<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">S</font> be  non  <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a>  non  <a href="struct_0.html#V11" title="STRUCT_0:attr.11">void</a>   <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a> ; <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="relat_1.html#V2" title="RELAT_1:attr.2">V5</a>() <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> holds   <a href="msafree.html#K14" title="MSAFREE:func.14">FreeGen</a> <font color="Olive" title="b1">X</font> is  <a href="msafree.html#V1" title="MSAFREE:attr.1">free</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">X</font> be  <a href="relat_1.html#V2" title="RELAT_1:attr.2">V5</a>() <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="msafree.html#K14" title="MSAFREE:func.14">FreeGen</a> <font color="Maroon" title="c2">X</font> is  <a href="msafree.html#V1" title="MSAFREE:attr.1">free</a> </span><br/>



<span class="kw">set </span><font color="Maroon" title="c3">D</font> =  <a href="msafree.html#K6" title="MSAFREE:func.6">DTConMSA</a> <font color="Maroon" title="c2">X</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">FA</font> =  <a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <font color="Maroon" title="c2">X</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">FG</font> =  <a href="msafree.html#K14" title="MSAFREE:func.14">FreeGen</a> <font color="Maroon" title="c2">X</font>;<br/>
<span class="kw">let </span><font color="Maroon" title="c6">U1</font> be   <a href="msualg_1.html#V4" title="MSUALG_1:attr.4">non-empty</a>   <a href="msualg_1.html#L3" title="MSUALG_1:struct.3">MSAlgebra</a> of <font color="Maroon" title="c1">S</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="msafree.html#D5" target="_self" title="MSAFREE:def.5">MSAFREE:def 5</a> <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">f</font> being   <a href="pboole.html#M2" title="PBOOLE:mode.2">ManySortedFunction</a> of  <a href="msafree.html#K14" title="MSAFREE:func.14">FreeGen</a> <font color="Maroon" title="c2">X</font>,the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c6">U1</font>  ex <font color="Olive" title="b2">h</font> being  <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <span class="p1">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span>,<font color="Maroon" title="c6">U1</font> st <br/>( <font color="Olive" title="b2">h</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a>  <a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c6">U1</font> &amp; <font color="Olive" title="b2">h</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <span class="p1">(<span class="default"><a href="msafree.html#K14" title="MSAFREE:func.14">FreeGen</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Olive" title="b1">f</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c7">F</font> be    <a href="pboole.html#M2" title="PBOOLE:mode.2">ManySortedFunction</a> of  <a href="msafree.html#K14" title="MSAFREE:func.14">FreeGen</a> <font color="Maroon" title="c2">X</font>,the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c6">U1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">h</font> being  <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <span class="p1">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span>,<font color="Maroon" title="c6">U1</font> st <br/>( <font color="Olive" title="b1">h</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a>  <a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c6">U1</font> &amp; <font color="Olive" title="b1">h</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <span class="p1">(<span class="default"><a href="msafree.html#K14" title="MSAFREE:func.14">FreeGen</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c7">F</font> )</span><br/>



<span class="kw">set </span><font color="Maroon" title="c8">SA</font> = the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">AR</font> = the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">Arity</a> of <font color="Maroon" title="c1">S</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">S1</font> = the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c6">U1</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">O</font> = the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">S</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c12">RS</font> = the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <font color="Maroon" title="c1">S</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c13">DU</font> =  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c6">U1</font>;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(  <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" title="c2">X</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> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c6">U1</font> =  <a href="msafree.html#K17" title="MSAFREE:func.17">pi</a> <font color="Maroon" title="c7">F</font>,the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c6">U1</font>,$1;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>2</sub></font>(  <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" title="c2">X</font></span>)</span>,  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>,  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</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> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c6">U1</font> =  <a href="msafree.html#K19" title="MSAFREE:func.19">pi</a> <span class="p1">(<span class="default"><a href="msafree.html#K18" title="MSAFREE:func.18">@</a> <font color="Maroon" title="c2">X</font>,$1</span>)</span>,<font color="Maroon" title="c6">U1</font>,$3;<br/>
<span class="kw">consider </span><font color="Maroon" title="c14">G</font> being   <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" title="c2">X</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c6">U1</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E2:50"/><span class="lab"><font color="Green" title="E30">A1</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" title="c2">X</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" title="c2">X</font></span>)</span> holds <br/><font color="Maroon" title="c14">G</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>1</sub></font>(<font color="Olive" title="b1">t</font>)
 <span class="kw">and </span><br/><a NAME="E3:50"/><span class="lab"><font color="Green" title="E31">A2</font></span>: 
for <font color="Olive" title="b1">nt</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" title="c2">X</font></span>)</span><br/> for <font color="Olive" title="b2">ts</font> being   <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> <span class="p1">(<span class="default"><a href="msafree.html#K6" title="MSAFREE:func.6">DTConMSA</a> <font color="Maroon" title="c2">X</font></span>)</span>  st <font color="Olive" title="b1">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="Olive" title="b2">ts</font> holds <br/><font color="Maroon" title="c14">G</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#K12" title="TREES_4:func.12">-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>, <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Olive" title="b2">ts</font>,<font color="Maroon" title="c14">G</font> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Olive" title="b2">ts</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="dtconstr.html#S8" title="DTCONSTR:sch.8">DTCONSTR:sch 8</a>();<br/></span>
<span class="kw">deffunc </span><font color="Maroon">H<sub>3</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <span class="kw">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <span class="p3">(<span class="default"><a href="msafree.html#K6" title="MSAFREE:func.6">DTConMSA</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c6">U1</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> = <font color="Maroon" title="c14">G</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p2">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c15">h</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E5:50"/><span class="lab"><font color="Green" title="E32">A3</font></span>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c15">h</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" title="c1">S</font> &amp; ( for <font color="Olive" title="b1">s</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> holds <br/><font color="Maroon" title="c15">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b1">s</font>) ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_1.html#S3" title="FUNCT_1:sch.3">FUNCT_1:sch 3</a>();<br/></span>
<span class="kw">reconsider </span><font color="Maroon" title="c16">h</font> = <font color="Maroon" title="c15">h</font> as   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:50"><span class="lab"><font color="Green" title="E32">A3</font></span></a>, <a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a>, <a class="ref" href="relat_1.html#D18" title="RELAT_1:def.18">RELAT_1:def 18</a></span>;<br/>
<a NAME="E7:50"/>
for <font color="Olive" title="b1">s</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c16">h</font> holds <br/><font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s</font> is   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c17">s</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <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="c17">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c16">h</font> implies <font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">s</font> is   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> )</span><br/>

<span class="kw">assume </span><a NAME="E1:50_1"/>
<font color="Maroon" title="c17">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c16">h</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="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">s</font> is   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a></span><br/>

<a NAME="E2:50_1"/><span class="kw">then </span>
<font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">G</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p2">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">s</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:50"><span class="lab"><font color="Green" title="E32">A3</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:50_1"/>
<font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">s</font> is   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>
 ; <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>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c17">h</font> = <font color="Maroon" title="c16">h</font> as   <a href="pboole.html#NM2" title="PBOOLE:NM.2">ManySortedFunction</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="funcop_1.html#D6" title="FUNCOP_1:def.6">FUNCOP_1:def 6</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </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" title="c1">S</font>  st $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c17">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c6">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s</font>;<br/>
<a NAME="E9:50"/><span class="lab"><font color="Green" title="E33">A4</font></span>: 
for <font color="Olive" title="b1">nt</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" title="c2">X</font></span>)</span><br/> for <font color="Olive" title="b2">ts</font> being   <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> <span class="p1">(<span class="default"><a href="msafree.html#K6" title="MSAFREE:func.6">DTConMSA</a> <font color="Maroon" title="c2">X</font></span>)</span>  st <font color="Olive" title="b1">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="Olive" title="b2">ts</font> &amp; ( for <font color="Olive" title="b3">t</font> being  <a href="trees_2.html#NM5" title="TREES_2:NM.5">DecoratedTree</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" title="c2">X</font></span>)</span>  st <font color="Olive" title="b3">t</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="Olive" title="b2">ts</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b3">t</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Olive" title="b2">ts</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c18">nt</font> be   <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" title="c2">X</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="trees_4.html#M1" title="TREES_4: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" title="c2">X</font></span>)</span>  st <font color="Maroon" title="c18">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="Olive" title="b1">ts</font> &amp; ( for <font color="Olive" title="b2">t</font> being  <a href="trees_2.html#NM5" title="TREES_2:NM.5">DecoratedTree</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" title="c2">X</font></span>)</span>  st <font color="Olive" title="b2">t</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="Olive" title="b1">ts</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">t</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c18">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Olive" title="b1">ts</font>]</span><br/><span class="kw">let </span><font color="Maroon" title="c19">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> <span class="p1">(<span class="default"><a href="msafree.html#K6" title="MSAFREE:func.6">DTConMSA</a> <font color="Maroon" title="c2">X</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="c18">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="c19">ts</font> &amp; ( for <font color="Olive" title="b1">t</font> being  <a href="trees_2.html#NM5" title="TREES_2:NM.5">DecoratedTree</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" title="c2">X</font></span>)</span>  st <font color="Olive" title="b1">t</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="c19">ts</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">t</font>] ) implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c18">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c19">ts</font>] )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:50_2"/><span class="lab"><font color="Green" title="E34">A5</font></span>: 
<font color="Maroon" title="c18">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="c19">ts</font>
 <span class="kw">and </span><br/><a NAME="E2:50_2"/><span class="lab"><font color="Green" title="E35">A6</font></span>: 
for <font color="Olive" title="b1">t</font> being  <a href="trees_2.html#NM5" title="TREES_2:NM.5">DecoratedTree</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" title="c2">X</font></span>)</span>  st <font color="Olive" title="b1">t</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="c19">ts</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">t</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">S<sub>1</sub></font>[<font color="Maroon" title="c18">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c19">ts</font>]</span><br/>

<span class="kw">set </span><font color="Maroon" title="c20">p</font> = <font color="Maroon" title="c14">G</font> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c19">ts</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c21">o</font> =  <a href="msafree.html#K18" title="MSAFREE:func.18">@</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c18">nt</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c22">ar</font> =  <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <span class="p1">(<span class="default"><a href="msafree.html#K18" title="MSAFREE:func.18">@</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c18">nt</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c23">rs</font> =  <a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <span class="p1">(<span class="default"><a href="msafree.html#K18" title="MSAFREE:func.18">@</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c18">nt</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c24">OU</font> = <span class="p1"><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" title="c1">S</font>,<span class="p2"><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" title="c1">S</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="p1">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p2">(<span class="default"><a href="msafree.html#K4" title="MSAFREE:func.4">coprod</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c25">rt</font> =  <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c19">ts</font>;<br/>
<a NAME="E3:50_2"/><span class="lab"><font color="Green" title="E36">A7</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="msafree.html#K18" title="MSAFREE:func.18">@</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c18">nt</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">nt</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50_2"><span class="lab"><font color="Green" title="E34">A5</font></span></a>, <a class="ref" href="msafree.html#D22" target="_self" title="MSAFREE:def.22">Def22</a></span>;<br/>
<a NAME="E4:50_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A8</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p3">(<span class="default"><a href="msafree.html#K18" title="MSAFREE:func.18">@</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c18">nt</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c19">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> the <a href="lang1.html#U1" title="LANG1:sel.1">Rules</a> of <span class="p1">(<span class="default"><a href="msafree.html#K6" title="MSAFREE:func.6">DTConMSA</a> <font color="Maroon" title="c2">X</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50_2"><span class="lab"><font color="Green" title="E34">A5</font></span></a>, <a class="ref" href="lang1.html#D1" title="LANG1:def.1">LANG1:def 1</a></span>;<br/>
<a NAME="E5:50_2"/><span class="lab"><font color="Green" title="E38">A9</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="msafree.html#K18" title="MSAFREE:func.18">@</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c18">nt</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msafree.html#K7" title="MSAFREE:func.7">Sym</a> <span class="p1">(<span class="default"><a href="msafree.html#K18" title="MSAFREE:func.18">@</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c18">nt</font></span>)</span>,<font color="Maroon" title="c2">X</font>
 
;<br/>
<a NAME="E6:50_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A10</font></span>: 
<span class="p1">(<span class="default"><a href="msafree.html#K10" title="MSAFREE:func.10">DenOp</a> <span class="p2">(<span class="default"><a href="msafree.html#K18" title="MSAFREE:func.18">@</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c18">nt</font></span>)</span>,<font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">ts</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c19">ts</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50_2"><span class="lab"><font color="Green" title="E34">A5</font></span></a>, <a class="txt" href="#E3:50_2"><span class="lab"><font color="Green" title="E36">A7</font></span></a>, <a class="ref" href="msafree.html#D14" target="_self" title="MSAFREE:def.14">Def14</a></span>;<br/>
<a NAME="E7:50_2"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="msafree.html#K10" title="MSAFREE:func.10">DenOp</a> <span class="p2">(<span class="default"><a href="msafree.html#K18" title="MSAFREE:func.18">@</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c18">nt</font></span>)</span>,<font color="Maroon" title="c2">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="c2">X</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" title="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="msafree.html#K18" title="MSAFREE:func.18">@</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c18">nt</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E8:50_2"/><span class="kw">then </span>
<font color="Maroon" title="c19">ts</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="msafree.html#K10" title="MSAFREE:func.10">DenOp</a> <span class="p2">(<span class="default"><a href="msafree.html#K18" title="MSAFREE:func.18">@</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c18">nt</font></span>)</span>,<font color="Maroon" title="c2">X</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50_2"><span class="lab"><font color="Green" title="E34">A5</font></span></a>, <a class="txt" href="#E3:50_2"><span class="lab"><font color="Green" title="E36">A7</font></span></a>, <a class="txt" href="#E5:50_2"><span class="lab"><font color="Green" title="E38">A9</font></span></a>, <a class="ref" href="msafree.html#T10" target="_self" title="MSAFREE:th.10">Th10</a></span>;<br/>
<a NAME="E9:50_2"/><span class="kw">then </span>
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="msafree.html#K10" title="MSAFREE:func.10">DenOp</a> <span class="p2">(<span class="default"><a href="msafree.html#K18" title="MSAFREE:func.18">@</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c18">nt</font></span>)</span>,<font color="Maroon" title="c2">X</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <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" title="c2">X</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" title="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="msafree.html#K18" title="MSAFREE:func.18">@</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c18">nt</font></span>)</span> &amp; <font color="Maroon" title="c18">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c19">ts</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> <span class="p1">(<span class="default"><a href="msafree.html#K10" title="MSAFREE:func.10">DenOp</a> <span class="p2">(<span class="default"><a href="msafree.html#K18" title="MSAFREE:func.18">@</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c18">nt</font></span>)</span>,<font color="Maroon" title="c2">X</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:50_2"><span class="lab"><font color="Green" title="E39">A10</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a>, <a class="ref" href="relat_1.html#D19" title="RELAT_1:def.19">RELAT_1:def 19</a></span>;<br/>
<a NAME="E10:50_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A11</font></span>: 
<font color="Maroon" title="c18">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c19">ts</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p2">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <font color="Maroon" title="c2">X</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" title="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="msafree.html#K18" title="MSAFREE:func.18">@</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c18">nt</font></span>)</span>
 
;<br/>
<span class="kw">set </span><font color="Maroon" title="c26">ppi</font> =  <a href="msafree.html#K19" title="MSAFREE:func.19">pi</a> <span class="p1">(<span class="default"><a href="msafree.html#K18" title="MSAFREE:func.18">@</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c18">nt</font></span>)</span>,<font color="Maroon" title="c6">U1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">G</font> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c19">ts</font></span>)</span>;<br/>
<a NAME="E11:50_2"/><span class="lab"><font color="Green" title="E41">A12</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <font color="Maroon" title="c1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#D19" title="RELAT_1:def.19">RELAT_1:def 19</a></span>;<br/>
<a NAME="E12:50_2"/><span class="lab"><font color="Green" title="E42">A13</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <span class="p2">(<span class="default"><a href="msafree.html#K18" title="MSAFREE:func.18">@</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c18">nt</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c27">rt</font> =  <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c19">ts</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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" title="c1">S</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" title="c1">S</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" title="c2">X</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K13" title="FINSEQ_1:func.13">*</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:50_2"><span class="lab"><font color="Green" title="E37">A8</font></span></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<a NAME="E14:50_2"/><span class="lab"><font color="Green" title="E43">A14</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c27">rt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <span class="p2">(<span class="default"><a href="msafree.html#K18" title="MSAFREE:func.18">@</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c18">nt</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:50_2"><span class="lab"><font color="Green" title="E37">A8</font></span></a>, <a class="ref" href="msafree.html#T5" target="_self" title="MSAFREE:th.5">Th5</a></span>;<br/>
<a NAME="E15:50_2"/><span class="lab"><font color="Green" title="E44">A15</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c27">rt</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="c27">rt</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E16:50_2"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <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" title="c1">S</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/>
<a NAME="E17:50_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A16</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p2">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <span class="p3">(<span class="default"><a href="msafree.html#K18" title="MSAFREE:func.18">@</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c18">nt</font></span>)</span></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="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <span class="p2">(<span class="default"><a href="msafree.html#K18" title="MSAFREE:func.18">@</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c18">nt</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:50_2"><span class="lab"><font color="Green" title="E42">A13</font></span></a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></span>;<br/>
<a NAME="E18:50_2"/><span class="lab"><font color="Green" title="E46">A17</font></span>: 
 <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <span class="p1">(<span class="default"><a href="msafree.html#K18" title="MSAFREE:func.18">@</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c18">nt</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">Arity</a> of <font color="Maroon" title="c1">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="msafree.html#K18" title="MSAFREE:func.18">@</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c18">nt</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_1.html#D6" title="MSUALG_1:def.6">MSUALG_1:def 6</a></span>;<br/>
<a NAME="E19:50_2"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c6">U1</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" title="c1">S</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/>
<a NAME="E20:50_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A18</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c6">U1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <span class="p3">(<span class="default"><a href="msafree.html#K18" title="MSAFREE:func.18">@</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c18">nt</font></span>)</span></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="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <span class="p2">(<span class="default"><a href="msafree.html#K18" title="MSAFREE:func.18">@</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c18">nt</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:50_2"><span class="lab"><font color="Green" title="E42">A13</font></span></a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></span>;<br/>
<a NAME="E21:50_2"/><span class="lab"><font color="Green" title="E48">A19</font></span>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c27">rt</font> <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="c19">ts</font> &amp;  <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> <span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <span class="p3">(<span class="default"><a href="msafree.html#K18" title="MSAFREE:func.18">@</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c18">nt</font></span>)</span></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="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <span class="p2">(<span class="default"><a href="msafree.html#K18" title="MSAFREE:func.18">@</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c18">nt</font></span>)</span></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a>, <a class="ref" href="trees_3.html#D18" title="TREES_3:def.18">TREES_3:def 18</a></span>;<br/>
<a NAME="E22:50_2"/><span class="lab"><font color="Green" title="E49">A20</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">G</font> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c19">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="c19">ts</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="alg_1.html#T1" title="ALG_1:th.1">ALG_1:1</a></span>;<br/>
<a NAME="E23:50_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A21</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">G</font> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c19">ts</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c6">U1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <span class="p3">(<span class="default"><a href="msafree.html#K18" title="MSAFREE:func.18">@</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c18">nt</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:50_2"><span class="lab"><font color="Green" title="E47">A18</font></span></a>, <a class="txt" href="#E4:50_2"><span class="lab"><font color="Green" title="E37">A8</font></span></a>, <a class="txt" href="#E15:50_2"><span class="lab"><font color="Green" title="E44">A15</font></span></a>, <a class="txt" href="#E21:50_2"><span class="lab"><font color="Green" title="E48">A19</font></span></a>, <a class="ref" href="msafree.html#T5" target="_self" title="MSAFREE:th.5">Th5</a></span>;<br/>
<a NAME="E24:50_2"/><span class="lab"><font color="Green" title="E51">A22</font></span>: 
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c6">U1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <span class="p3">(<span class="default"><a href="msafree.html#K18" title="MSAFREE:func.18">@</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c18">nt</font></span>)</span></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c14">G</font> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c19">ts</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c6">U1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <span class="p3">(<span class="default"><a href="msafree.html#K18" title="MSAFREE:func.18">@</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c18">nt</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c28">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <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="c28">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c6">U1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <span class="p3">(<span class="default"><a href="msafree.html#K18" title="MSAFREE:func.18">@</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c18">nt</font></span>)</span></span>)</span></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c14">G</font> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c19">ts</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c6">U1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <span class="p3">(<span class="default"><a href="msafree.html#K18" title="MSAFREE:func.18">@</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c18">nt</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:50_2_1"/><span class="lab"><font color="Green" title="E52">A23</font></span>: 
<font color="Maroon" title="c28">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c6">U1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <span class="p3">(<span class="default"><a href="msafree.html#K18" title="MSAFREE:func.18">@</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c18">nt</font></span>)</span></span>)</span></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"><font color="Maroon" title="c14">G</font> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c19">ts</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c6">U1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <span class="p3">(<span class="default"><a href="msafree.html#K18" title="MSAFREE:func.18">@</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c18">nt</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">x</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c29">n</font> = <font color="Maroon" title="c28">x</font> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:50_2"><span class="lab"><font color="Green" title="E47">A18</font></span></a></span>;<br/>
<a NAME="E3:50_2_1"/><span class="lab"><font color="Green" title="E53">A24</font></span>: 
<font color="Maroon" title="c19">ts</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">n</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="c19">ts</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:50_2"><span class="lab"><font color="Green" title="E47">A18</font></span></a>, <a class="txt" href="#E14:50_2"><span class="lab"><font color="Green" title="E43">A14</font></span></a>, <a class="txt" href="#E15:50_2"><span class="lab"><font color="Green" title="E44">A15</font></span></a>, <a class="txt" href="#E21:50_2"><span class="lab"><font color="Green" title="E48">A19</font></span></a>, <a class="txt" href="#E1:50_2_1"><span class="lab"><font color="Green" title="E52">A23</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<a NAME="E4:50_2_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c19">ts</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <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" title="c2">X</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c30">t</font> = <font color="Maroon" title="c19">ts</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">n</font> as    <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</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" title="c2">X</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:50_2_1"><span class="lab"><font color="Green" title="E53">A24</font></span></a></span>;<br/>
<a NAME="E6:50_2_1"/><span class="lab"><font color="Green" title="E54">A25</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c14">G</font> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c19">ts</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">ts</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">n</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:50_2"><span class="lab"><font color="Green" title="E50">A21</font></span></a>, <a class="txt" href="#E1:50_2_1"><span class="lab"><font color="Green" title="E52">A23</font></span></a>, <a class="ref" href="alg_1.html#T1" title="ALG_1:th.1">ALG_1:1</a></span>;<br/>
<a NAME="E7:50_2_1"/>
<span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <span class="p2">(<span class="default"><a href="msafree.html#K18" title="MSAFREE:func.18">@</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c18">nt</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">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> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <span class="p2">(<span class="default"><a href="msafree.html#K18" title="MSAFREE:func.18">@</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c18">nt</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:50_2"><span class="lab"><font color="Green" title="E47">A18</font></span></a>, <a class="txt" href="#E1:50_2_1"><span class="lab"><font color="Green" title="E52">A23</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c31">s</font> = <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <span class="p2">(<span class="default"><a href="msafree.html#K18" title="MSAFREE:func.18">@</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c18">nt</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">x</font> as   <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c1">S</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:50_2"><span class="lab"><font color="Green" title="E42">A13</font></span></a></span>;<br/>
<a NAME="E9:50_2_1"/><span class="lab"><font color="Green" title="E55">A26</font></span>: 
<font color="Maroon" title="c17">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">G</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p2">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">s</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:50"><span class="lab"><font color="Green" title="E32">A3</font></span></a></span>;<br/>
<a NAME="E10:50_2_1"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <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" title="c1">S</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/>
<a NAME="E11:50_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A27</font></span>: 
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">s</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> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c14">G</font> = 
 <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" title="c2">X</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>
<br/>.= 
 <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> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p2">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="msafree.html#T12" target="_self" title="MSAFREE:th.12">Th12</a></span>
;<br/>
<a NAME="E13:50_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A28</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:50_2_1"><span class="lab"><font color="Green" title="E55">A26</font></span></a>, <a class="txt" href="#E11:50_2_1"><span class="lab"><font color="Green" title="E56">A27</font></span></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></span>;<br/>
<a NAME="E14:50_2_1"/>
<font color="Maroon" title="c19">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" title="c2">X</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" title="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="msafree.html#K18" title="MSAFREE:func.18">@</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c18">nt</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50_2"><span class="lab"><font color="Green" title="E34">A5</font></span></a>, <a class="txt" href="#E3:50_2"><span class="lab"><font color="Green" title="E36">A7</font></span></a>, <a class="txt" href="#E5:50_2"><span class="lab"><font color="Green" title="E38">A9</font></span></a>, <a class="ref" href="msafree.html#T10" target="_self" title="MSAFREE:th.10">Th10</a></span>;<br/>
<a NAME="E15:50_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c19">ts</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <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" title="c2">X</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <span class="p3">(<span class="default"><a href="msafree.html#K18" title="MSAFREE:func.18">@</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c18">nt</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:50_2"><span class="lab"><font color="Green" title="E46">A17</font></span></a>, <a class="ref" href="msafree.html#T1" target="_self" title="MSAFREE:th.1">Th1</a></span>;<br/>
<a NAME="E16:50_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c19">ts</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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" title="c2">X</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <span class="p3">(<span class="default"><a href="msafree.html#K18" title="MSAFREE:func.18">@</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c18">nt</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:50_2"><span class="lab"><font color="Green" title="E47">A18</font></span></a>, <a class="txt" href="#E17:50_2"><span class="lab"><font color="Green" title="E45">A16</font></span></a>, <a class="txt" href="#E1:50_2_1"><span class="lab"><font color="Green" title="E52">A23</font></span></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></span>;<br/>
<a NAME="E17:50_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A29</font></span>: 
<font color="Maroon" title="c19">ts</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">x</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" title="c2">X</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="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <span class="p3">(<span class="default"><a href="msafree.html#K18" title="MSAFREE:func.18">@</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c18">nt</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">x</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:50_2"><span class="lab"><font color="Green" title="E47">A18</font></span></a>, <a class="txt" href="#E17:50_2"><span class="lab"><font color="Green" title="E45">A16</font></span></a>, <a class="txt" href="#E1:50_2_1"><span class="lab"><font color="Green" title="E52">A23</font></span></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></span>;<br/>
<a NAME="E18:50_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c17">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c30">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c6">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:50_2"><span class="lab"><font color="Green" title="E35">A6</font></span></a>, <a class="txt" href="#E3:50_2_1"><span class="lab"><font color="Green" title="E53">A24</font></span></a></span>;<br/>
<a NAME="E19:50_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">G</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c30">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c6">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c31">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:50_2_1"><span class="lab"><font color="Green" title="E58">A29</font></span></a>, <a class="txt" href="#E9:50_2_1"><span class="lab"><font color="Green" title="E55">A26</font></span></a>, <a class="txt" href="#E13:50_2_1"><span class="lab"><font color="Green" title="E57">A28</font></span></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></span>;<br/>
<span class="kw">hence </span><a NAME="E20:50_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c14">G</font> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c19">ts</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c6">U1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <span class="p3">(<span class="default"><a href="msafree.html#K18" title="MSAFREE:func.18">@</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c18">nt</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50_2_1"><span class="lab"><font color="Green" title="E52">A23</font></span></a>, <a class="txt" href="#E6:50_2_1"><span class="lab"><font color="Green" title="E54">A25</font></span></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</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="E25:50_2"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c6">U1</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" title="c1">S</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/>
<a NAME="E26:50_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A30</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c6">U1</font> <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" title="c1">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <font color="Maroon" title="c1">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:50_2"><span class="lab"><font color="Green" title="E41">A12</font></span></a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c28">s</font> be   <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c1">S</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="c18">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c19">ts</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">s</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c17">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">s</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="c18">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c19">ts</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c6">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">s</font> )</span><br/>

<a NAME="E27:50_2"/><span class="lab"><font color="Green" title="E53">A31</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <span class="p2">(<span class="default"><a href="msafree.html#K18" title="MSAFREE:func.18">@</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c18">nt</font></span>)</span>,<font color="Maroon" title="c6">U1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <span class="p1">(<span class="default"><a href="msafree.html#K18" title="MSAFREE:func.18">@</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c18">nt</font></span>)</span>,<font color="Maroon" title="c6">U1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E28:50_2"/><span class="lab"><font color="Green" title="E54">A32</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <font color="Maroon" title="c1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E29:50_2"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <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" title="c1">S</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/>
<a NAME="E30:50_2"/><span class="kw">then </span>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p2">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <font color="Maroon" title="c2">X</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" title="c1">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <font color="Maroon" title="c1">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:50_2"><span class="lab"><font color="Green" title="E41">A12</font></span></a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></span>;<br/>
<a NAME="E31:50_2"/><span class="kw">then </span>
<font color="Maroon" title="c18">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c19">ts</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <font color="Maroon" title="c1">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="msafree.html#K18" title="MSAFREE:func.18">@</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c18">nt</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:50_2"><span class="lab"><font color="Green" title="E54">A32</font></span></a>, <a class="txt" href="#E10:50_2"><span class="lab"><font color="Green" title="E40">A11</font></span></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></span>;<br/>
<a NAME="E32:50_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A33</font></span>: 
<font color="Maroon" title="c18">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c19">ts</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <font color="Maroon" title="c2">X</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> <span class="p2">(<span class="default"><a href="msafree.html#K18" title="MSAFREE:func.18">@</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c18">nt</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_1.html#D7" title="MSUALG_1:def.7">MSUALG_1:def 7</a></span>;<br/>
 <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <span class="p1">(<span class="default"><a href="msafree.html#K18" title="MSAFREE:func.18">@</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c18">nt</font></span>)</span>,<font color="Maroon" title="c6">U1</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c6">U1</font> <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" title="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="msafree.html#K18" title="MSAFREE:func.18">@</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c18">nt</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_1.html#D9" title="MSUALG_1:def.9">MSUALG_1:def 9</a></span>
<br/>.= 
 <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c6">U1</font> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <span class="p3">(<span class="default"><a href="msafree.html#K18" title="MSAFREE:func.18">@</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c18">nt</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:50_2"><span class="lab"><font color="Green" title="E46">A17</font></span></a>, <a class="ref" href="msafree.html#T1" target="_self" title="MSAFREE:th.1">Th1</a></span>
;<br/>
<a NAME="E34:50_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A34</font></span>: 
<font color="Maroon" title="c14">G</font> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c19">ts</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <span class="p1">(<span class="default"><a href="msafree.html#K18" title="MSAFREE:func.18">@</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c18">nt</font></span>)</span>,<font color="Maroon" title="c6">U1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:50_2"><span class="lab"><font color="Green" title="E49">A20</font></span></a>, <a class="txt" href="#E20:50_2"><span class="lab"><font color="Green" title="E47">A18</font></span></a>, <a class="txt" href="#E14:50_2"><span class="lab"><font color="Green" title="E43">A14</font></span></a>, <a class="txt" href="#E15:50_2"><span class="lab"><font color="Green" title="E44">A15</font></span></a>, <a class="txt" href="#E21:50_2"><span class="lab"><font color="Green" title="E48">A19</font></span></a>, <a class="txt" href="#E24:50_2"><span class="lab"><font color="Green" title="E51">A22</font></span></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></span>;<br/>
<a NAME="E35:50_2"/><span class="kw">then </span>
 <a href="msafree.html#K19" title="MSAFREE:func.19">pi</a> <span class="p1">(<span class="default"><a href="msafree.html#K18" title="MSAFREE:func.18">@</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c18">nt</font></span>)</span>,<font color="Maroon" title="c6">U1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">G</font> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c19">ts</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <span class="p2">(<span class="default"><a href="msafree.html#K18" title="MSAFREE:func.18">@</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c18">nt</font></span>)</span>,<font color="Maroon" title="c6">U1</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="c14">G</font> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c19">ts</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msafree.html#D23" target="_self" title="MSAFREE:def.23">Def23</a></span>;<br/>
<a NAME="E36:50_2"/><span class="kw">then </span>
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <span class="p2">(<span class="default"><a href="msafree.html#K18" title="MSAFREE:func.18">@</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c18">nt</font></span>)</span>,<font color="Maroon" title="c6">U1</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="msualg_1.html#K4" title="MSUALG_1:func.4">Result</a> <span class="p1">(<span class="default"><a href="msafree.html#K18" title="MSAFREE:func.18">@</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c18">nt</font></span>)</span>,<font color="Maroon" title="c6">U1</font> &amp;  <a href="msafree.html#K19" title="MSAFREE:func.19">pi</a> <span class="p1">(<span class="default"><a href="msafree.html#K18" title="MSAFREE:func.18">@</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c18">nt</font></span>)</span>,<font color="Maroon" title="c6">U1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">G</font> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c19">ts</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <span class="p2">(<span class="default"><a href="msafree.html#K18" title="MSAFREE:func.18">@</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c18">nt</font></span>)</span>,<font color="Maroon" title="c6">U1</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:50_2"><span class="lab"><font color="Green" title="E56">A34</font></span></a>, <a class="txt" href="#E27:50_2"><span class="lab"><font color="Green" title="E53">A31</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a>, <a class="ref" href="relat_1.html#D19" title="RELAT_1:def.19">RELAT_1:def 19</a></span>;<br/>
<a NAME="E37:50_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A35</font></span>: 
 <a href="msafree.html#K19" title="MSAFREE:func.19">pi</a> <span class="p1">(<span class="default"><a href="msafree.html#K18" title="MSAFREE:func.18">@</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c18">nt</font></span>)</span>,<font color="Maroon" title="c6">U1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">G</font> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c19">ts</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msualg_1.html#K4" title="MSUALG_1:func.4">Result</a> <span class="p1">(<span class="default"><a href="msafree.html#K18" title="MSAFREE:func.18">@</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c18">nt</font></span>)</span>,<font color="Maroon" title="c6">U1</font>
 
;<br/>
<span class="kw">assume </span><a NAME="E38:50_2"/><span class="lab"><font color="Green" title="E58">A36</font></span>: 
<font color="Maroon" title="c18">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c19">ts</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">s</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"><font color="Maroon" title="c17">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">s</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="c18">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c19">ts</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c6">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">s</font></span><br/>

<div><span class="lab"><font color="Green" title="E59">A37</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="50_2_3"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:50_2_3"/><span class="lab"><font color="Green" title="E60">A38</font></span>: 
 <a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <span class="p1">(<span class="default"><a href="msafree.html#K18" title="MSAFREE:func.18">@</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c18">nt</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c28">s</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:50_2_3"/>
<span class="p1">(<span class="default"><a href="msafree.html#K9" title="MSAFREE:func.9">FreeSort</a> <font color="Maroon" title="c2">X</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> <span class="p2">(<span class="default"><a href="msafree.html#K18" title="MSAFREE:func.18">@</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c18">nt</font></span>)</span></span>)</span> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a> <span class="p1">(<span class="default"><a href="msafree.html#K9" title="MSAFREE:func.9">FreeSort</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:50_2"><span class="lab"><font color="Green" title="E55">A33</font></span></a>, <a class="txt" href="#E38:50_2"><span class="lab"><font color="Green" title="E58">A36</font></span></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/><span class="kw">hence </span><a NAME="E3:50_2_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50_2_3"><span class="lab"><font color="Green" title="E60">A38</font></span></a>, <a class="ref" href="msafree.html#T13" target="_self" title="MSAFREE:th.13">Th13</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="E40:50_2"/>
<font color="Maroon" title="c14">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c19">ts</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msafree.html#K19" title="MSAFREE:func.19">pi</a> <span class="p1">(<span class="default"><a href="msafree.html#K18" title="MSAFREE:func.18">@</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c18">nt</font></span>)</span>,<font color="Maroon" title="c6">U1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">G</font> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c19">ts</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:50"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="txt" href="#E1:50_2"><span class="lab"><font color="Green" title="E34">A5</font></span></a></span>;<br/>
<a NAME="E41:50_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A39</font></span>: 
 <a href="msafree.html#K19" title="MSAFREE:func.19">pi</a> <span class="p1">(<span class="default"><a href="msafree.html#K18" title="MSAFREE:func.18">@</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c18">nt</font></span>)</span>,<font color="Maroon" title="c6">U1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">G</font> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c19">ts</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">G</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p2">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p3">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <span class="p4">(<span class="default"><a href="msafree.html#K18" title="MSAFREE:func.18">@</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c18">nt</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c19">ts</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:50_2"><span class="lab"><font color="Green" title="E55">A33</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>;<br/>
 <a href="msualg_1.html#K4" title="MSUALG_1:func.4">Result</a> <span class="p1">(<span class="default"><a href="msafree.html#K18" title="MSAFREE:func.18">@</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c18">nt</font></span>)</span>,<font color="Maroon" title="c6">U1</font> = 
<span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c6">U1</font> <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" title="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="msafree.html#K18" title="MSAFREE:func.18">@</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c18">nt</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_1.html#D10" title="MSUALG_1:def.10">MSUALG_1:def 10</a></span>
<br/>.= 
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c6">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <font color="Maroon" title="c1">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="msafree.html#K18" title="MSAFREE:func.18">@</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c18">nt</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:50_2"><span class="lab"><font color="Green" title="E52">A30</font></span></a>, <a class="txt" href="#E28:50_2"><span class="lab"><font color="Green" title="E54">A32</font></span></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></span>
<br/>.= 
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c6">U1</font> <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> <span class="p2">(<span class="default"><a href="msafree.html#K18" title="MSAFREE:func.18">@</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c18">nt</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_1.html#D7" title="MSUALG_1:def.7">MSUALG_1:def 7</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E43:50_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c17">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">s</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="c18">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c19">ts</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c6">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:50"><span class="lab"><font color="Green" title="E32">A3</font></span></a>, <a class="txt" href="#E37:50_2"><span class="lab"><font color="Green" title="E57">A35</font></span></a>, <a class="txt" href="#E41:50_2"><span class="lab"><font color="Green" title="E60">A39</font></span></a>, <a class="txt" href="#E39:50_2"><span class="lab"><font color="Green" title="E59">A37</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="E10:50"/><span class="lab"><font color="Green" title="E34">A40</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" title="c2">X</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" title="c2">X</font></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Olive" title="b1">t</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c18">t</font> be   <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" title="c2">X</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="c18">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" title="c2">X</font></span>)</span> implies <font color="Maroon">S<sub>1</sub></font>[ <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Maroon" title="c18">t</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:50_3"/><span class="lab"><font color="Green" title="E35">A41</font></span>: 
<font color="Maroon" title="c18">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" title="c2">X</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">S<sub>1</sub></font>[ <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Maroon" title="c18">t</font>]</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c19">s</font> being   <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c1">S</font>, <font color="Maroon" title="c20">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:50_3"/>
<font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">s</font>
 <span class="kw">and </span><br/><a NAME="E4:50_3"/><span class="lab"><font color="Green" title="E36">A42</font></span>: 
<font color="Maroon" title="c18">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"><font color="Maroon" title="c20">x</font>,<font color="Maroon" title="c19">s</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="msafree.html#T7" target="_self" title="MSAFREE:th.7">Th7</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c21">E</font> = <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> <font color="Olive" title="b1">tt</font></span>)</span> where <font color="Olive" title="b1">tt</font> is   <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" title="c2">X</font></span>)</span> : ( <font color="Olive" title="b1">tt</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" title="c2">X</font></span>)</span> &amp; <font color="Olive" title="b1">tt</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">s</font> ) </span>}</span> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c22">a</font> =  <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Maroon" title="c18">t</font>;<br/>
<a NAME="E5:50_3"/><span class="lab"><font color="Green" title="E37">A43</font></span>: 
<font color="Maroon" title="c18">t</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:50_3"><span class="lab"><font color="Green" title="E36">A42</font></span></a>, <a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></span>;<br/>
<a NAME="E6:50_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A44</font></span>: 
 <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Maroon" title="c18">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> <font color="Olive" title="b1">tt</font></span>)</span> where <font color="Olive" title="b1">tt</font> is   <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" title="c2">X</font></span>)</span> : ( <font color="Olive" title="b1">tt</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" title="c2">X</font></span>)</span> &amp; <font color="Olive" title="b1">tt</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">s</font> ) </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50_3"><span class="lab"><font color="Green" title="E35">A41</font></span></a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c23">s1</font> be   <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c1">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Maroon" title="c18">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">s1</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c17">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">s1</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="c18">t</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c6">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">s1</font> )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c24">f</font> = <font color="Maroon" title="c7">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c19">s</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree.html#K14" title="MSAFREE:func.14">FreeGen</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">s</font></span>)</span>,<span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c6">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">s</font></span>)</span> ;<br/>
<a NAME="E8:50_3"/><span class="lab"><font color="Green" title="E39">A45</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c24">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msafree.html#K14" title="MSAFREE:func.14">FreeGen</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E9:50_3"/><span class="lab"><font color="Green" title="E40">A46</font></span>: 
<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> <font color="Olive" title="b1">tt</font></span>)</span> where <font color="Olive" title="b1">tt</font> is   <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" title="c2">X</font></span>)</span> : ( <font color="Olive" title="b1">tt</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" title="c2">X</font></span>)</span> &amp; <font color="Olive" title="b1">tt</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">s</font> ) </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msafree.html#K13" title="MSAFREE:func.13">FreeGen</a> <font color="Maroon" title="c19">s</font>,<font color="Maroon" title="c2">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msafree.html#T14" target="_self" title="MSAFREE:th.14">Th14</a></span>;<br/>
<a NAME="E10:50_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="msafree.html#K14" title="MSAFREE:func.14">FreeGen</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">s</font> <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> <font color="Olive" title="b1">tt</font></span>)</span> where <font color="Olive" title="b1">tt</font> is   <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" title="c2">X</font></span>)</span> : ( <font color="Olive" title="b1">tt</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" title="c2">X</font></span>)</span> &amp; <font color="Olive" title="b1">tt</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">s</font> ) </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msafree.html#D18" target="_self" title="MSAFREE:def.18">Def18</a></span>;<br/>
<a NAME="E11:50_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A47</font></span>: 
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c24">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c6">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">s</font> &amp; <font color="Maroon" title="c24">f</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="c18">t</font></span>)</span> <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="c24">f</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:50_3"><span class="lab"><font color="Green" title="E38">A44</font></span></a>, <a class="txt" href="#E8:50_3"><span class="lab"><font color="Green" title="E39">A45</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a>, <a class="ref" href="relat_1.html#D19" title="RELAT_1:def.19">RELAT_1:def 19</a></span>;<br/>
<span class="kw">assume </span><a NAME="E12:50_3"/><span class="lab"><font color="Green" title="E42">A48</font></span>: 
 <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Maroon" title="c18">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">s1</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"><font color="Maroon" title="c17">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">s1</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="c18">t</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c6">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">s1</font></span><br/>

<div><span class="lab"><font color="Green" title="E43">A49</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="50_3_1"><span class="kw">now </span></a><div class="add"><a NAME="E1:50_3_1"/>
 <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Maroon" title="c18">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">s</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <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" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">s1</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:50_3"><span class="lab"><font color="Green" title="E38">A44</font></span></a>, <a class="txt" href="#E9:50_3"><span class="lab"><font color="Green" title="E40">A46</font></span></a>, <a class="txt" href="#E12:50_3"><span class="lab"><font color="Green" title="E42">A48</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E2:50_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A50</font></span>: 
<span class="p1">(<span class="default"><a href="msafree.html#K9" title="MSAFREE:func.9">FreeSort</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">s</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a> <span class="p1">(<span class="default"><a href="msafree.html#K9" title="MSAFREE:func.9">FreeSort</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">s1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#T4" title="XBOOLE_0:th.4">XBOOLE_0:4</a></span>;<br/><span class="kw">assume </span><a NAME="E3:50_3_1"/>
<font color="Maroon" title="c19">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c23">s1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">hence </span><a NAME="E4:50_3_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:50_3_1"><span class="lab"><font color="Green" title="E44">A50</font></span></a>, <a class="ref" href="msafree.html#T13" target="_self" title="MSAFREE:th.13">Th13</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="E14:50_3"/>
<font color="Maroon" title="c17">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">G</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p2">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">s</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:50"><span class="lab"><font color="Green" title="E32">A3</font></span></a></span>;<br/>
<span class="kw">then </span><span class="p1">(<span class="default"><font color="Maroon" title="c17">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">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#K2" title="TREES_4:func.2">root-tree</a> <font color="Maroon" title="c18">t</font></span>)</span> = 
<font color="Maroon" title="c14">G</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="c18">t</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:50_3"><span class="lab"><font color="Green" title="E38">A44</font></span></a>, <a class="txt" href="#E9:50_3"><span class="lab"><font color="Green" title="E40">A46</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></span>
<br/>.= 
 <a href="msafree.html#K17" title="MSAFREE:func.17">pi</a> <font color="Maroon" title="c7">F</font>,the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c6">U1</font>,<font color="Maroon" title="c18">t</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:50"><span class="lab"><font color="Green" title="E30">A1</font></span></a>, <a class="txt" href="#E1:50_3"><span class="lab"><font color="Green" title="E35">A41</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c24">f</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="c18">t</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50_3"><span class="lab"><font color="Green" title="E35">A41</font></span></a>, <a class="txt" href="#E5:50_3"><span class="lab"><font color="Green" title="E37">A43</font></span></a>, <a class="ref" href="msafree.html#D21" target="_self" title="MSAFREE:def.21">Def21</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E16:50_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c17">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">s1</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="c18">t</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c6">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c23">s1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:50_3"><span class="lab"><font color="Green" title="E41">A47</font></span></a>, <a class="txt" href="#E13:50_3"><span class="lab"><font color="Green" title="E43">A49</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="E11:50"/><span class="lab"><font color="Green" title="E35">A51</font></span>: 
for <font color="Olive" title="b1">t</font> being  <a href="trees_2.html#NM5" title="TREES_2:NM.5">DecoratedTree</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" title="c2">X</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#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" title="c2">X</font></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">t</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="dtconstr.html#S7" title="DTCONSTR:sch.7">DTCONSTR:sch 7</a>(<a class="txt" href="#E10:50"><span class="lab"><font color="Green" title="E34">A40</font></span></a>, <a class="txt" href="#E9:50"><span class="lab"><font color="Green" title="E33">A4</font></span></a>);<br/></span>
<a NAME="E12:50"/>
for <font color="Olive" title="b1">s</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> holds <br/><font color="Maroon" title="c17">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p2">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s</font></span>)</span>,<span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c6">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">s</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c18">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <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="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> implies <font color="Maroon" title="c17">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p2">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font></span>)</span>,<span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c6">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:50_4"/>
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</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="c17">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p2">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font></span>)</span>,<span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c6">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c19">s</font> = <font color="Maroon" title="c18">x</font> as   <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c1">S</font> ;<br/>
<span class="lab"><font color="Green" title="E36">A52</font></span>:  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c14">G</font> = 
 <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" title="c2">X</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>
<br/>.= 
 <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> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p2">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="msafree.html#T12" target="_self" title="MSAFREE:th.12">Th12</a></span>
;<br/>
<a NAME="E4:50_4"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <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" title="c1">S</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/>
<a NAME="E5:50_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A53</font></span>: 
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">s</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> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<a NAME="E6:50_4"/><span class="lab"><font color="Green" title="E38">A54</font></span>: 
<font color="Maroon" title="c17">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">G</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p2">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">s</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:50"><span class="lab"><font color="Green" title="E32">A3</font></span></a></span>;<br/>
<a NAME="E7:50_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A55</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:50_4"><span class="lab"><font color="Green" title="E36">A52</font></span></a>, <a class="txt" href="#E5:50_4"><span class="lab"><font color="Green" title="E37">A53</font></span></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></span>;<br/>
<a NAME="E8:50_4"/><span class="lab"><font color="Green" title="E40">A56</font></span>: 
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c14">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:50_4"><span class="lab"><font color="Green" title="E36">A52</font></span></a>, <a class="txt" href="#E5:50_4"><span class="lab"><font color="Green" title="E37">A53</font></span></a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></span>;<br/>
<a NAME="E9:50_4"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">s</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c6">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">s</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_4_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c20">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c20">a</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> <span class="p1">(<span class="default"><font color="Maroon" title="c17">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">s</font></span>)</span> or <font color="Maroon" title="c20">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c6">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">s</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:50_4_2"/>
<font color="Maroon" title="c20">a</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> <span class="p1">(<span class="default"><font color="Maroon" title="c17">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">s</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="c20">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c6">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">s</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c21">T</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:50_4_2"/><span class="lab"><font color="Green" title="E41">A57</font></span>: 
<font color="Maroon" title="c21">T</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">s</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:50_4_2"/><span class="lab"><font color="Green" title="E42">A58</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c17">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c22">T</font> = <font color="Maroon" title="c21">T</font> as    <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</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" title="c2">X</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:50_4"><span class="lab"><font color="Green" title="E40">A56</font></span></a>, <a class="txt" href="#E7:50_4"><span class="lab"><font color="Green" title="E39">A55</font></span></a>, <a class="txt" href="#E3:50_4_2"><span class="lab"><font color="Green" title="E41">A57</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E6:50_4_2"/>
<font color="Maroon" title="c22">T</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:50_4"><span class="lab"><font color="Green" title="E38">A54</font></span></a>, <a class="txt" href="#E3:50_4"><span class="lab"><font color="Green" title="E36">A52</font></span></a>, <a class="txt" href="#E5:50_4"><span class="lab"><font color="Green" title="E37">A53</font></span></a>, <a class="txt" href="#E3:50_4_2"><span class="lab"><font color="Green" title="E41">A57</font></span></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:50_4_2"/>
<font color="Maroon" title="c20">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c6">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:50"><span class="lab"><font color="Green" title="E35">A51</font></span></a>, <a class="txt" href="#E4:50_4_2"><span class="lab"><font color="Green" title="E42">A58</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>
<span class="kw">hence </span><a NAME="E10:50_4"/>
<font color="Maroon" title="c17">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p2">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font></span>)</span>,<span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c6">U1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">x</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:50_4"><span class="lab"><font color="Green" title="E39">A55</font></span></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></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>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c18">h</font> = <font color="Maroon" title="c17">h</font> as   <a href="msualg_3.html#NM1" title="MSUALG_3:NM.1">ManySortedFunction</a> of <span class="p1">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span>,<font color="Maroon" title="c6">U1</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#D18" title="PBOOLE:def.18">PBOOLE:def 18</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c18">h</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="c18">h</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a>  <a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c6">U1</font> &amp; <font color="Maroon" title="c18">h</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <span class="p1">(<span class="default"><a href="msafree.html#K14" title="MSAFREE:func.14">FreeGen</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c7">F</font> )</span><br/>

<span class="kw">thus </span><a NAME="E14:50"/>
<font color="Maroon" title="c18">h</font> <a href="msualg_3.html#R1" title="MSUALG_3:pred.1">is_homomorphism</a>  <a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c6">U1</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="c18">h</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <span class="p1">(<span class="default"><a href="msafree.html#K14" title="MSAFREE:func.14">FreeGen</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c7">F</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_5"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:50_5"/>
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <font color="Maroon" title="c1">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <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" title="c1">S</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a>, <a class="ref" href="relat_1.html#D19" title="RELAT_1:def.19">RELAT_1:def 19</a></span>;<br/>
<a NAME="E2:50_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A59</font></span>: 
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <font color="Maroon" title="c1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U4" title="STRUCT_0:sel.4">carrier'</a> of <font color="Maroon" title="c1">S</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p2">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <font color="Maroon" title="c2">X</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" title="c1">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <font color="Maroon" title="c1">S</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c19">o</font> be   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="msualg_3.html#D9" title="MSUALG_3:def.9">MSUALG_3:def 9</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c19">o</font>,<span class="p1">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or for <font color="Olive">b<sub>1</sub></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="Maroon" title="c19">o</font>,<span class="p1">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c18">h</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="c19">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="c19">o</font>,<span class="p3">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c19">o</font>,<font color="Maroon" title="c6">U1</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="c18">h</font> <a href="msualg_3.html#K5" title="MSUALG_3:func.5">#</a> <font color="Olive">b<sub>1</sub></font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E3:50_5"/>
 <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c19">o</font>,<span class="p1">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <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">b<sub>1</sub></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="Maroon" title="c19">o</font>,<span class="p1">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span> holds  <span class="p1">(<span class="default"><font color="Maroon" title="c18">h</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="c19">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="c19">o</font>,<span class="p3">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c19">o</font>,<font color="Maroon" title="c6">U1</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="c18">h</font> <a href="msualg_3.html#K5" title="MSUALG_3:func.5">#</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><br/>

<span class="kw">let </span><font color="Maroon" title="c20">x</font> be    <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="Maroon" title="c19">o</font>,<span class="p1">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <font color="Maroon" title="c2">X</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"><font color="Maroon" title="c18">h</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="c19">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="c19">o</font>,<span class="p3">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c19">o</font>,<font color="Maroon" title="c6">U1</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="c18">h</font> <a href="msualg_3.html#K5" title="MSUALG_3:func.5">#</a> <font color="Maroon" title="c20">x</font></span>)</span></span><br/>

<span class="kw">set </span><font color="Maroon" title="c21">rs</font> =  <a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c19">o</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c22">DA</font> =  <a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c19">o</font>,<span class="p1">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c23">D1</font> =  <a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c19">o</font>,<font color="Maroon" title="c6">U1</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c24">OU</font> = <span class="p1"><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" title="c1">S</font>,<span class="p2"><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" title="c1">S</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="p1">(<span class="default"><a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p2">(<span class="default"><a href="msafree.html#K4" title="MSAFREE:func.4">coprod</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c25">ar</font> =  <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c19">o</font>;<br/>
<a NAME="E4:50_5"/><span class="lab"><font color="Green" title="E37">A60</font></span>: 
 <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c19">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U1" title="MSUALG_1:sel.1">Arity</a> of <font color="Maroon" title="c1">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">o</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_1.html#D6" title="MSUALG_1:def.6">MSUALG_1:def 6</a></span>;<br/>
<a NAME="E5:50_5"/><span class="lab"><font color="Green" title="E38">A61</font></span>: 
 <a href="msualg_1.html#K3" title="MSUALG_1:func.3">Args</a> <font color="Maroon" title="c19">o</font>,<span class="p1">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="c2">X</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" title="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">o</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_1.html#D9" title="MSUALG_1:def.9">MSUALG_1:def 9</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c26">p</font> = <font color="Maroon" title="c20">x</font> as    <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> <span class="p1">(<span class="default"><a href="msafree.html#K6" title="MSAFREE:func.6">DTConMSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="msafree.html#T8" target="_self" title="MSAFREE:th.8">Th8</a></span>;<br/>
<a NAME="E7:50_5"/><span class="lab"><font color="Green" title="E39">A62</font></span>: 
 <a href="msafree.html#K7" title="MSAFREE:func.7">Sym</a> <font color="Maroon" title="c19">o</font>,<font color="Maroon" title="c2">X</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="c26">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:50_5"><span class="lab"><font color="Green" title="E38">A61</font></span></a>, <a class="ref" href="msafree.html#T10" target="_self" title="MSAFREE:th.10">Th10</a></span>;<br/>
<a NAME="E8:50_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A63</font></span>: 
 <a href="msafree.html#K18" title="MSAFREE:func.18">@</a> <font color="Maroon" title="c2">X</font>,<span class="p1">(<span class="default"><a href="msafree.html#K7" title="MSAFREE:func.7">Sym</a> <font color="Maroon" title="c19">o</font>,<font color="Maroon" title="c2">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">o</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msafree.html#D22" target="_self" title="MSAFREE:def.22">Def22</a></span>;<br/>
<a NAME="E9:50_5"/><span class="lab"><font color="Green" title="E41">A64</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">G</font> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c26">p</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="c26">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="alg_1.html#T1" title="ALG_1:th.1">ALG_1:1</a></span>;<br/>
<a NAME="E10:50_5"/><span class="lab"><font color="Green" title="E42">A65</font></span>: 
<font color="Maroon" title="c20">x</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" title="c2">X</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" title="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">o</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:50_5"><span class="lab"><font color="Green" title="E38">A61</font></span></a></span>;<br/>
<a NAME="E11:50_5"/><span class="lab"><font color="Green" title="E43">A66</font></span>: 
for <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">a</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="c26">p</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c14">G</font> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c26">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">h</font> <a href="msualg_3.html#K6" title="MSUALG_3:func.6">#</a> <font color="Maroon" title="c20">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_5_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:50_5_1"/>
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c19">o</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <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" title="c1">S</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a>, <a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a></span>;<br/>
<a NAME="E2:50_5_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A67</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p2">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c19">o</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="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c19">o</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c27">rt</font> =  <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c26">p</font>;<br/>
<span class="kw">let </span><font color="Maroon" title="c28">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <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="c28">a</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="c26">p</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c14">G</font> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c26">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">h</font> <a href="msualg_3.html#K6" title="MSUALG_3:func.6">#</a> <font color="Maroon" title="c20">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">a</font> )</span><br/>

<span class="kw">assume </span><a NAME="E3:50_5_1"/><span class="lab"><font color="Green" title="E45">A68</font></span>: 
<font color="Maroon" title="c28">a</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="c26">p</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"><font color="Maroon" title="c14">G</font> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c26">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">h</font> <a href="msualg_3.html#K6" title="MSUALG_3:func.6">#</a> <font color="Maroon" title="c20">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">a</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c29">n</font> = <font color="Maroon" title="c28">a</font> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> ;<br/>
<a NAME="E5:50_5_1"/><span class="lab"><font color="Green" title="E46">A69</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">o</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<span class="p2">(<span class="default"><a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c26">p</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> the <a href="lang1.html#U1" title="LANG1:sel.1">Rules</a> of <span class="p1">(<span class="default"><a href="msafree.html#K6" title="MSAFREE:func.6">DTConMSA</a> <font color="Maroon" title="c2">X</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:50_5"><span class="lab"><font color="Green" title="E39">A62</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">reconsider </span><font color="Maroon" title="c30">rt</font> =  <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c26">p</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <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" title="c1">S</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" title="c1">S</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" title="c2">X</font></span>)</span></span>)</span></span>)</span> <a href="finseq_1.html#K13" title="FINSEQ_1:func.13">*</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></span>;<br/>
<a NAME="E7:50_5_1"/><span class="lab"><font color="Green" title="E47">A70</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c30">rt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c19">o</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:50_5_1"><span class="lab"><font color="Green" title="E46">A69</font></span></a>, <a class="ref" href="msafree.html#T5" target="_self" title="MSAFREE:th.5">Th5</a></span>;<br/>
<a NAME="E8:50_5_1"/><span class="lab"><font color="Green" title="E48">A71</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c14">G</font> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c26">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c18">h</font> <a href="msualg_3.html#K6" title="MSUALG_3:func.6">#</a> <font color="Maroon" title="c20">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c19">o</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c29">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">n</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:50_5"><span class="lab"><font color="Green" title="E41">A64</font></span></a>, <a class="txt" href="#E3:50_5_1"><span class="lab"><font color="Green" title="E45">A68</font></span></a>, <a class="ref" href="alg_1.html#T1" title="ALG_1:th.1">ALG_1:1</a>, <a class="ref" href="msualg_3.html#D8" title="MSUALG_3:def.8">MSUALG_3:def 8</a></span>;<br/>
<a NAME="E9:50_5_1"/><span class="lab"><font color="Green" title="E49">A72</font></span>: 
<font color="Maroon" title="c18">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c19">o</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c29">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">G</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p2">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c19">o</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c29">n</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:50"><span class="lab"><font color="Green" title="E32">A3</font></span></a></span>;<br/>
<a NAME="E10:50_5_1"/><span class="lab"><font color="Green" title="E50">A73</font></span>: 
 <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="c30">rt</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="c30">rt</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E11:50_5_1"/><span class="lab"><font color="Green" title="E51">A74</font></span>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c30">rt</font> <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="c26">p</font> &amp;  <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> <span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c19">o</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="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c19">o</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a>, <a class="ref" href="trees_3.html#D18" title="TREES_3:def.18">TREES_3:def 18</a></span>;<br/>
<a NAME="E12:50_5_1"/>
<font color="Maroon" title="c26">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <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" title="c2">X</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c19">o</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:50_5"><span class="lab"><font color="Green" title="E42">A65</font></span></a>, <a class="txt" href="#E4:50_5"><span class="lab"><font color="Green" title="E37">A60</font></span></a>, <a class="ref" href="msafree.html#T1" target="_self" title="MSAFREE:th.1">Th1</a></span>;<br/>
<a NAME="E13:50_5_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A75</font></span>: 
<font color="Maroon" title="c26">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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" title="c2">X</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c19">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:50_5_1"><span class="lab"><font color="Green" title="E45">A68</font></span></a>, <a class="txt" href="#E2:50_5_1"><span class="lab"><font color="Green" title="E44">A67</font></span></a>, <a class="txt" href="#E7:50_5_1"><span class="lab"><font color="Green" title="E47">A70</font></span></a>, <a class="txt" href="#E10:50_5_1"><span class="lab"><font color="Green" title="E50">A73</font></span></a>, <a class="txt" href="#E11:50_5_1"><span class="lab"><font color="Green" title="E51">A74</font></span></a>, <a class="ref" href="card_3.html#T18" title="CARD_3:th.18">CARD_3:18</a></span>;<br/>
<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" title="c2">X</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c19">o</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">n</font> = 
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <font color="Maroon" title="c2">X</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="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c19">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c29">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:50_5_1"><span class="lab"><font color="Green" title="E45">A68</font></span></a>, <a class="txt" href="#E2:50_5_1"><span class="lab"><font color="Green" title="E44">A67</font></span></a>, <a class="txt" href="#E7:50_5_1"><span class="lab"><font color="Green" title="E47">A70</font></span></a>, <a class="txt" href="#E10:50_5_1"><span class="lab"><font color="Green" title="E50">A73</font></span></a>, <a class="txt" href="#E11:50_5_1"><span class="lab"><font color="Green" title="E51">A74</font></span></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></span>
<br/>.= 
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <font color="Maroon" title="c2">X</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="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c19">o</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c29">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:50_5_1"><span class="lab"><font color="Green" title="E45">A68</font></span></a>, <a class="txt" href="#E7:50_5_1"><span class="lab"><font color="Green" title="E47">A70</font></span></a>, <a class="txt" href="#E10:50_5_1"><span class="lab"><font color="Green" title="E50">A73</font></span></a>, <a class="txt" href="#E11:50_5_1"><span class="lab"><font color="Green" title="E51">A74</font></span></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E15:50_5_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c14">G</font> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c26">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">h</font> <a href="msualg_3.html#K6" title="MSUALG_3:func.6">#</a> <font color="Maroon" title="c20">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c28">a</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:50_5_1"><span class="lab"><font color="Green" title="E48">A71</font></span></a>, <a class="txt" href="#E9:50_5_1"><span class="lab"><font color="Green" title="E49">A72</font></span></a>, <a class="txt" href="#E13:50_5_1"><span class="lab"><font color="Green" title="E52">A75</font></span></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</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="E12:50_5"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">h</font> <a href="msualg_3.html#K6" title="MSUALG_3:func.6">#</a> <font color="Maroon" title="c20">x</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> <span class="p1">(<span class="default"><a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c19">o</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_3.html#T6" title="MSUALG_3:th.6">MSUALG_3:6</a></span>;<br/>
<a NAME="E13:50_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A76</font></span>: 
<font color="Maroon" title="c14">G</font> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c26">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">h</font> <a href="msualg_3.html#K6" title="MSUALG_3:func.6">#</a> <font color="Maroon" title="c20">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:50_5"><span class="lab"><font color="Green" title="E41">A64</font></span></a>, <a class="txt" href="#E11:50_5"><span class="lab"><font color="Green" title="E43">A66</font></span></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a>, <a class="ref" href="msualg_3.html#T6" title="MSUALG_3:th.6">MSUALG_3:6</a></span>;<br/>
<a NAME="E14:50_5"/><span class="lab"><font color="Green" title="E45">A77</font></span>: 
<font color="Maroon" title="c18">h</font> <a href="msualg_3.html#K1" title="MSUALG_3: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="c19">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">G</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p2">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1: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="c19">o</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:50"><span class="lab"><font color="Green" title="E32">A3</font></span></a></span>;<br/>
<a NAME="E15:50_5"/><span class="lab"><font color="Green" title="E46">A78</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="msafree.html#K10" title="MSAFREE:func.10">DenOp</a> <font color="Maroon" title="c19">o</font>,<font color="Maroon" title="c2">X</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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" title="c2">X</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" title="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">o</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E16:50_5"/>
<span class="p1">(<span class="default"><a href="msafree.html#K10" title="MSAFREE:func.10">DenOp</a> <font color="Maroon" title="c19">o</font>,<font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msafree.html#K7" title="MSAFREE:func.7">Sym</a> <font color="Maroon" title="c19">o</font>,<font color="Maroon" title="c2">X</font></span>)</span> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c26">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:50_5"><span class="lab"><font color="Green" title="E39">A62</font></span></a>, <a class="ref" href="msafree.html#D14" target="_self" title="MSAFREE:def.14">Def14</a></span>;<br/>
<a NAME="E17:50_5"/><span class="kw">then </span>
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="msafree.html#K10" title="MSAFREE:func.10">DenOp</a> <font color="Maroon" title="c19">o</font>,<font color="Maroon" title="c2">X</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <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" title="c2">X</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" title="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">o</font> &amp; <span class="p1">(<span class="default"><a href="msafree.html#K7" title="MSAFREE:func.7">Sym</a> <font color="Maroon" title="c19">o</font>,<font color="Maroon" title="c2">X</font></span>)</span> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c26">p</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> <span class="p1">(<span class="default"><a href="msafree.html#K10" title="MSAFREE:func.10">DenOp</a> <font color="Maroon" title="c19">o</font>,<font color="Maroon" title="c2">X</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:50_5"><span class="lab"><font color="Green" title="E38">A61</font></span></a>, <a class="txt" href="#E15:50_5"><span class="lab"><font color="Green" title="E46">A78</font></span></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a>, <a class="ref" href="relat_1.html#D19" title="RELAT_1:def.19">RELAT_1:def 19</a></span>;<br/>
<a NAME="E18:50_5"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="msafree.html#K7" title="MSAFREE:func.7">Sym</a> <font color="Maroon" title="c19">o</font>,<font color="Maroon" title="c2">X</font></span>)</span> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c26">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p2">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <font color="Maroon" title="c2">X</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" title="c1">S</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">o</font>
 
;<br/>
<a NAME="E19:50_5"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="msafree.html#K7" title="MSAFREE:func.7">Sym</a> <font color="Maroon" title="c19">o</font>,<font color="Maroon" title="c2">X</font></span>)</span> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c26">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U2" title="MSUALG_1:sel.2">ResultSort</a> of <font color="Maroon" title="c1">S</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">o</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:50_5"><span class="lab"><font color="Green" title="E36">A59</font></span></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></span>;<br/>
<a NAME="E20:50_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A79</font></span>: 
<span class="p1">(<span class="default"><a href="msafree.html#K7" title="MSAFREE:func.7">Sym</a> <font color="Maroon" title="c19">o</font>,<font color="Maroon" title="c2">X</font></span>)</span> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c26">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <font color="Maroon" title="c2">X</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="c19">o</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_1.html#D7" title="MSUALG_1:def.7">MSUALG_1:def 7</a></span>;<br/>
<a NAME="E21:50_5"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <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" title="c1">S</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/>
<a NAME="E22:50_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A80</font></span>: 
the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <font color="Maroon" title="c2">X</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="c19">o</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c14">G</font> = 
 <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" title="c2">X</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>
<br/>.= 
 <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> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p2">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="msafree.html#T12" target="_self" title="MSAFREE:th.12">Th12</a></span>
;<br/>
<a NAME="E24:50_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A81</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">h</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="c19">o</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <font color="Maroon" title="c2">X</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="c19">o</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:50_5"><span class="lab"><font color="Green" title="E45">A77</font></span></a>, <a class="txt" href="#E22:50_5"><span class="lab"><font color="Green" title="E48">A80</font></span></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a>, <a class="ref" href="zfmisc_1.html#T92" title="ZFMISC_1:th.92">ZFMISC_1:92</a></span>;<br/>
 <a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c19">o</font>,<span class="p1">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="msafree.html#K11" title="MSAFREE:func.11">FreeOper</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c19">o</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="msualg_1.html#D11" title="MSUALG_1:def.11">MSUALG_1:def 11</a></span>
<br/>.= 
 <a href="msafree.html#K10" title="MSAFREE:func.10">DenOp</a> <font color="Maroon" title="c19">o</font>,<font color="Maroon" title="c2">X</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="msafree.html#D15" target="_self" title="MSAFREE:def.15">Def15</a></span>
;<br/>
<a NAME="E26:50_5"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c19">o</font>,<span class="p2">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msafree.html#K7" title="MSAFREE:func.7">Sym</a> <font color="Maroon" title="c19">o</font>,<font color="Maroon" title="c2">X</font></span>)</span> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c26">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:50_5"><span class="lab"><font color="Green" title="E39">A62</font></span></a>, <a class="ref" href="msafree.html#D14" target="_self" title="MSAFREE:def.14">Def14</a></span>;<br/>
<span class="kw">hence </span><span class="p1">(<span class="default"><font color="Maroon" title="c18">h</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="c19">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="c19">o</font>,<span class="p3">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">x</font></span>)</span> = 
<font color="Maroon" title="c14">G</font> <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#K7" title="MSAFREE:func.7">Sym</a> <font color="Maroon" title="c19">o</font>,<font color="Maroon" title="c2">X</font></span>)</span> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c26">p</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:50_5"><span class="lab"><font color="Green" title="E47">A79</font></span></a>, <a class="txt" href="#E14:50_5"><span class="lab"><font color="Green" title="E45">A77</font></span></a>, <a class="txt" href="#E24:50_5"><span class="lab"><font color="Green" title="E49">A81</font></span></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></span>
<br/>.= 
 <a href="msafree.html#K19" title="MSAFREE:func.19">pi</a> <span class="p1">(<span class="default"><a href="msafree.html#K18" title="MSAFREE:func.18">@</a> <font color="Maroon" title="c2">X</font>,<span class="p2">(<span class="default"><a href="msafree.html#K7" title="MSAFREE:func.7">Sym</a> <font color="Maroon" title="c19">o</font>,<font color="Maroon" title="c2">X</font></span>)</span></span>)</span>,<font color="Maroon" title="c6">U1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">G</font> <a href="lang1.html#K11" title="LANG1:func.11">*</a> <font color="Maroon" title="c26">p</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:50"><span class="lab"><font color="Green" title="E31">A2</font></span></a>, <a class="txt" href="#E7:50_5"><span class="lab"><font color="Green" title="E39">A62</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="msualg_1.html#K5" title="MSUALG_1:func.5">Den</a> <font color="Maroon" title="c19">o</font>,<font color="Maroon" title="c6">U1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">h</font> <a href="msualg_3.html#K6" title="MSUALG_3:func.6">#</a> <font color="Maroon" title="c20">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:50_5"><span class="lab"><font color="Green" title="E40">A63</font></span></a>, <a class="txt" href="#E13:50_5"><span class="lab"><font color="Green" title="E44">A76</font></span></a>, <a class="ref" href="msafree.html#D23" target="_self" title="MSAFREE:def.23">Def23</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="E15:50"/>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c18">h</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <span class="p2">(<span class="default"><a href="msafree.html#K14" title="MSAFREE:func.14">FreeGen</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c19">hf</font> = <font color="Maroon" title="c18">h</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <span class="p1">(<span class="default"><a href="msafree.html#K14" title="MSAFREE:func.14">FreeGen</a> <font color="Maroon" title="c2">X</font></span>)</span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c20">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <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="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c18">h</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <span class="p2">(<span class="default"><a href="msafree.html#K14" title="MSAFREE:func.14">FreeGen</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:50_6"/>
<font color="Maroon" title="c20">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</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"><font color="Maroon" title="c18">h</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <span class="p2">(<span class="default"><a href="msafree.html#K14" title="MSAFREE:func.14">FreeGen</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x</font></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c21">s</font> = <font color="Maroon" title="c20">x</font> as   <a href="msualg_1.html#NM1" title="MSUALG_1:NM.1">SortSymbol</a> of <font color="Maroon" title="c1">S</font> ;<br/>
<a NAME="E3:50_6"/><span class="lab"><font color="Green" title="E36">A82</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c21">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E4:50_6"/><span class="lab"><font color="Green" title="E37">A83</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">h</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <span class="p3">(<span class="default"><a href="msafree.html#K14" title="MSAFREE:func.14">FreeGen</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c21">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msafree.html#K14" title="MSAFREE:func.14">FreeGen</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E5:50_6"/><span class="lab"><font color="Green" title="E38">A84</font></span>: 
<span class="p1">(<span class="default"><a href="msafree.html#K14" title="MSAFREE:func.14">FreeGen</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msafree.html#K13" title="MSAFREE:func.13">FreeGen</a> <font color="Maroon" title="c21">s</font>,<font color="Maroon" title="c2">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msafree.html#D18" target="_self" title="MSAFREE:def.18">Def18</a></span>;<br/>
<a NAME="E6:50_6"/><span class="lab"><font color="Green" title="E39">A85</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c18">h</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <span class="p2">(<span class="default"><a href="msafree.html#K14" title="MSAFREE:func.14">FreeGen</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c21">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c21">s</font></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree.html#K14" title="MSAFREE:func.14">FreeGen</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">s</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msafree.html#D1" target="_self" title="MSAFREE:def.1">Def1</a></span>;<br/>
<a NAME="E7:50_6"/><span class="lab"><font color="Green" title="E40">A86</font></span>: 
for <font color="Olive" title="b1">a</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="msafree.html#K14" title="MSAFREE:func.14">FreeGen</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">s</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">h</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <span class="p3">(<span class="default"><a href="msafree.html#K14" title="MSAFREE:func.14">FreeGen</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c21">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c21">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_6_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c22">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <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="c22">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="msafree.html#K14" title="MSAFREE:func.14">FreeGen</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">s</font> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">h</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <span class="p3">(<span class="default"><a href="msafree.html#K14" title="MSAFREE:func.14">FreeGen</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c21">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c21">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">a</font> )</span><br/>

<a NAME="E1:50_6_1"/><span class="lab"><font color="Green" title="E41">A87</font></span>: 
<font color="Maroon" title="c18">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c21">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">G</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default">the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p2">(<span class="default"><a href="msafree.html#K12" title="MSAFREE:func.12">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">s</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:50"><span class="lab"><font color="Green" title="E32">A3</font></span></a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:50_6_1"/><span class="lab"><font color="Green" title="E42">A88</font></span>: 
<font color="Maroon" title="c22">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="msafree.html#K14" title="MSAFREE:func.14">FreeGen</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">s</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"><span class="p2">(<span class="default"><font color="Maroon" title="c18">h</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <span class="p3">(<span class="default"><a href="msafree.html#K14" title="MSAFREE:func.14">FreeGen</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c21">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c21">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">a</font></span><br/>

<a NAME="E3:50_6_1"/><span class="kw">then </span>
<font color="Maroon" title="c22">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> <font color="Olive" title="b1">t</font></span>)</span> where <font color="Olive" title="b1">t</font> is   <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" title="c2">X</font></span>)</span> : ( <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" title="c2">X</font></span>)</span> &amp; <font color="Olive" title="b1">t</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">s</font> ) </span>}</span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:50_6"><span class="lab"><font color="Green" title="E38">A84</font></span></a>, <a class="ref" href="msafree.html#T14" target="_self" title="MSAFREE:th.14">Th14</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c23">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" title="c2">X</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E5:50_6_1"/><span class="lab"><font color="Green" title="E43">A89</font></span>: 
( <font color="Maroon" title="c22">a</font> <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="c23">t</font> &amp; <font color="Maroon" title="c23">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" title="c2">X</font></span>)</span> )
 <span class="kw">and </span><br/><a NAME="E6:50_6_1"/><span class="lab"><font color="Green" title="E44">A90</font></span>: 
<font color="Maroon" title="c23">t</font> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">s</font>
 ;<br/>
<span class="kw">thus </span><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">h</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <span class="p3">(<span class="default"><a href="msafree.html#K14" title="MSAFREE:func.14">FreeGen</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c21">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">a</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c18">h</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c21">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:50_6"><span class="lab"><font color="Green" title="E39">A85</font></span></a>, <a class="txt" href="#E4:50_6"><span class="lab"><font color="Green" title="E37">A83</font></span></a>, <a class="txt" href="#E2:50_6_1"><span class="lab"><font color="Green" title="E42">A88</font></span></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></span>
<br/>.= 
<font color="Maroon" title="c14">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:50_6"><span class="lab"><font color="Green" title="E36">A82</font></span></a>, <a class="txt" href="#E5:50_6"><span class="lab"><font color="Green" title="E38">A84</font></span></a>, <a class="txt" href="#E2:50_6_1"><span class="lab"><font color="Green" title="E42">A88</font></span></a>, <a class="txt" href="#E1:50_6_1"><span class="lab"><font color="Green" title="E41">A87</font></span></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></span>
<br/>.= 
 <a href="msafree.html#K17" title="MSAFREE:func.17">pi</a> <font color="Maroon" title="c7">F</font>,the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <font color="Maroon" title="c6">U1</font>,<font color="Maroon" title="c23">t</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:50"><span class="lab"><font color="Green" title="E30">A1</font></span></a>, <a class="txt" href="#E5:50_6_1"><span class="lab"><font color="Green" title="E43">A89</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c21">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">a</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:50_6_1"><span class="lab"><font color="Green" title="E43">A89</font></span></a>, <a class="txt" href="#E6:50_6_1"><span class="lab"><font color="Green" title="E44">A90</font></span></a>, <a class="ref" href="msafree.html#D21" target="_self" title="MSAFREE:def.21">Def21</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="E8:50_6"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="msualg_3.html#K1" title="MSUALG_3:func.1">.</a> <font color="Maroon" title="c21">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="msafree.html#K14" title="MSAFREE:func.14">FreeGen</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:50_6"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c18">h</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <span class="p2">(<span class="default"><a href="msafree.html#K14" title="MSAFREE:func.14">FreeGen</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c20">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:50_6"><span class="lab"><font color="Green" title="E37">A83</font></span></a>, <a class="txt" href="#E7:50_6"><span class="lab"><font color="Green" title="E40">A86</font></span></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</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>
<span class="kw">hence </span><a NAME="E16:50"/>
<font color="Maroon" title="c18">h</font> <a href="msafree.html#K1" title="MSAFREE:func.1">||</a> <span class="p1">(<span class="default"><a href="msafree.html#K14" title="MSAFREE:func.14">FreeGen</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <font color="Maroon" title="c7">F</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pboole.html#T3" title="PBOOLE:th.3">PBOOLE:3</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>
