color="Maroon" title="c7">nt</font>,<font color="Maroon" title="c8">ts</font>,<font color="Maroon" title="c11">x</font> )</span><br/><span class="kw">assume </span><a NAME="E6:19_1_2"/><span class="lab"><font color="Green" title="E19">A11</font></span>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msafree1.html#K1" title="MSAFREE1:func.1">Flatten</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c8">ts</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="msafree1.html#K1" title="MSAFREE1:func.1">Flatten</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">nt</font> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c8">ts</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">H</font> <a href="multop_1.html#K2" title="MULTOP_1:func.2">.</a> <font color="Maroon" title="c7">nt</font>,<font color="Maroon" title="c8">ts</font>,<font color="Maroon" title="c11">x</font></span><br/><a NAME="E7:19_1_2"/><span class="lab"><font color="Green" title="E20">A12</font></span>: 
 <a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <font color="Maroon">F<sub>2</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#G3" title="MSUALG_1:aggr.3">MSAlgebra</a>(# <span class="p1">(<span class="default"><a href="msafree.html#K9" title="MSAFREE:func.9">FreeSort</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span>,<span class="p1">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeOper</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> #)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="msafree.html#D16" title="MSAFREE:def.16">MSAFREE:def 16</a></span>;<br/><a NAME="E8:19_1_2"/><span class="lab"><font color="Green" title="E21">A13</font></span>: 
<font color="Maroon" title="c10">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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 class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E9:19_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A14</font></span>: 
<font color="Maroon" title="c7">nt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msafree.html#K7" title="MSAFREE:func.7">Sym</a> <font color="Maroon" title="c9">o</font>,<font color="Maroon">F<sub>2</sub></font>()
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:19_1_2"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="ref" href="msafree.html#D11" title="MSAFREE:def.11">MSAFREE:def 11</a></span>;<br/><a NAME="E10:19_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A15</font></span>: 
<font color="Maroon" title="c8">ts</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msafree.html#K9" title="MSAFREE:func.9">FreeSort</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="pboole.html#K6" title="PBOOLE:func.6">#</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">Arity</a> of <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">o</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:19_1_2"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="ref" href="msafree.html#T10" title="MSAFREE:th.10">MSAFREE:10</a></span>;<br/><a NAME="E11:19_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree.html#K9" title="MSAFREE:func.9">FreeSort</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <font color="Maroon">F<sub>1</sub></font>()</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msafree.html#K9" title="MSAFREE:func.9">FreeSort</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c9">o</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></span>;<br/><a NAME="E12:19_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A16</font></span>: 
<span class="p1">(<span class="default"><a href="msafree.html#K10" title="MSAFREE:func.10">DenOp</a> <font color="Maroon" title="c9">o</font>,<font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">ts</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="msafree.html#K9" title="MSAFREE:func.9">FreeSort</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c9">o</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:19_1_2"><span class="lab"><font color="Green" title="E23">A15</font></span></a>, <a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></span>;<br/><span class="p1">(<span class="default"><a href="msafree1.html#K1" title="MSAFREE1:func.1">Flatten</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">o</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">F<sub>1</sub></font>()</span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c8">ts</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="msafree1.html#K1" title="MSAFREE1:func.1">Flatten</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree.html#K10" title="MSAFREE:func.10">DenOp</a> <font color="Maroon" title="c9">o</font>,<font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">ts</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:19_1_2"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="txt" href="#E5:19_1_2"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="txt" href="#E8:19_1_2"><span class="lab"><font color="Green" title="E21">A13</font></span></a>, <a class="txt" href="#E9:19_1_2"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="ref" href="msafree.html#D14" title="MSAFREE:def.14">MSAFREE:def 14</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c9">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree.html#K10" title="MSAFREE:func.10">DenOp</a> <font color="Maroon" title="c9">o</font>,<font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">ts</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:19_1_2"><span class="lab"><font color="Green" title="E24">A16</font></span></a>, <a class="ref" href="msafree1.html#D1" target="_self" title="MSAFREE1:def.1">Def1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c9">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c9">o</font>,<span class="p3">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">ts</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:19_1_2"><span class="lab"><font color="Green" title="E20">A12</font></span></a>, <a class="ref" href="msafree.html#D15" title="MSAFREE:def.15">MSAFREE:def 15</a></span>
<br/>.= 
<font color="Maroon">F<sub>5</sub></font>(<font color="Maroon" title="c9">o</font>,<font color="Maroon" title="c8">ts</font>,<font color="Maroon" title="c11">x</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19_1_2"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="txt" href="#E7:19_1_2"><span class="lab"><font color="Green" title="E20">A12</font></span></a>, <a class="txt" href="#E10:19_1_2"><span class="lab"><font color="Green" title="E23">A15</font></span></a>, <a class="txt" href="#E6:19_1_2"><span class="lab"><font color="Green" title="E19">A11</font></span></a></span>
<br/>.= 
<font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c7">nt</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span>,<font color="Maroon" title="c8">ts</font>,<font color="Maroon" title="c11">x</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:19_1_2"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>
;<br/><span class="kw">hence </span><a NAME="E14:19_1_2"/>
<span class="p1">(<span class="default"><a href="msafree1.html#K1" title="MSAFREE1:func.1">Flatten</a> <font color="Maroon" title="c6">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">nt</font> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c8">ts</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">H</font> <a href="multop_1.html#K2" title="MULTOP_1:func.2">.</a> <font color="Maroon" title="c7">nt</font>,<font color="Maroon" title="c8">ts</font>,<font color="Maroon" title="c11">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:19_1"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="txt" href="#E5:19_1_2"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="txt" href="#E8:19_1_2"><span class="lab"><font color="Green" title="E21">A13</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E9:19_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A17</font></span>: 
for <font color="Olive" title="b1">nt</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="msafree.html#K6" title="MSAFREE:func.6">DTConMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span><br/> for <font color="Olive" title="b2">ts</font> being   <a href="msafree1.html#M1" title="MSAFREE1:mode.1">FinSequence</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="msafree.html#K6" title="MSAFREE:func.6">DTConMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span>  st <font color="Olive" title="b1">nt</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="trees_3.html#K16" title="TREES_3:func.16">roots</a> <font color="Olive" title="b2">ts</font> holds <br/>for <font color="Olive" title="b3">x</font> being   <a href="msafree1.html#M1" title="MSAFREE1:mode.1">FinSequence</a> of <font color="Maroon" title="c1">D</font>  st <font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">f1</font> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Olive" title="b2">ts</font> holds <br/><font color="Maroon" title="c2">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">nt</font> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Olive" title="b2">ts</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">nt</font>,<font color="Olive" title="b2">ts</font>,<font color="Olive" title="b3">x</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19"><span class="lab"><font color="Green" title="E9">A1</font></span></a></span>;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>3</sub></font>(  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="msafree.html#K6" title="MSAFREE:func.6">DTConMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span>) <span class="kw">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon">F<sub>3</sub></font>() = <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> $1</span>)</span>);<br/>
<a NAME="E10:19_1"/><span class="lab"><font color="Green" title="E17">A18</font></span>: 
 <a href="dtconstr.html#K6" title="DTCONSTR:func.6">Terminals</a> <span class="p1">(<span class="default"><a href="msafree.html#K6" title="MSAFREE:func.6">DTConMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="msafree.html#K4" title="MSAFREE:func.4">coprod</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msafree.html#T6" title="MSAFREE:th.6">MSAFREE:6</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">G</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="msafree.html#K6" title="MSAFREE:func.6">DTConMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span>,<span class="p1">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <font color="Maroon">F<sub>3</sub></font>()</span>)</span><span class="kw"> such that </span><br/><a NAME="E12:19_1"/><span class="lab"><font color="Green" title="E18">A19</font></span>: 
for <font color="Olive" title="b1">t</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="msafree.html#K6" title="MSAFREE:func.6">DTConMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> holds  <font color="Maroon" title="c6">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b1">t</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S4" title="FUNCT_2:sch.4">FUNCT_2:sch 4</a>();<br/></span>
<span class="kw">reconsider </span><font color="Maroon" title="c7">G</font> = <font color="Maroon" title="c6">G</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="msafree.html#K6" title="MSAFREE:func.6">DTConMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span>,<font color="Maroon" title="c1">D</font> ;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>4</sub></font>(  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="msafree.html#K6" title="MSAFREE:func.6">DTConMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span>) <span class="kw">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">D</font> = <font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1;<br/>
<a NAME="E14:19_1"/><span class="lab"><font color="Green" title="E19">A20</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon">F<sub>2</sub></font>() <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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 class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E20">A21</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">f</font> be    <a href="pboole.html#M2" title="PBOOLE:mode.2">ManySortedFunction</a> of  <a href="msafree.html#K9" title="MSAFREE:func.9">FreeSort</a> <font color="Maroon">F<sub>2</sub></font>(),<font color="Maroon">F<sub>3</sub></font>(); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( for <font color="Olive" title="b1">s</font> being  <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon">F<sub>1</sub></font>()<br/> for <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msafree.html#K13" title="MSAFREE:func.13">FreeGen</a> <font color="Olive" title="b1">s</font>,<font color="Maroon">F<sub>2</sub></font>() holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Olive" title="b1">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b2">y</font>) ) implies for <font color="Olive" title="b1">t</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="msafree.html#K6" title="MSAFREE:func.6">DTConMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span>  st <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dtconstr.html#K6" title="DTCONSTR:func.6">Terminals</a> <span class="p1">(<span class="default"><a href="msafree.html#K6" title="MSAFREE:func.6">DTConMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> holds <br/><span class="p1">(<span class="default"><a href="msafree1.html#K1" title="MSAFREE1:func.1">Flatten</a> <font color="Maroon" title="c8">f</font></span>)</span> <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="Olive" title="b1">t</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">t</font> )</span><br/><span class="kw">assume </span><a NAME="E1:19_1_3"/><span class="lab"><font color="Green" title="E21">A22</font></span>: 
for <font color="Olive" title="b1">s</font> being  <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon">F<sub>1</sub></font>()<br/> for <font color="Olive" title="b2">y</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msafree.html#K13" title="MSAFREE:func.13">FreeGen</a> <font color="Olive" title="b1">s</font>,<font color="Maroon">F<sub>2</sub></font>() holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Olive" title="b1">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive" title="b2">y</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">t</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="msafree.html#K6" title="MSAFREE:func.6">DTConMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span>  st <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dtconstr.html#K6" title="DTCONSTR:func.6">Terminals</a> <span class="p1">(<span class="default"><a href="msafree.html#K6" title="MSAFREE:func.6">DTConMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> holds <br/><span class="p1">(<span class="default"><a href="msafree1.html#K1" title="MSAFREE1:func.1">Flatten</a> <font color="Maroon" title="c8">f</font></span>)</span> <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="Olive" title="b1">t</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">t</font></span><br/><span class="kw">let </span><font color="Maroon" title="c9">t</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="msafree.html#K6" title="MSAFREE:func.6">DTConMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dtconstr.html#K6" title="DTCONSTR:func.6">Terminals</a> <span class="p1">(<span class="default"><a href="msafree.html#K6" title="MSAFREE:func.6">DTConMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> implies <span class="p1">(<span class="default"><a href="msafree1.html#K1" title="MSAFREE1:func.1">Flatten</a> <font color="Maroon" title="c8">f</font></span>)</span> <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="c9">t</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">t</font> )</span><br/><span class="kw">assume </span><a NAME="E2:19_1_3"/><span class="lab"><font color="Green" title="E22">A23</font></span>: 
<font color="Maroon" title="c9">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dtconstr.html#K6" title="DTCONSTR:func.6">Terminals</a> <span class="p1">(<span class="default"><a href="msafree.html#K6" title="MSAFREE:func.6">DTConMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="msafree1.html#K1" title="MSAFREE1:func.1">Flatten</a> <font color="Maroon" title="c8">f</font></span>)</span> <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="c9">t</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">t</font></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">s</font> = <font color="Maroon" title="c9">t</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  as   <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon">F<sub>1</sub></font>() <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:19_1"><span class="lab"><font color="Green" title="E17">A18</font></span></a>, <a class="txt" href="#E14:19_1"><span class="lab"><font color="Green" title="E19">A20</font></span></a>, <a class="ref" href="card_3.html#T33" title="CARD_3:th.33">CARD_3:33</a></span>;<br/><a NAME="E4:19_1_3"/>
<font color="Maroon" title="c9">t</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:19_1"><span class="lab"><font color="Green" title="E17">A18</font></span></a>, <a class="txt" href="#E2:19_1_3"><span class="lab"><font color="Green" title="E22">A23</font></span></a>, <a class="ref" href="card_3.html#T33" title="CARD_3:th.33">CARD_3:33</a></span>;<br/><a NAME="E5:19_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A24</font></span>: 
 <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span>,<font color="Maroon" title="c10">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msafree.html#K13" title="MSAFREE:func.13">FreeGen</a> <font color="Maroon" title="c10">s</font>,<font color="Maroon">F<sub>2</sub></font>()
 <span class="kw">by</span> <span class="lab"><a class="ref" href="msafree.html#D17" title="MSAFREE:def.17">MSAFREE:def 17</a></span>;<br/><a NAME="E6:19_1_3"/><span class="lab"><font color="Green" title="E24">A25</font></span>: 
<font color="Maroon" title="c9">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:19_1"><span class="lab"><font color="Green" title="E17">A18</font></span></a>, <a class="txt" href="#E2:19_1_3"><span class="lab"><font color="Green" title="E22">A23</font></span></a>, <a class="ref" href="card_3.html#T33" title="CARD_3:th.33">CARD_3:33</a></span>;<br/><span class="kw">hence </span><span class="p1">(<span class="default"><a href="msafree1.html#K1" title="MSAFREE1:func.1">Flatten</a> <font color="Maroon" title="c8">f</font></span>)</span> <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="c9">t</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c10">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">t</font> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span>,<font color="Maroon" title="c10">s</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:19_1_3"><span class="lab"><font color="Green" title="E23">A24</font></span></a>, <a class="ref" href="msafree1.html#D1" target="_self" title="MSAFREE1:def.1">Def1</a></span>
<br/>.= 
<font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Maroon" title="c9">t</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19_1_3"><span class="lab"><font color="Green" title="E21">A22</font></span></a>, <a class="txt" href="#E6:19_1_3"><span class="lab"><font color="Green" title="E24">A25</font></span></a>, <a class="txt" href="#E5:19_1_3"><span class="lab"><font color="Green" title="E23">A24</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c7">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">t</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:19_1"><span class="lab"><font color="Green" title="E18">A19</font></span></a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E16:19_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A26</font></span>: 
for <font color="Olive" title="b1">t</font> being  <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <span class="p1">(<span class="default"><a href="msafree.html#K6" title="MSAFREE:func.6">DTConMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span>  st <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dtconstr.html#K6" title="DTCONSTR:func.6">Terminals</a> <span class="p1">(<span class="default"><a href="msafree.html#K6" title="MSAFREE:func.6">DTConMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> holds <br/><font color="Maroon" title="c3">f2</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="Olive" title="b1">t</font></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">t</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:19"><span class="lab"><font color="Green" title="E12">A4</font></span></a></span>;<br/>
<a NAME="E17:19_1"/><span class="lab"><font color="Green" title="E22">A27</font></span>: 
for <font color="Olive" title="b1">nt</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="msafree.html#K6" title="MSAFREE:func.6">DTConMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span><br/> for <font color="Olive" title="b2">ts</font> being   <a href="msafree1.html#M1" title="MSAFREE1:mode.1">FinSequence</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p1">(<span class="default"><a href="msafree.html#K6" title="MSAFREE:func.6">DTConMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span>  st <font color="Olive" title="b1">nt</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="trees_3.html#K16" title="TREES_3:func.16">roots</a> <font color="Olive" title="b2">ts</font> holds <br/>for <font color="Olive" title="b3">x</font> being   <a href="msafree1.html#M1" title="MSAFREE1:mode.1">FinSequence</a> of <font color="Maroon" title="c1">D</font>  st <font color="Olive" title="b3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">f2</font> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Olive" title="b2">ts</font> holds <br/><font color="Maroon" title="c3">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">nt</font> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Olive" title="b2">ts</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">nt</font>,<font color="Olive" title="b2">ts</font>,<font color="Olive" title="b3">x</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:19"><span class="lab"><font color="Green" title="E11">A3</font></span></a>, <a class="txt" href="#E8:19_1"><span class="lab"><font color="Green" title="E15">A7</font></span></a></span>;<br/>
<a NAME="E18:19_1"/><span class="lab"><font color="Green" title="E23">A28</font></span>: 
for <font color="Olive" title="b1">t</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="msafree.html#K6" title="MSAFREE:func.6">DTConMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span>  st <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dtconstr.html#K6" title="DTCONSTR:func.6">Terminals</a> <span class="p1">(<span class="default"><a href="msafree.html#K6" title="MSAFREE:func.6">DTConMSA</a> <font color="Maroon">F<sub>2</sub></font>()</span>)</span> holds <br/><font color="Maroon" title="c2">f1</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="Olive" title="b1">t</font></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">t</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:19"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="txt" href="#E15:19_1"><span class="lab"><font color="Green" title="E20">A21</font></span></a></span>;<br/>
<a NAME="E19:19_1"/>
<font color="Maroon" title="c2">f1</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">f2</font>
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="msafree1.html#S1" target="_self" title="MSAFREE1:sch.1">MSAFREE1:sch 1</a>(<a class="txt" href="#E18:19_1"><span class="lab"><font color="Green" title="E23">A28</font></span></a>, <a class="txt" href="#E9:19_1"><span class="lab"><font color="Green" title="E16">A17</font></span></a>, <a class="txt" href="#E16:19_1"><span class="lab"><font color="Green" title="E21">A26</font></span></a>, <a class="txt" href="#E17:19_1"><span class="lab"><font color="Green" title="E22">A27</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E20:19_1"/>
<font color="Maroon">F<sub>6</sub></font>() <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon">F<sub>7</sub></font>()
 <span class="kw">by</span> <span class="lab"><a class="ref" href="msafree1.html#T4" target="_self" title="MSAFREE1:th.4">Th4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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