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<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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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#K13" title="MSAFREE:func.13">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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="msafree.html#K13" title="MSAFREE:func.13">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#K5" title="MSAFREE:func.5">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#K11" title="MSAFREE:func.11">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#K13" title="MSAFREE:func.13">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> over <font color="Maroon" title="c1">S</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="msafree.html#D5" target="_self" onmouseover="rs('msafree/D5')" onmouseout="rh()">MSAFREE:def 5</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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#K13" title="MSAFREE:func.13">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#K11" title="MSAFREE:func.11">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#K11" title="MSAFREE:func.11">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#K13" title="MSAFREE:func.13">FreeGen</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</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#K13" title="MSAFREE:func.13">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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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#K11" title="MSAFREE:func.11">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#K11" title="MSAFREE:func.11">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#K13" title="MSAFREE:func.13">FreeGen</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</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#K11" title="MSAFREE:func.11">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#K5" title="MSAFREE:func.5">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#K16" title="MSAFREE:func.16">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#K5" title="MSAFREE:func.5">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#K18" title="MSAFREE:func.18">pi</a> (<span class="p1">(<span class="default"><a href="msafree.html#K17" title="MSAFREE:func.17">@</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#K5" title="MSAFREE:func.5">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="E2:48"/><span class="lab"><font color="Green" title="E29">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#K5" title="MSAFREE:func.5">DTConMSA</a> <font color="Maroon" title="c2">X</font></span>)</span>  st <font color="Olive" title="b1">t</font> <a href="tarski.html#R2" title="TARSKI: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#K5" title="MSAFREE:func.5">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="E3:48"/><span class="lab"><font color="Green" title="E30">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#K5" title="MSAFREE:func.5">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#K5" title="MSAFREE:func.5">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#K8" title="TREES_4:func.8">-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" onmouseover="rs('dtconstr/S8')" onmouseout="rh()">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">object</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#K5" title="MSAFREE:func.5">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#K11" title="MSAFREE:func.11">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="E5:48"/><span class="lab"><font color="Green" title="E31">A3</font></span>: 
(  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">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">object</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" onmouseover="rs('funct_1/S3')" onmouseout="rh()">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="#E5:48"><span class="lab"><font color="Green" title="E31">A3</font></span></a>, <a class="ref" href="partfun1.html#D2" onmouseover="rs('partfun1/D2')" onmouseout="rh()">PARTFUN1:def 2</a>, <a class="ref" href="relat_1.html#D18" onmouseover="rs('relat_1/D18')" onmouseout="rh()">RELAT_1:def 18</a></span>;<br/>
<a NAME="E7:48"/>
 for <font color="Olive" title="b1">s</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">s</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> <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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="48_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">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="relset_1.html#K1" title="RELSET_1:func.1">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="E1:48_1"/>
<font color="Maroon" title="c17">s</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> <font color="Maroon" title="c16">h</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="E2:48_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#K11" title="MSAFREE:func.11">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="#E5:48"><span class="lab"><font color="Green" title="E31">A3</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:48_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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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" onmouseover="rs('funcop_1/D6')" onmouseout="rh()">FUNCOP_1:def 6</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M2" title="HIDDEN:mode.2">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="tarski.html#R2" title="TARSKI: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#K11" title="MSAFREE:func.11">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="tarski.html#R2" title="TARSKI: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="E9:48"/><span class="lab"><font color="Green" title="E32">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#K5" title="MSAFREE:func.5">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#K5" title="MSAFREE:func.5">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#K5" title="MSAFREE:func.5">DTConMSA</a> <font color="Maroon" title="c2">X</font></span>)</span>  st <font color="Olive" title="b3">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">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#K8" title="TREES_4:func.8">-tree</a> <font color="Olive" title="b2">ts</font>]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="48_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#K5" title="MSAFREE:func.5">DTConMSA</a> <font color="Maroon" title="c2">X</font></span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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#K5" title="MSAFREE:func.5">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#K5" title="MSAFREE:func.5">DTConMSA</a> <font color="Maroon" title="c2">X</font></span>)</span>  st <font color="Olive" title="b2">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">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#K8" title="TREES_4:func.8">-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#K5" title="MSAFREE:func.5">DTConMSA</a> <font color="Maroon" title="c2">X</font></span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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#K5" title="MSAFREE:func.5">DTConMSA</a> <font color="Maroon" title="c2">X</font></span>)</span>  st <font color="Olive" title="b1">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">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#K8" title="TREES_4:func.8">-tree</a> <font color="Maroon" title="c19">ts</font>] )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:48_2"/><span class="lab"><font color="Green" title="E33">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="E2:48_2"/><span class="lab"><font color="Green" title="E34">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#K5" title="MSAFREE:func.5">DTConMSA</a> <font color="Maroon" title="c2">X</font></span>)</span>  st <font color="Olive" title="b1">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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#K8" title="TREES_4:func.8">-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#K17" title="MSAFREE:func.17">@</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#K17" title="MSAFREE:func.17">@</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#K17" title="MSAFREE:func.17">@</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#K3" title="MSAFREE:func.3">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="E3:48_2"/><span class="lab"><font color="Green" title="E35">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#K17" title="MSAFREE:func.17">@</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="#E1:48_2"><span class="lab"><font color="Green" title="E33">A5</font></span></a>, <a class="ref" href="msafree.html#D20" target="_self" onmouseover="rs('msafree/D20')" onmouseout="rh()">Def20</a></span>;<br/>
<a NAME="E4:48_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">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#K17" title="MSAFREE:func.17">@</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#K5" title="MSAFREE:func.5">DTConMSA</a> <font color="Maroon" title="c2">X</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_2"><span class="lab"><font color="Green" title="E33">A5</font></span></a>, <a class="ref" href="lang1.html#D1" onmouseover="rs('lang1/D1')" onmouseout="rh()">LANG1:def 1</a></span>;<br/>
<a NAME="E5:48_2"/><span class="lab"><font color="Green" title="E37">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#K17" title="MSAFREE:func.17">@</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#K6" title="MSAFREE:func.6">Sym</a> (<span class="p1">(<span class="default"><a href="msafree.html#K17" title="MSAFREE:func.17">@</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="E6:48_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A10</font></span>: 
<span class="p1">(<span class="default"><a href="msafree.html#K9" title="MSAFREE:func.9">DenOp</a> (<span class="p2">(<span class="default"><a href="msafree.html#K17" title="MSAFREE:func.17">@</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#K8" title="TREES_4:func.8">-tree</a> <font color="Maroon" title="c19">ts</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_2"><span class="lab"><font color="Green" title="E33">A5</font></span></a>, <a class="txt" href="#E3:48_2"><span class="lab"><font color="Green" title="E35">A7</font></span></a>, <a class="ref" href="msafree.html#D12" target="_self" onmouseover="rs('msafree/D12')" onmouseout="rh()">Def12</a></span>;<br/>
<a NAME="E7:48_2"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="msafree.html#K9" title="MSAFREE:func.9">DenOp</a> (<span class="p2">(<span class="default"><a href="msafree.html#K17" title="MSAFREE:func.17">@</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#K8" title="MSAFREE:func.8">FreeSort</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="finseq_2.html#K6" title="FINSEQ_2: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#K17" title="MSAFREE:func.17">@</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" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E8:48_2"/><span class="kw">then </span>
<font color="Maroon" title="c19">ts</font> <a href="tarski.html#R2" title="TARSKI: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#K9" title="MSAFREE:func.9">DenOp</a> (<span class="p2">(<span class="default"><a href="msafree.html#K17" title="MSAFREE:func.17">@</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="#E1:48_2"><span class="lab"><font color="Green" title="E33">A5</font></span></a>, <a class="txt" href="#E3:48_2"><span class="lab"><font color="Green" title="E35">A7</font></span></a>, <a class="txt" href="#E5:48_2"><span class="lab"><font color="Green" title="E37">A9</font></span></a>, <a class="ref" href="msafree.html#T10" target="_self" onmouseover="rs('msafree/T10')" onmouseout="rh()">Th10</a></span>;<br/>
<a NAME="E9:48_2"/><span class="kw">then </span>
(  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><a href="msafree.html#K9" title="MSAFREE:func.9">DenOp</a> (<span class="p2">(<span class="default"><a href="msafree.html#K17" title="MSAFREE:func.17">@</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#K8" title="MSAFREE:func.8">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#K17" title="MSAFREE:func.17">@</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#K8" title="TREES_4:func.8">-tree</a> <font color="Maroon" title="c19">ts</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><a href="msafree.html#K9" title="MSAFREE:func.9">DenOp</a> (<span class="p2">(<span class="default"><a href="msafree.html#K17" title="MSAFREE:func.17">@</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="#E6:48_2"><span class="lab"><font color="Green" title="E38">A10</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a>, <a class="ref" href="relat_1.html#D19" onmouseover="rs('relat_1/D19')" onmouseout="rh()">RELAT_1:def 19</a></span>;<br/>
<a NAME="E10:48_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A11</font></span>: 
<font color="Maroon" title="c18">nt</font> <a href="trees_4.html#K8" title="TREES_4:func.8">-tree</a> <font color="Maroon" title="c19">ts</font> <a href="tarski.html#R2" title="TARSKI: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#K11" title="MSAFREE:func.11">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#K17" title="MSAFREE:func.17">@</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#K18" title="MSAFREE:func.18">pi</a> (<span class="p1">(<span class="default"><a href="msafree.html#K17" title="MSAFREE:func.17">@</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="E11:48_2"/><span class="lab"><font color="Green" title="E40">A12</font></span>: 
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">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" onmouseover="rs('relat_1/D19')" onmouseout="rh()">RELAT_1:def 19</a></span>;<br/>
<a NAME="E12:48_2"/><span class="lab"><font color="Green" title="E41">A13</font></span>: 
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">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#K17" title="MSAFREE:func.17">@</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" onmouseover="rs('finseq_1/D4')" onmouseout="rh()">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="finseq_2.html#M2" title="FINSEQ_2:mode.2">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#K3" title="MSAFREE:func.3">coprod</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:48_2"><span class="lab"><font color="Green" title="E36">A8</font></span></a>, <a class="ref" href="zfmisc_1.html#T87" onmouseover="rs('zfmisc_1/T87')" onmouseout="rh()">ZFMISC_1:87</a></span>;<br/>
<a NAME="E14:48_2"/><span class="lab"><font color="Green" title="E42">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#K17" title="MSAFREE:func.17">@</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="#E4:48_2"><span class="lab"><font color="Green" title="E36">A8</font></span></a>, <a class="ref" href="msafree.html#T5" target="_self" onmouseover="rs('msafree/T5')" onmouseout="rh()">Th5</a></span>;<br/>
<a NAME="E15:48_2"/><span class="lab"><font color="Green" title="E43">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" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E16:48_2"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">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#K11" title="MSAFREE:func.11">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#D2" onmouseover="rs('partfun1/D2')" onmouseout="rh()">PARTFUN1:def 2</a></span>;<br/>
<a NAME="E17:48_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A16</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">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#K11" title="MSAFREE:func.11">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#K17" title="MSAFREE:func.17">@</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#K17" title="MSAFREE:func.17">@</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="#E12:48_2"><span class="lab"><font color="Green" title="E41">A13</font></span></a>, <a class="ref" href="relat_1.html#T27" onmouseover="rs('relat_1/T27')" onmouseout="rh()">RELAT_1:27</a></span>;<br/>
<a NAME="E18:48_2"/><span class="lab"><font color="Green" title="E45">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#K17" title="MSAFREE:func.17">@</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#K17" title="MSAFREE:func.17">@</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#D1" onmouseover="rs('msualg_1/D1')" onmouseout="rh()">MSUALG_1:def 1</a></span>;<br/>
<a NAME="E19:48_2"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">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#D2" onmouseover="rs('partfun1/D2')" onmouseout="rh()">PARTFUN1:def 2</a></span>;<br/>
<a NAME="E20:48_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A18</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">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#K17" title="MSAFREE:func.17">@</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#K17" title="MSAFREE:func.17">@</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="#E12:48_2"><span class="lab"><font color="Green" title="E41">A13</font></span></a>, <a class="ref" href="relat_1.html#T27" onmouseover="rs('relat_1/T27')" onmouseout="rh()">RELAT_1:27</a></span>;<br/>
<a NAME="E21:48_2"/><span class="lab"><font color="Green" title="E47">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#K17" title="MSAFREE:func.17">@</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#K17" title="MSAFREE:func.17">@</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" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a>, <a class="ref" href="trees_3.html#D18" onmouseover="rs('trees_3/D18')" onmouseout="rh()">TREES_3:def 18</a></span>;<br/>
<a NAME="E22:48_2"/><span class="lab"><font color="Green" title="E48">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="finseq_3.html#T120" onmouseover="rs('finseq_3/T120')" onmouseout="rh()">FINSEQ_3:120</a></span>;<br/>
<a NAME="E23:48_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">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="relset_1.html#K1" title="RELSET_1:func.1">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#K17" title="MSAFREE:func.17">@</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="#E20:48_2"><span class="lab"><font color="Green" title="E46">A18</font></span></a>, <a class="txt" href="#E4:48_2"><span class="lab"><font color="Green" title="E36">A8</font></span></a>, <a class="txt" href="#E15:48_2"><span class="lab"><font color="Green" title="E43">A15</font></span></a>, <a class="txt" href="#E21:48_2"><span class="lab"><font color="Green" title="E47">A19</font></span></a>, <a class="ref" href="msafree.html#T5" target="_self" onmouseover="rs('msafree/T5')" onmouseout="rh()">Th5</a></span>;<br/>
<a NAME="E24:48_2"/><span class="lab"><font color="Green" title="E50">A22</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</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"> 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#K17" title="MSAFREE:func.17">@</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="tarski.html#R2" title="TARSKI: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#K17" title="MSAFREE:func.17">@</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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="48_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">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="relset_1.html#K1" title="RELSET_1:func.1">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#K17" title="MSAFREE:func.17">@</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="tarski.html#R2" title="TARSKI: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#K17" title="MSAFREE:func.17">@</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="E1:48_2_1"/><span class="lab"><font color="Green" title="E51">A23</font></span>: 
<font color="Maroon" title="c28">x</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"> 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#K17" title="MSAFREE:func.17">@</a> (<font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c18">nt</font>)</span>)</span></span>)</span></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="tarski.html#R2" title="TARSKI: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#K17" title="MSAFREE:func.17">@</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="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> ;<br/>
<a NAME="E3:48_2_1"/><span class="lab"><font color="Green" title="E52">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="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c19">ts</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:48_2"><span class="lab"><font color="Green" title="E46">A18</font></span></a>, <a class="txt" href="#E14:48_2"><span class="lab"><font color="Green" title="E42">A14</font></span></a>, <a class="txt" href="#E15:48_2"><span class="lab"><font color="Green" title="E43">A15</font></span></a>, <a class="txt" href="#E21:48_2"><span class="lab"><font color="Green" title="E47">A19</font></span></a>, <a class="txt" href="#E1:48_2_1"><span class="lab"><font color="Green" title="E51">A23</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<a NAME="E4:48_2_1"/>
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">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#K5" title="MSAFREE:func.5">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" onmouseover="rs('finseq_1/D4')" onmouseout="rh()">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#K5" title="MSAFREE:func.5">DTConMSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_2_1"><span class="lab"><font color="Green" title="E52">A24</font></span></a></span>;<br/>
<a NAME="E6:48_2_1"/><span class="lab"><font color="Green" title="E53">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="#E23:48_2"><span class="lab"><font color="Green" title="E49">A21</font></span></a>, <a class="txt" href="#E1:48_2_1"><span class="lab"><font color="Green" title="E51">A23</font></span></a>, <a class="ref" href="finseq_3.html#T120" onmouseover="rs('finseq_3/T120')" onmouseout="rh()">FINSEQ_3:120</a></span>;<br/>
<a NAME="E7:48_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#K17" title="MSAFREE:func.17">@</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="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">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#K17" title="MSAFREE:func.17">@</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="#E20:48_2"><span class="lab"><font color="Green" title="E46">A18</font></span></a>, <a class="txt" href="#E1:48_2_1"><span class="lab"><font color="Green" title="E51">A23</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</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#K17" title="MSAFREE:func.17">@</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="#E12:48_2"><span class="lab"><font color="Green" title="E41">A13</font></span></a></span>;<br/>
<a NAME="E9:48_2_1"/><span class="lab"><font color="Green" title="E54">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#K11" title="MSAFREE:func.11">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="#E5:48"><span class="lab"><font color="Green" title="E31">A3</font></span></a></span>;<br/>
<a NAME="E10:48_2_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">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#K11" title="MSAFREE:func.11">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#D2" onmouseover="rs('partfun1/D2')" onmouseout="rh()">PARTFUN1:def 2</a></span>;<br/>
<a NAME="E11:48_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">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#K11" title="MSAFREE:func.11">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="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">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#K11" title="MSAFREE:func.11">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#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</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#K5" title="MSAFREE:func.5">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" onmouseover="rs('funct_2/D1')" onmouseout="rh()">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="xtuple_0.html#K10" title="XTUPLE_0:func.10">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#K11" title="MSAFREE:func.11">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#T11" target="_self" onmouseover="rs('msafree/T11')" onmouseout="rh()">Th11</a></span>
;<br/>
<a NAME="E13:48_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A28</font></span>: 
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">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#K11" title="MSAFREE:func.11">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="#E9:48_2_1"><span class="lab"><font color="Green" title="E54">A26</font></span></a>, <a class="txt" href="#E11:48_2_1"><span class="lab"><font color="Green" title="E55">A27</font></span></a>, <a class="ref" href="relat_1.html#T62" onmouseover="rs('relat_1/T62')" onmouseout="rh()">RELAT_1:62</a>, <a class="ref" href="zfmisc_1.html#T74" onmouseover="rs('zfmisc_1/T74')" onmouseout="rh()">ZFMISC_1:74</a></span>;<br/>
<a NAME="E14:48_2_1"/>
<font color="Maroon" title="c19">ts</font> <a href="tarski.html#R2" title="TARSKI: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#K8" title="MSAFREE:func.8">FreeSort</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="finseq_2.html#K6" title="FINSEQ_2: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#K17" title="MSAFREE:func.17">@</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="#E1:48_2"><span class="lab"><font color="Green" title="E33">A5</font></span></a>, <a class="txt" href="#E3:48_2"><span class="lab"><font color="Green" title="E35">A7</font></span></a>, <a class="txt" href="#E5:48_2"><span class="lab"><font color="Green" title="E37">A9</font></span></a>, <a class="ref" href="msafree.html#T10" target="_self" onmouseover="rs('msafree/T10')" onmouseout="rh()">Th10</a></span>;<br/>
<a NAME="E15:48_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c19">ts</font> <a href="tarski.html#R2" title="TARSKI: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#K8" title="MSAFREE:func.8">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#K17" title="MSAFREE:func.17">@</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="#E18:48_2"><span class="lab"><font color="Green" title="E45">A17</font></span></a>, <a class="ref" href="msafree.html#T1" target="_self" onmouseover="rs('msafree/T1')" onmouseout="rh()">Th1</a></span>;<br/>
<a NAME="E16:48_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="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree.html#K8" title="MSAFREE:func.8">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#K17" title="MSAFREE:func.17">@</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="#E20:48_2"><span class="lab"><font color="Green" title="E46">A18</font></span></a>, <a class="txt" href="#E17:48_2"><span class="lab"><font color="Green" title="E44">A16</font></span></a>, <a class="txt" href="#E1:48_2_1"><span class="lab"><font color="Green" title="E51">A23</font></span></a>, <a class="ref" href="card_3.html#T9" onmouseover="rs('card_3/T9')" onmouseout="rh()">CARD_3:9</a></span>;<br/>
<a NAME="E17:48_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">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="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="msafree.html#K8" title="MSAFREE:func.8">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#K17" title="MSAFREE:func.17">@</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="#E20:48_2"><span class="lab"><font color="Green" title="E46">A18</font></span></a>, <a class="txt" href="#E17:48_2"><span class="lab"><font color="Green" title="E44">A16</font></span></a>, <a class="txt" href="#E1:48_2_1"><span class="lab"><font color="Green" title="E51">A23</font></span></a>, <a class="ref" href="funct_1.html#T12" onmouseover="rs('funct_1/T12')" onmouseout="rh()">FUNCT_1:12</a></span>;<br/>
<a NAME="E18:48_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="tarski.html#R2" title="TARSKI: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="#E2:48_2"><span class="lab"><font color="Green" title="E34">A6</font></span></a>, <a class="txt" href="#E3:48_2_1"><span class="lab"><font color="Green" title="E52">A24</font></span></a></span>;<br/>
<a NAME="E19:48_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="tarski.html#R2" title="TARSKI: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="#E17:48_2_1"><span class="lab"><font color="Green" title="E57">A29</font></span></a>, <a class="txt" href="#E9:48_2_1"><span class="lab"><font color="Green" title="E54">A26</font></span></a>, <a class="txt" href="#E13:48_2_1"><span class="lab"><font color="Green" title="E56">A28</font></span></a>, <a class="ref" href="funct_1.html#T47" onmouseover="rs('funct_1/T47')" onmouseout="rh()">FUNCT_1:47</a></span>;<br/>
<span class="kw">hence </span><a NAME="E20:48_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="tarski.html#R2" title="TARSKI: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#K17" title="MSAFREE:func.17">@</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="#E1:48_2_1"><span class="lab"><font color="Green" title="E51">A23</font></span></a>, <a class="txt" href="#E6:48_2_1"><span class="lab"><font color="Green" title="E53">A25</font></span></a>, <a class="ref" href="funct_1.html#T12" onmouseover="rs('funct_1/T12')" onmouseout="rh()">FUNCT_1:12</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E25:48_2"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">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#D2" onmouseover="rs('partfun1/D2')" onmouseout="rh()">PARTFUN1:def 2</a></span>;<br/>
<a NAME="E26:48_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A30</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">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="#E11:48_2"><span class="lab"><font color="Green" title="E40">A12</font></span></a>, <a class="ref" href="relat_1.html#T27" onmouseover="rs('relat_1/T27')" onmouseout="rh()">RELAT_1:27</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c18">nt</font> <a href="trees_4.html#K8" title="TREES_4:func.8">-tree</a> <font color="Maroon" title="c19">ts</font> <a href="tarski.html#R2" title="TARSKI: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#K11" title="MSAFREE:func.11">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#K8" title="TREES_4:func.8">-tree</a> <font color="Maroon" title="c19">ts</font></span>)</span> <a href="tarski.html#R2" title="TARSKI: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="E27:48_2"/><span class="lab"><font color="Green" title="E52">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#K17" title="MSAFREE:func.17">@</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#K17" title="MSAFREE:func.17">@</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" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E28:48_2"/><span class="lab"><font color="Green" title="E53">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" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E29:48_2"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">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#K11" title="MSAFREE:func.11">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#D2" onmouseover="rs('partfun1/D2')" onmouseout="rh()">PARTFUN1:def 2</a></span>;<br/>
<a NAME="E30:48_2"/><span class="kw">then </span>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">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#K11" title="MSAFREE:func.11">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="#E11:48_2"><span class="lab"><font color="Green" title="E40">A12</font></span></a>, <a class="ref" href="relat_1.html#T27" onmouseover="rs('relat_1/T27')" onmouseout="rh()">RELAT_1:27</a></span>;<br/>
<a NAME="E31:48_2"/><span class="kw">then </span>
<font color="Maroon" title="c18">nt</font> <a href="trees_4.html#K8" title="TREES_4:func.8">-tree</a> <font color="Maroon" title="c19">ts</font> <a href="tarski.html#R2" title="TARSKI: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#K11" title="MSAFREE:func.11">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#K17" title="MSAFREE:func.17">@</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="#E28:48_2"><span class="lab"><font color="Green" title="E53">A32</font></span></a>, <a class="txt" href="#E10:48_2"><span class="lab"><font color="Green" title="E39">A11</font></span></a>, <a class="ref" href="funct_1.html#T12" onmouseover="rs('funct_1/T12')" onmouseout="rh()">FUNCT_1:12</a></span>;<br/>
<a NAME="E32:48_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A33</font></span>: 
<font color="Maroon" title="c18">nt</font> <a href="trees_4.html#K8" title="TREES_4:func.8">-tree</a> <font color="Maroon" title="c19">ts</font> <a href="tarski.html#R2" title="TARSKI: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#K11" title="MSAFREE:func.11">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#K17" title="MSAFREE:func.17">@</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#D2" onmouseover="rs('msualg_1/D2')" onmouseout="rh()">MSUALG_1:def 2</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#K17" title="MSAFREE:func.17">@</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="finseq_2.html#K6" title="FINSEQ_2: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#K17" title="MSAFREE:func.17">@</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#D4" onmouseover="rs('msualg_1/D4')" onmouseout="rh()">MSUALG_1:def 4</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#K17" title="MSAFREE:func.17">@</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="#E18:48_2"><span class="lab"><font color="Green" title="E45">A17</font></span></a>, <a class="ref" href="msafree.html#T1" target="_self" onmouseover="rs('msafree/T1')" onmouseout="rh()">Th1</a></span>
;<br/>
<a NAME="E34:48_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">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="tarski.html#R2" title="TARSKI: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#K17" title="MSAFREE:func.17">@</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="#E22:48_2"><span class="lab"><font color="Green" title="E48">A20</font></span></a>, <a class="txt" href="#E20:48_2"><span class="lab"><font color="Green" title="E46">A18</font></span></a>, <a class="txt" href="#E14:48_2"><span class="lab"><font color="Green" title="E42">A14</font></span></a>, <a class="txt" href="#E15:48_2"><span class="lab"><font color="Green" title="E43">A15</font></span></a>, <a class="txt" href="#E21:48_2"><span class="lab"><font color="Green" title="E47">A19</font></span></a>, <a class="txt" href="#E24:48_2"><span class="lab"><font color="Green" title="E50">A22</font></span></a>, <a class="ref" href="card_3.html#T9" onmouseover="rs('card_3/T9')" onmouseout="rh()">CARD_3:9</a></span>;<br/>
<a NAME="E35:48_2"/><span class="kw">then </span>
 <a href="msafree.html#K18" title="MSAFREE:func.18">pi</a> (<span class="p1">(<span class="default"><a href="msafree.html#K17" title="MSAFREE:func.17">@</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#K17" title="MSAFREE:func.17">@</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#D21" target="_self" onmouseover="rs('msafree/D21')" onmouseout="rh()">Def21</a></span>;<br/>
<a NAME="E36:48_2"/><span class="kw">then </span>
(  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">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#K17" title="MSAFREE:func.17">@</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#K17" title="MSAFREE:func.17">@</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#K18" title="MSAFREE:func.18">pi</a> (<span class="p1">(<span class="default"><a href="msafree.html#K17" title="MSAFREE:func.17">@</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="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">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#K17" title="MSAFREE:func.17">@</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="#E34:48_2"><span class="lab"><font color="Green" title="E55">A34</font></span></a>, <a class="txt" href="#E27:48_2"><span class="lab"><font color="Green" title="E52">A31</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a>, <a class="ref" href="relat_1.html#D19" onmouseover="rs('relat_1/D19')" onmouseout="rh()">RELAT_1:def 19</a></span>;<br/>
<a NAME="E37:48_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A35</font></span>: 
 <a href="msafree.html#K18" title="MSAFREE:func.18">pi</a> (<span class="p1">(<span class="default"><a href="msafree.html#K17" title="MSAFREE:func.17">@</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="tarski.html#R2" title="TARSKI: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#K17" title="MSAFREE:func.17">@</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="E38:48_2"/><span class="lab"><font color="Green" title="E57">A36</font></span>: 
<font color="Maroon" title="c18">nt</font> <a href="trees_4.html#K8" title="TREES_4:func.8">-tree</a> <font color="Maroon" title="c19">ts</font> <a href="tarski.html#R2" title="TARSKI: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#K11" title="MSAFREE:func.11">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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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#K8" title="TREES_4:func.8">-tree</a> <font color="Maroon" title="c19">ts</font></span>)</span> <a href="tarski.html#R2" title="TARSKI: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="E39:48_2"/><span class="lab"><font color="Green" title="E58">A37</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#K17" title="MSAFREE:func.17">@</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> <font color="Maroon" title="c28">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:48_2"><span class="lab"><font color="Green" title="E54">A33</font></span></a>, <a class="txt" href="#E38:48_2"><span class="lab"><font color="Green" title="E57">A36</font></span></a>, <a class="ref" href="xboole_0.html#T3" onmouseover="rs('xboole_0/T3')" onmouseout="rh()">XBOOLE_0:3</a>, <a class="ref" href="msafree.html#T12" target="_self" onmouseover="rs('msafree/T12')" onmouseout="rh()">Th12</a></span>;<br/>
<a NAME="E40:48_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#K8" title="TREES_4:func.8">-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#K18" title="MSAFREE:func.18">pi</a> (<span class="p1">(<span class="default"><a href="msafree.html#K17" title="MSAFREE:func.17">@</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="#E3:48"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="txt" href="#E1:48_2"><span class="lab"><font color="Green" title="E33">A5</font></span></a></span>;<br/>
<a NAME="E41:48_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A38</font></span>: 
 <a href="msafree.html#K18" title="MSAFREE:func.18">pi</a> (<span class="p1">(<span class="default"><a href="msafree.html#K17" title="MSAFREE:func.17">@</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#K11" title="MSAFREE:func.11">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#K17" title="MSAFREE:func.17">@</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#K8" title="TREES_4:func.8">-tree</a> <font color="Maroon" title="c19">ts</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:48_2"><span class="lab"><font color="Green" title="E54">A33</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</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#K17" title="MSAFREE:func.17">@</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#K17" title="MSAFREE:func.17">@</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#D5" onmouseover="rs('msualg_1/D5')" onmouseout="rh()">MSUALG_1:def 5</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#K17" title="MSAFREE:func.17">@</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="#E26:48_2"><span class="lab"><font color="Green" title="E51">A30</font></span></a>, <a class="txt" href="#E28:48_2"><span class="lab"><font color="Green" title="E53">A32</font></span></a>, <a class="ref" href="funct_1.html#T12" onmouseover="rs('funct_1/T12')" onmouseout="rh()">FUNCT_1:12</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#K17" title="MSAFREE:func.17">@</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#D2" onmouseover="rs('msualg_1/D2')" onmouseout="rh()">MSUALG_1:def 2</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E43:48_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#K8" title="TREES_4:func.8">-tree</a> <font color="Maroon" title="c19">ts</font></span>)</span> <a href="tarski.html#R2" title="TARSKI: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="#E5:48"><span class="lab"><font color="Green" title="E31">A3</font></span></a>, <a class="txt" href="#E37:48_2"><span class="lab"><font color="Green" title="E56">A35</font></span></a>, <a class="txt" href="#E41:48_2"><span class="lab"><font color="Green" title="E59">A38</font></span></a>, <a class="txt" href="#E39:48_2"><span class="lab"><font color="Green" title="E58">A37</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E10:48"/><span class="lab"><font color="Green" title="E33">A39</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#K5" title="MSAFREE:func.5">DTConMSA</a> <font color="Maroon" title="c2">X</font></span>)</span>  st <font color="Olive" title="b1">t</font> <a href="tarski.html#R2" title="TARSKI: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#K5" title="MSAFREE:func.5">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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="48_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#K5" title="MSAFREE:func.5">DTConMSA</a> <font color="Maroon" title="c2">X</font></span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c18">t</font> <a href="tarski.html#R2" title="TARSKI: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#K5" title="MSAFREE:func.5">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="E1:48_3"/><span class="lab"><font color="Green" title="E34">A40</font></span>: 
<font color="Maroon" title="c18">t</font> <a href="tarski.html#R2" title="TARSKI: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#K5" title="MSAFREE:func.5">DTConMSA</a> <font color="Maroon" title="c2">X</font></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E3:48_3"/>
<font color="Maroon" title="c20">x</font> <a href="tarski.html#R2" title="TARSKI: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="E4:48_3"/><span class="lab"><font color="Green" title="E35">A41</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" onmouseover="rs('msafree/T7')" onmouseout="rh()">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#K5" title="MSAFREE:func.5">DTConMSA</a> <font color="Maroon" title="c2">X</font></span>)</span> : ( <font color="Olive" title="b1">tt</font> <a href="tarski.html#R2" title="TARSKI: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#K5" title="MSAFREE:func.5">DTConMSA</a> <font color="Maroon" title="c2">X</font></span>)</span> &amp; <font color="Olive" title="b1">tt</font> <a href="xtuple_0.html#K2" title="XTUPLE_0: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="E5:48_3"/><span class="lab"><font color="Green" title="E36">A42</font></span>: 
<font color="Maroon" title="c18">t</font> <a href="xtuple_0.html#K2" title="XTUPLE_0: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="#E4:48_3"><span class="lab"><font color="Green" title="E35">A41</font></span></a></span>;<br/>
<a NAME="E6:48_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A43</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="tarski.html#R2" title="TARSKI: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#K5" title="MSAFREE:func.5">DTConMSA</a> <font color="Maroon" title="c2">X</font></span>)</span> : ( <font color="Olive" title="b1">tt</font> <a href="tarski.html#R2" title="TARSKI: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#K5" title="MSAFREE:func.5">DTConMSA</a> <font color="Maroon" title="c2">X</font></span>)</span> &amp; <font color="Olive" title="b1">tt</font> <a href="xtuple_0.html#K2" title="XTUPLE_0: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="#E1:48_3"><span class="lab"><font color="Green" title="E34">A40</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="tarski.html#R2" title="TARSKI: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#K11" title="MSAFREE:func.11">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="tarski.html#R2" title="TARSKI: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#K13" title="MSAFREE:func.13">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="E8:48_3"/><span class="lab"><font color="Green" title="E38">A44</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#K13" title="MSAFREE:func.13">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" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E9:48_3"/><span class="lab"><font color="Green" title="E39">A45</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#K5" title="MSAFREE:func.5">DTConMSA</a> <font color="Maroon" title="c2">X</font></span>)</span> : ( <font color="Olive" title="b1">tt</font> <a href="tarski.html#R2" title="TARSKI: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#K5" title="MSAFREE:func.5">DTConMSA</a> <font color="Maroon" title="c2">X</font></span>)</span> &amp; <font color="Olive" title="b1">tt</font> <a href="xtuple_0.html#K2" title="XTUPLE_0: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#K12" title="MSAFREE:func.12">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#T13" target="_self" onmouseover="rs('msafree/T13')" onmouseout="rh()">Th13</a></span>;<br/>
<a NAME="E10:48_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="msafree.html#K13" title="MSAFREE:func.13">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#K5" title="MSAFREE:func.5">DTConMSA</a> <font color="Maroon" title="c2">X</font></span>)</span> : ( <font color="Olive" title="b1">tt</font> <a href="tarski.html#R2" title="TARSKI: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#K5" title="MSAFREE:func.5">DTConMSA</a> <font color="Maroon" title="c2">X</font></span>)</span> &amp; <font color="Olive" title="b1">tt</font> <a href="xtuple_0.html#K2" title="XTUPLE_0: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#D16" target="_self" onmouseover="rs('msafree/D16')" onmouseout="rh()">Def16</a></span>;<br/>
<a NAME="E11:48_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A46</font></span>: 
(  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">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="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c24">f</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:48_3"><span class="lab"><font color="Green" title="E37">A43</font></span></a>, <a class="txt" href="#E8:48_3"><span class="lab"><font color="Green" title="E38">A44</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a>, <a class="ref" href="relat_1.html#D19" onmouseover="rs('relat_1/D19')" onmouseout="rh()">RELAT_1:def 19</a></span>;<br/>
<span class="kw">assume </span><a NAME="E12:48_3"/><span class="lab"><font color="Green" title="E41">A47</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="tarski.html#R2" title="TARSKI: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#K11" title="MSAFREE:func.11">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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="tarski.html#R2" title="TARSKI: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="E42">A48</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="48_3_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <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></span></a><div class="add"><a NAME="E1:48_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="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree.html#K8" title="MSAFREE:func.8">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#K8" title="MSAFREE:func.8">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="#E6:48_3"><span class="lab"><font color="Green" title="E37">A43</font></span></a>, <a class="txt" href="#E9:48_3"><span class="lab"><font color="Green" title="E39">A45</font></span></a>, <a class="txt" href="#E12:48_3"><span class="lab"><font color="Green" title="E41">A47</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/><a NAME="E2:48_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A49</font></span>: 
<span class="p1">(<span class="default"><a href="msafree.html#K8" title="MSAFREE:func.8">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#K8" title="MSAFREE:func.8">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>
 ;<br/><span class="kw">assume </span><a NAME="E3:48_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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">hence </span><a NAME="E4:48_3_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48_3_1"><span class="lab"><font color="Green" title="E43">A49</font></span></a>, <a class="ref" href="msafree.html#T12" target="_self" onmouseover="rs('msafree/T12')" onmouseout="rh()">Th12</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E14:48_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#K11" title="MSAFREE:func.11">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="#E5:48"><span class="lab"><font color="Green" title="E31">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="#E6:48_3"><span class="lab"><font color="Green" title="E37">A43</font></span></a>, <a class="txt" href="#E9:48_3"><span class="lab"><font color="Green" title="E39">A45</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>
<br/>.= 
 <a href="msafree.html#K16" title="MSAFREE:func.16">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="#E2:48"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E1:48_3"><span class="lab"><font color="Green" title="E34">A40</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="#E1:48_3"><span class="lab"><font color="Green" title="E34">A40</font></span></a>, <a class="txt" href="#E5:48_3"><span class="lab"><font color="Green" title="E36">A42</font></span></a>, <a class="ref" href="msafree.html#D19" target="_self" onmouseover="rs('msafree/D19')" onmouseout="rh()">Def19</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E16:48_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="tarski.html#R2" title="TARSKI: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="#E11:48_3"><span class="lab"><font color="Green" title="E40">A46</font></span></a>, <a class="txt" href="#E13:48_3"><span class="lab"><font color="Green" title="E42">A48</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E11:48"/><span class="lab"><font color="Green" title="E34">A50</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#K5" title="MSAFREE:func.5">DTConMSA</a> <font color="Maroon" title="c2">X</font></span>)</span>  st <font color="Olive" title="b1">t</font> <a href="tarski.html#R2" title="TARSKI: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#K5" title="MSAFREE:func.5">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" onmouseover="rs('dtconstr/S7')" onmouseout="rh()">DTCONSTR:sch 7</a>(<a class="txt" href="#E10:48"><span class="lab"><font color="Green" title="E33">A39</font></span></a>, <a class="txt" href="#E9:48"><span class="lab"><font color="Green" title="E32">A4</font></span></a>);<br/></span>
<a NAME="E12:48"/>
 for <font color="Olive" title="b1">s</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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#K11" title="MSAFREE:func.11">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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="48_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">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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#K11" title="MSAFREE:func.11">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="E1:48_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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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#K11" title="MSAFREE:func.11">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="E35">A51</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#K5" title="MSAFREE:func.5">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" onmouseover="rs('funct_2/D1')" onmouseout="rh()">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="xtuple_0.html#K10" title="XTUPLE_0:func.10">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#K11" title="MSAFREE:func.11">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#T11" target="_self" onmouseover="rs('msafree/T11')" onmouseout="rh()">Th11</a></span>
;<br/>
<a NAME="E4:48_4"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">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#K11" title="MSAFREE:func.11">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#D2" onmouseover="rs('partfun1/D2')" onmouseout="rh()">PARTFUN1:def 2</a></span>;<br/>
<a NAME="E5:48_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A52</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#K11" title="MSAFREE:func.11">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#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">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#K11" title="MSAFREE:func.11">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#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<a NAME="E6:48_4"/><span class="lab"><font color="Green" title="E37">A53</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#K11" title="MSAFREE:func.11">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="#E5:48"><span class="lab"><font color="Green" title="E31">A3</font></span></a></span>;<br/>
<a NAME="E7:48_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A54</font></span>: 
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">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#K11" title="MSAFREE:func.11">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="#E3:48_4"><span class="lab"><font color="Green" title="E35">A51</font></span></a>, <a class="txt" href="#E5:48_4"><span class="lab"><font color="Green" title="E36">A52</font></span></a>, <a class="ref" href="relat_1.html#T62" onmouseover="rs('relat_1/T62')" onmouseout="rh()">RELAT_1:62</a>, <a class="ref" href="zfmisc_1.html#T74" onmouseover="rs('zfmisc_1/T74')" onmouseout="rh()">ZFMISC_1:74</a></span>;<br/>
<a NAME="E8:48_4"/><span class="lab"><font color="Green" title="E39">A55</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#K11" title="MSAFREE:func.11">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="#E3:48_4"><span class="lab"><font color="Green" title="E35">A51</font></span></a>, <a class="txt" href="#E5:48_4"><span class="lab"><font color="Green" title="E36">A52</font></span></a>, <a class="ref" href="zfmisc_1.html#T74" onmouseover="rs('zfmisc_1/T74')" onmouseout="rh()">ZFMISC_1:74</a></span>;<br/>
<a NAME="E9:48_4"/>
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="48_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">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="xtuple_0.html#K10" title="XTUPLE_0:func.10">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="E1:48_4_2"/>
<font color="Maroon" title="c20">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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">object</a> <span class="kw"> such that </span><br/><a NAME="E3:48_4_2"/><span class="lab"><font color="Green" title="E40">A56</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#NK1" title="RELAT_1:NK.1">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="E4:48_4_2"/><span class="lab"><font color="Green" title="E41">A57</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#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</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#K5" title="MSAFREE:func.5">DTConMSA</a> <font color="Maroon" title="c2">X</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:48_4"><span class="lab"><font color="Green" title="E39">A55</font></span></a>, <a class="txt" href="#E7:48_4"><span class="lab"><font color="Green" title="E38">A54</font></span></a>, <a class="txt" href="#E3:48_4_2"><span class="lab"><font color="Green" title="E40">A56</font></span></a>, <a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E6:48_4_2"/>
<font color="Maroon" title="c22">T</font> <a href="tarski.html#R2" title="TARSKI: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#K11" title="MSAFREE:func.11">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="#E6:48_4"><span class="lab"><font color="Green" title="E37">A53</font></span></a>, <a class="txt" href="#E3:48_4"><span class="lab"><font color="Green" title="E35">A51</font></span></a>, <a class="txt" href="#E5:48_4"><span class="lab"><font color="Green" title="E36">A52</font></span></a>, <a class="txt" href="#E3:48_4_2"><span class="lab"><font color="Green" title="E40">A56</font></span></a>, <a class="ref" href="relat_1.html#T62" onmouseover="rs('relat_1/T62')" onmouseout="rh()">RELAT_1:62</a>, <a class="ref" href="zfmisc_1.html#T74" onmouseover="rs('zfmisc_1/T74')" onmouseout="rh()">ZFMISC_1:74</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:48_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="#E11:48"><span class="lab"><font color="Green" title="E34">A50</font></span></a>, <a class="txt" href="#E4:48_4_2"><span class="lab"><font color="Green" title="E41">A57</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E10:48_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#K11" title="MSAFREE:func.11">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="#E7:48_4"><span class="lab"><font color="Green" title="E38">A54</font></span></a>, <a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T4" onmouseover="rs('relset_1/T4')" onmouseout="rh()">RELSET_1:4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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#K11" title="MSAFREE:func.11">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#D15" onmouseover="rs('pboole/D15')" onmouseout="rh()">PBOOLE:def 15</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c18">h</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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#K11" title="MSAFREE:func.11">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#K13" title="MSAFREE:func.13">FreeGen</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Maroon" title="c7">F</font> )</span><br/>

<span class="kw">thus </span><a NAME="E14:48"/>
<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#K11" title="MSAFREE:func.11">FreeMSA</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c6">U1</font>
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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#K13" title="MSAFREE:func.13">FreeGen</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Maroon" title="c7">F</font></span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="48_5"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:48_5"/>
(  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">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="relset_1.html#K1" title="RELSET_1:func.1">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#K11" title="MSAFREE:func.11">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#D2" onmouseover="rs('partfun1/D2')" onmouseout="rh()">PARTFUN1:def 2</a>, <a class="ref" href="relat_1.html#D19" onmouseover="rs('relat_1/D19')" onmouseout="rh()">RELAT_1:def 19</a></span>;<br/>
<a NAME="E2:48_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A58</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="relset_1.html#K1" title="RELSET_1:func.1">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#K11" title="MSAFREE:func.11">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" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a>, <a class="ref" href="relat_1.html#T27" onmouseover="rs('relat_1/T27')" onmouseout="rh()">RELAT_1:27</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="firebrick">:: according to </font></span><a class="ref" href="msualg_3.html#D7" onmouseover="rs('msualg_3/D7')" onmouseout="rh()">MSUALG_3:def 7</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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#K11" title="MSAFREE:func.11">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#K11" title="MSAFREE:func.11">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#K11" title="MSAFREE:func.11">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="E3:48_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#K11" title="MSAFREE:func.11">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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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#K11" title="MSAFREE:func.11">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#K11" title="MSAFREE:func.11">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#K11" title="MSAFREE:func.11">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span>); <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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#K11" title="MSAFREE:func.11">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#K11" title="MSAFREE:func.11">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#K3" title="MSAFREE:func.3">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="E4:48_5"/><span class="lab"><font color="Green" title="E36">A59</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#D1" onmouseover="rs('msualg_1/D1')" onmouseout="rh()">MSUALG_1:def 1</a></span>;<br/>
<a NAME="E5:48_5"/><span class="lab"><font color="Green" title="E37">A60</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#K11" title="MSAFREE:func.11">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#K8" title="MSAFREE:func.8">FreeSort</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="finseq_2.html#K6" title="FINSEQ_2: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#D4" onmouseover="rs('msualg_1/D4')" onmouseout="rh()">MSUALG_1:def 4</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#K5" title="MSAFREE:func.5">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" onmouseover="rs('msafree/T8')" onmouseout="rh()">Th8</a></span>;<br/>
<a NAME="E7:48_5"/><span class="lab"><font color="Green" title="E38">A61</font></span>: 
 <a href="msafree.html#K6" title="MSAFREE:func.6">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="#E5:48_5"><span class="lab"><font color="Green" title="E37">A60</font></span></a>, <a class="ref" href="msafree.html#T10" target="_self" onmouseover="rs('msafree/T10')" onmouseout="rh()">Th10</a></span>;<br/>
<a NAME="E8:48_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A62</font></span>: 
 <a href="msafree.html#K17" title="MSAFREE:func.17">@</a> (<font color="Maroon" title="c2">X</font>,<span class="p1">(<span class="default"><a href="msafree.html#K6" title="MSAFREE:func.6">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#D20" target="_self" onmouseover="rs('msafree/D20')" onmouseout="rh()">Def20</a></span>;<br/>
<a NAME="E9:48_5"/><span class="lab"><font color="Green" title="E40">A63</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="finseq_3.html#T120" onmouseover="rs('finseq_3/T120')" onmouseout="rh()">FINSEQ_3:120</a></span>;<br/>
<a NAME="E10:48_5"/><span class="lab"><font color="Green" title="E41">A64</font></span>: 
<font color="Maroon" title="c20">x</font> <a href="tarski.html#R2" title="TARSKI: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#K8" title="MSAFREE:func.8">FreeSort</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="finseq_2.html#K6" title="FINSEQ_2: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="#E5:48_5"><span class="lab"><font color="Green" title="E37">A60</font></span></a></span>;<br/>
<a NAME="E11:48_5"/><span class="lab"><font color="Green" title="E42">A65</font></span>: 
 for <font color="Olive" title="b1">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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#K5" title="MSUALG_3:func.5">#</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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="48_5_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:48_5_1"/>
(  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">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="relset_1.html#K1" title="RELSET_1:func.1">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#K11" title="MSAFREE:func.11">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" onmouseover="rs('finseq_1/D4')" onmouseout="rh()">FINSEQ_1:def 4</a>, <a class="ref" href="partfun1.html#D2" onmouseover="rs('partfun1/D2')" onmouseout="rh()">PARTFUN1:def 2</a></span>;<br/>
<a NAME="E2:48_5_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A66</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">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#K11" title="MSAFREE:func.11">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#T27" onmouseover="rs('relat_1/T27')" onmouseout="rh()">RELAT_1:27</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">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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#K5" title="MSUALG_3:func.5">#</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="E3:48_5_1"/><span class="lab"><font color="Green" title="E44">A67</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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#K5" title="MSUALG_3:func.5">#</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="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> ;<br/>
<a NAME="E5:48_5_1"/><span class="lab"><font color="Green" title="E45">A68</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#K5" title="MSAFREE:func.5">DTConMSA</a> <font color="Maroon" title="c2">X</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:48_5"><span class="lab"><font color="Green" title="E38">A61</font></span></a>, <a class="ref" href="lang1.html#D1" onmouseover="rs('lang1/D1')" onmouseout="rh()">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="finseq_2.html#M2" title="FINSEQ_2:mode.2">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#K3" title="MSAFREE:func.3">coprod</a> <font color="Maroon" title="c2">X</font></span>)</span></span>)</span></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T87" onmouseover="rs('zfmisc_1/T87')" onmouseout="rh()">ZFMISC_1:87</a></span>;<br/>
<a NAME="E7:48_5_1"/><span class="lab"><font color="Green" title="E46">A69</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="#E5:48_5_1"><span class="lab"><font color="Green" title="E45">A68</font></span></a>, <a class="ref" href="msafree.html#T5" target="_self" onmouseover="rs('msafree/T5')" onmouseout="rh()">Th5</a></span>;<br/>
<a NAME="E8:48_5_1"/><span class="lab"><font color="Green" title="E47">A70</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#K5" title="MSUALG_3:func.5">#</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="#E9:48_5"><span class="lab"><font color="Green" title="E40">A63</font></span></a>, <a class="txt" href="#E3:48_5_1"><span class="lab"><font color="Green" title="E44">A67</font></span></a>, <a class="ref" href="finseq_3.html#T120" onmouseover="rs('finseq_3/T120')" onmouseout="rh()">FINSEQ_3:120</a>, <a class="ref" href="msualg_3.html#D6" onmouseover="rs('msualg_3/D6')" onmouseout="rh()">MSUALG_3:def 6</a></span>;<br/>
<a NAME="E9:48_5_1"/><span class="lab"><font color="Green" title="E48">A71</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#K11" title="MSAFREE:func.11">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="#E5:48"><span class="lab"><font color="Green" title="E31">A3</font></span></a></span>;<br/>
<a NAME="E10:48_5_1"/><span class="lab"><font color="Green" title="E49">A72</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" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E11:48_5_1"/><span class="lab"><font color="Green" title="E50">A73</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" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a>, <a class="ref" href="trees_3.html#D18" onmouseover="rs('trees_3/D18')" onmouseout="rh()">TREES_3:def 18</a></span>;<br/>
<a NAME="E12:48_5_1"/>
<font color="Maroon" title="c26">p</font> <a href="tarski.html#R2" title="TARSKI: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#K8" title="MSAFREE:func.8">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="#E10:48_5"><span class="lab"><font color="Green" title="E41">A64</font></span></a>, <a class="txt" href="#E4:48_5"><span class="lab"><font color="Green" title="E36">A59</font></span></a>, <a class="ref" href="msafree.html#T1" target="_self" onmouseover="rs('msafree/T1')" onmouseout="rh()">Th1</a></span>;<br/>
<a NAME="E13:48_5_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A74</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="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree.html#K8" title="MSAFREE:func.8">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="#E3:48_5_1"><span class="lab"><font color="Green" title="E44">A67</font></span></a>, <a class="txt" href="#E2:48_5_1"><span class="lab"><font color="Green" title="E43">A66</font></span></a>, <a class="txt" href="#E7:48_5_1"><span class="lab"><font color="Green" title="E46">A69</font></span></a>, <a class="txt" href="#E10:48_5_1"><span class="lab"><font color="Green" title="E49">A72</font></span></a>, <a class="txt" href="#E11:48_5_1"><span class="lab"><font color="Green" title="E50">A73</font></span></a>, <a class="ref" href="card_3.html#T9" onmouseover="rs('card_3/T9')" onmouseout="rh()">CARD_3:9</a></span>;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree.html#K8" title="MSAFREE:func.8">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#K11" title="MSAFREE:func.11">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="#E3:48_5_1"><span class="lab"><font color="Green" title="E44">A67</font></span></a>, <a class="txt" href="#E2:48_5_1"><span class="lab"><font color="Green" title="E43">A66</font></span></a>, <a class="txt" href="#E7:48_5_1"><span class="lab"><font color="Green" title="E46">A69</font></span></a>, <a class="txt" href="#E10:48_5_1"><span class="lab"><font color="Green" title="E49">A72</font></span></a>, <a class="txt" href="#E11:48_5_1"><span class="lab"><font color="Green" title="E50">A73</font></span></a>, <a class="ref" href="funct_1.html#T12" onmouseover="rs('funct_1/T12')" onmouseout="rh()">FUNCT_1:12</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#K11" title="MSAFREE:func.11">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="#E3:48_5_1"><span class="lab"><font color="Green" title="E44">A67</font></span></a>, <a class="txt" href="#E7:48_5_1"><span class="lab"><font color="Green" title="E46">A69</font></span></a>, <a class="txt" href="#E10:48_5_1"><span class="lab"><font color="Green" title="E49">A72</font></span></a>, <a class="txt" href="#E11:48_5_1"><span class="lab"><font color="Green" title="E50">A73</font></span></a>, <a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E15:48_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#K5" title="MSUALG_3:func.5">#</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="#E8:48_5_1"><span class="lab"><font color="Green" title="E47">A70</font></span></a>, <a class="txt" href="#E9:48_5_1"><span class="lab"><font color="Green" title="E48">A71</font></span></a>, <a class="txt" href="#E13:48_5_1"><span class="lab"><font color="Green" title="E51">A74</font></span></a>, <a class="ref" href="funct_1.html#T49" onmouseover="rs('funct_1/T49')" onmouseout="rh()">FUNCT_1:49</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E12:48_5"/>
 <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</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> <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" onmouseover="rs('msualg_3/T6')" onmouseout="rh()">MSUALG_3:6</a></span>;<br/>
<a NAME="E13:48_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A75</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#K5" title="MSUALG_3:func.5">#</a> <font color="Maroon" title="c20">x</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:48_5"><span class="lab"><font color="Green" title="E40">A63</font></span></a>, <a class="txt" href="#E11:48_5"><span class="lab"><font color="Green" title="E42">A65</font></span></a>, <a class="ref" href="funct_1.html#T2" onmouseover="rs('funct_1/T2')" onmouseout="rh()">FUNCT_1:2</a>, <a class="ref" href="msualg_3.html#T6" onmouseover="rs('msualg_3/T6')" onmouseout="rh()">MSUALG_3:6</a></span>;<br/>
<a NAME="E14:48_5"/><span class="lab"><font color="Green" title="E44">A76</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#K11" title="MSAFREE:func.11">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="#E5:48"><span class="lab"><font color="Green" title="E31">A3</font></span></a></span>;<br/>
<a NAME="E15:48_5"/><span class="lab"><font color="Green" title="E45">A77</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#K9" title="MSAFREE:func.9">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#K8" title="MSAFREE:func.8">FreeSort</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="finseq_2.html#K6" title="FINSEQ_2: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" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E16:48_5"/>
<span class="p1">(<span class="default"><a href="msafree.html#K9" title="MSAFREE:func.9">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#K6" title="MSAFREE:func.6">Sym</a> (<font color="Maroon" title="c19">o</font>,<font color="Maroon" title="c2">X</font>)</span>)</span> <a href="trees_4.html#K8" title="TREES_4:func.8">-tree</a> <font color="Maroon" title="c26">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:48_5"><span class="lab"><font color="Green" title="E38">A61</font></span></a>, <a class="ref" href="msafree.html#D12" target="_self" onmouseover="rs('msafree/D12')" onmouseout="rh()">Def12</a></span>;<br/>
<a NAME="E17:48_5"/><span class="kw">then </span>
(  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><a href="msafree.html#K9" title="MSAFREE:func.9">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#K8" title="MSAFREE:func.8">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#K6" title="MSAFREE:func.6">Sym</a> (<font color="Maroon" title="c19">o</font>,<font color="Maroon" title="c2">X</font>)</span>)</span> <a href="trees_4.html#K8" title="TREES_4:func.8">-tree</a> <font color="Maroon" title="c26">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><a href="msafree.html#K9" title="MSAFREE:func.9">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="#E5:48_5"><span class="lab"><font color="Green" title="E37">A60</font></span></a>, <a class="txt" href="#E15:48_5"><span class="lab"><font color="Green" title="E45">A77</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a>, <a class="ref" href="relat_1.html#D19" onmouseover="rs('relat_1/D19')" onmouseout="rh()">RELAT_1:def 19</a></span>;<br/>
<a NAME="E18:48_5"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="msafree.html#K6" title="MSAFREE:func.6">Sym</a> (<font color="Maroon" title="c19">o</font>,<font color="Maroon" title="c2">X</font>)</span>)</span> <a href="trees_4.html#K8" title="TREES_4:func.8">-tree</a> <font color="Maroon" title="c26">p</font> <a href="tarski.html#R2" title="TARSKI: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#K11" title="MSAFREE:func.11">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="E19:48_5"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="msafree.html#K6" title="MSAFREE:func.6">Sym</a> (<font color="Maroon" title="c19">o</font>,<font color="Maroon" title="c2">X</font>)</span>)</span> <a href="trees_4.html#K8" title="TREES_4:func.8">-tree</a> <font color="Maroon" title="c26">p</font> <a href="tarski.html#R2" title="TARSKI: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#K11" title="MSAFREE:func.11">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="#E2:48_5"><span class="lab"><font color="Green" title="E35">A58</font></span></a>, <a class="ref" href="funct_1.html#T12" onmouseover="rs('funct_1/T12')" onmouseout="rh()">FUNCT_1:12</a></span>;<br/>
<a NAME="E20:48_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A78</font></span>: 
<span class="p1">(<span class="default"><a href="msafree.html#K6" title="MSAFREE:func.6">Sym</a> (<font color="Maroon" title="c19">o</font>,<font color="Maroon" title="c2">X</font>)</span>)</span> <a href="trees_4.html#K8" title="TREES_4:func.8">-tree</a> <font color="Maroon" title="c26">p</font> <a href="tarski.html#R2" title="TARSKI: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#K11" title="MSAFREE:func.11">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#D2" onmouseover="rs('msualg_1/D2')" onmouseout="rh()">MSUALG_1:def 2</a></span>;<br/>
<a NAME="E21:48_5"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">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#K11" title="MSAFREE:func.11">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#D2" onmouseover="rs('partfun1/D2')" onmouseout="rh()">PARTFUN1:def 2</a></span>;<br/>
<a NAME="E22:48_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A79</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#K11" title="MSAFREE:func.11">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="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">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#K11" title="MSAFREE:func.11">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#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</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#K5" title="MSAFREE:func.5">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" onmouseover="rs('funct_2/D1')" onmouseout="rh()">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="xtuple_0.html#K10" title="XTUPLE_0:func.10">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#K11" title="MSAFREE:func.11">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#T11" target="_self" onmouseover="rs('msafree/T11')" onmouseout="rh()">Th11</a></span>
;<br/>
<a NAME="E24:48_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A80</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#K11" title="MSAFREE:func.11">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="#E14:48_5"><span class="lab"><font color="Green" title="E44">A76</font></span></a>, <a class="txt" href="#E22:48_5"><span class="lab"><font color="Green" title="E47">A79</font></span></a>, <a class="ref" href="relat_1.html#T62" onmouseover="rs('relat_1/T62')" onmouseout="rh()">RELAT_1:62</a>, <a class="ref" href="zfmisc_1.html#T74" onmouseover="rs('zfmisc_1/T74')" onmouseout="rh()">ZFMISC_1:74</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#K11" title="MSAFREE:func.11">FreeMSA</a> <font color="Maroon" title="c2">X</font></span>)</span>) = 
<span class="p1">(<span class="default"><a href="msafree.html#K10" title="MSAFREE:func.10">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#D6" onmouseover="rs('msualg_1/D6')" onmouseout="rh()">MSUALG_1:def 6</a></span>
<br/>.= 
 <a href="msafree.html#K9" title="MSAFREE:func.9">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#D13" target="_self" onmouseover="rs('msafree/D13')" onmouseout="rh()">Def13</a></span>
;<br/>
<a NAME="E26:48_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#K11" title="MSAFREE:func.11">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#K6" title="MSAFREE:func.6">Sym</a> (<font color="Maroon" title="c19">o</font>,<font color="Maroon" title="c2">X</font>)</span>)</span> <a href="trees_4.html#K8" title="TREES_4:func.8">-tree</a> <font color="Maroon" title="c26">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:48_5"><span class="lab"><font color="Green" title="E38">A61</font></span></a>, <a class="ref" href="msafree.html#D12" target="_self" onmouseover="rs('msafree/D12')" onmouseout="rh()">Def12</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#K11" title="MSAFREE:func.11">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#K6" title="MSAFREE:func.6">Sym</a> (<font color="Maroon" title="c19">o</font>,<font color="Maroon" title="c2">X</font>)</span>)</span> <a href="trees_4.html#K8" title="TREES_4:func.8">-tree</a> <font color="Maroon" title="c26">p</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:48_5"><span class="lab"><font color="Green" title="E46">A78</font></span></a>, <a class="txt" href="#E14:48_5"><span class="lab"><font color="Green" title="E44">A76</font></span></a>, <a class="txt" href="#E24:48_5"><span class="lab"><font color="Green" title="E48">A80</font></span></a>, <a class="ref" href="funct_1.html#T47" onmouseover="rs('funct_1/T47')" onmouseout="rh()">FUNCT_1:47</a></span>
<br/>.= 
 <a href="msafree.html#K18" title="MSAFREE:func.18">pi</a> (<span class="p1">(<span class="default"><a href="msafree.html#K17" title="MSAFREE:func.17">@</a> (<font color="Maroon" title="c2">X</font>,<span class="p2">(<span class="default"><a href="msafree.html#K6" title="MSAFREE:func.6">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="#E3:48"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="txt" href="#E7:48_5"><span class="lab"><font color="Green" title="E38">A61</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#K5" title="MSUALG_3:func.5">#</a> <font color="Maroon" title="c20">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:48_5"><span class="lab"><font color="Green" title="E39">A62</font></span></a>, <a class="txt" href="#E13:48_5"><span class="lab"><font color="Green" title="E43">A75</font></span></a>, <a class="ref" href="msafree.html#D21" target="_self" onmouseover="rs('msafree/D21')" onmouseout="rh()">Def21</a></span>
;<br/> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<a NAME="E15:48"/>
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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#K13" title="MSAFREE:func.13">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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="48_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#K13" title="MSAFREE:func.13">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">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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#K13" title="MSAFREE:func.13">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="E1:48_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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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#K13" title="MSAFREE:func.13">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="E3:48_6"/><span class="lab"><font color="Green" title="E35">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> <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#K11" title="MSAFREE:func.11">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" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E4:48_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"><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#K13" title="MSAFREE:func.13">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#K13" title="MSAFREE:func.13">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" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E5:48_6"/><span class="lab"><font color="Green" title="E37">A83</font></span>: 
<span class="p1">(<span class="default"><a href="msafree.html#K13" title="MSAFREE:func.13">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#K12" title="MSAFREE:func.12">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#D16" target="_self" onmouseover="rs('msafree/D16')" onmouseout="rh()">Def16</a></span>;<br/>
<a NAME="E6:48_6"/><span class="lab"><font color="Green" title="E38">A84</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#K13" title="MSAFREE:func.13">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#K13" title="MSAFREE:func.13">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" onmouseover="rs('msafree/D1')" onmouseout="rh()">Def1</a></span>;<br/>
<a NAME="E7:48_6"/><span class="lab"><font color="Green" title="E39">A85</font></span>: 
 for <font color="Olive" title="b1">a</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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#K13" title="MSAFREE:func.13">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#K13" title="MSAFREE:func.13">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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="48_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">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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#K13" title="MSAFREE:func.13">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#K13" title="MSAFREE:func.13">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="E1:48_6_1"/><span class="lab"><font color="Green" title="E40">A86</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#K11" title="MSAFREE:func.11">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="#E5:48"><span class="lab"><font color="Green" title="E31">A3</font></span></a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:48_6_1"/><span class="lab"><font color="Green" title="E41">A87</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#K13" title="MSAFREE:func.13">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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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#K13" title="MSAFREE:func.13">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="E3:48_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#K5" title="MSAFREE:func.5">DTConMSA</a> <font color="Maroon" title="c2">X</font></span>)</span> : ( <font color="Olive" title="b1">t</font> <a href="tarski.html#R2" title="TARSKI: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#K5" title="MSAFREE:func.5">DTConMSA</a> <font color="Maroon" title="c2">X</font></span>)</span> &amp; <font color="Olive" title="b1">t</font> <a href="xtuple_0.html#K2" title="XTUPLE_0: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="#E5:48_6"><span class="lab"><font color="Green" title="E37">A83</font></span></a>, <a class="ref" href="msafree.html#T13" target="_self" onmouseover="rs('msafree/T13')" onmouseout="rh()">Th13</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#K5" title="MSAFREE:func.5">DTConMSA</a> <font color="Maroon" title="c2">X</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E5:48_6_1"/><span class="lab"><font color="Green" title="E42">A88</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="tarski.html#R2" title="TARSKI: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#K5" title="MSAFREE:func.5">DTConMSA</a> <font color="Maroon" title="c2">X</font></span>)</span> )
 <span class="kw">and </span><br/><a NAME="E6:48_6_1"/><span class="lab"><font color="Green" title="E43">A89</font></span>: 
<font color="Maroon" title="c23">t</font> <a href="xtuple_0.html#K2" title="XTUPLE_0: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#K13" title="MSAFREE:func.13">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="#E6:48_6"><span class="lab"><font color="Green" title="E38">A84</font></span></a>, <a class="txt" href="#E4:48_6"><span class="lab"><font color="Green" title="E36">A82</font></span></a>, <a class="txt" href="#E2:48_6_1"><span class="lab"><font color="Green" title="E41">A87</font></span></a>, <a class="ref" href="funct_1.html#T47" onmouseover="rs('funct_1/T47')" onmouseout="rh()">FUNCT_1:47</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="#E3:48_6"><span class="lab"><font color="Green" title="E35">A81</font></span></a>, <a class="txt" href="#E5:48_6"><span class="lab"><font color="Green" title="E37">A83</font></span></a>, <a class="txt" href="#E2:48_6_1"><span class="lab"><font color="Green" title="E41">A87</font></span></a>, <a class="txt" href="#E1:48_6_1"><span class="lab"><font color="Green" title="E40">A86</font></span></a>, <a class="ref" href="funct_1.html#T47" onmouseover="rs('funct_1/T47')" onmouseout="rh()">FUNCT_1:47</a></span>
<br/>.= 
 <a href="msafree.html#K16" title="MSAFREE:func.16">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="#E2:48"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E5:48_6_1"><span class="lab"><font color="Green" title="E42">A88</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="#E5:48_6_1"><span class="lab"><font color="Green" title="E42">A88</font></span></a>, <a class="txt" href="#E6:48_6_1"><span class="lab"><font color="Green" title="E43">A89</font></span></a>, <a class="ref" href="msafree.html#D19" target="_self" onmouseover="rs('msafree/D19')" onmouseout="rh()">Def19</a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E8:48_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#K13" title="MSAFREE:func.13">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" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:48_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#K13" title="MSAFREE:func.13">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="#E4:48_6"><span class="lab"><font color="Green" title="E36">A82</font></span></a>, <a class="txt" href="#E7:48_6"><span class="lab"><font color="Green" title="E39">A85</font></span></a>, <a class="ref" href="funct_1.html#T2" onmouseover="rs('funct_1/T2')" onmouseout="rh()">FUNCT_1:2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E16:48"/>
<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#K13" title="MSAFREE:func.13">FreeGen</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Maroon" title="c7">F</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
