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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><br/>  for <font color="Olive" title="b2">o</font> being   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font><br/>  for <font color="Olive" title="b3">p</font> being    <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <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="Olive" title="b1">X</font></span>)</span> holds <br/> (  <a href="msafree.html#K6" title="MSAFREE:func.6">Sym</a> (<font color="Olive" title="b2">o</font>,<font color="Olive" title="b1">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="Olive" title="b3">p</font> iff <font color="Olive" title="b3">p</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="Olive" title="b1">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="Olive" title="b2">o</font> )</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">  for <font color="Olive" title="b1">o</font> being   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font><br/>  for <font color="Olive" title="b2">p</font> being    <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <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/> (  <a href="msafree.html#K6" title="MSAFREE:func.6">Sym</a> (<font color="Olive" title="b1">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="Olive" title="b2">p</font> iff <font color="Olive" title="b2">p</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="Olive" title="b1">o</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">o</font> be   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</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">  for <font color="Olive" title="b1">p</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> holds <br/> (  <a href="msafree.html#K6" title="MSAFREE:func.6">Sym</a> (<font color="Maroon" title="c3">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="Olive" title="b1">p</font> iff <font color="Olive" title="b1">p</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="c3">o</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">p</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"> (  <a href="msafree.html#K6" title="MSAFREE:func.6">Sym</a> (<font color="Maroon" title="c3">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="c4">p</font> iff <font color="Maroon" title="c4">p</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="c3">o</font> )</span><br/>







<span class="kw">set </span><font color="Maroon" title="c5">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="c6">ar</font> =  <a href="msualg_1.html#K1" title="MSUALG_1:func.1">the_arity_of</a> <font color="Maroon" title="c3">o</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">r</font> =  <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c4">p</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">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/>
<a NAME="E1:30"/><span class="lab"><font color="Green" title="E16">A1</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c4">p</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="trees_3.html#D18" onmouseover="rs('trees_3/D18')" onmouseout="rh()">TREES_3:def 18</a></span>;<br/>
<span class="kw">thus </span><a NAME="E2:30"/>
(  <a href="msafree.html#K6" title="MSAFREE:func.6">Sym</a> (<font color="Maroon" title="c3">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="c4">p</font> implies <font color="Maroon" title="c4">p</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="c3">o</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="c4">p</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="c3">o</font> implies  <a href="msafree.html#K6" title="MSAFREE:func.6">Sym</a> (<font color="Maroon" title="c3">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="c4">p</font> )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="30_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:30_1"/>
 <a href="msafree.html#K6" title="MSAFREE:func.6">Sym</a> (<font color="Maroon" title="c3">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="c4">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"> <font color="Maroon" title="c4">p</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="c3">o</font></span><br/>

<a NAME="E2:30_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A2</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="c3">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="c4">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>  <a href="msafree.html#K4" title="MSAFREE:func.4">REL</a> <font color="Maroon" title="c2">X</font>
 
<span class="kw">by</span> <span class="lab"><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="c9">r</font> =  <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c4">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="E4:30_1"/><span class="lab"><font color="Green" title="E18">A3</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">p</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="c9">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="trees_3.html#D18" onmouseover="rs('trees_3/D18')" onmouseout="rh()">TREES_3:def 18</a></span>;<br/>
<a NAME="E5:30_1"/><span class="lab"><font color="Green" title="E19">A4</font></span>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c9">r</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="c9">r</font></span>)</span> &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="c3">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="c3">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></span>;<br/>
<a NAME="E6:30_1"/><span class="lab"><font color="Green" title="E20">A5</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c9">r</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="c3">o</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30_1"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="ref" href="msafree.html#T5" target="_self" onmouseover="rs('msafree/T5')" onmouseout="rh()">Th5</a></span>;<br/>
<a NAME="E7:30_1"/>
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">p</font> holds <br/><font color="Maroon" title="c4">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="msafree.html#K7" title="MSAFREE:func.7">FreeSort</a> (<font color="Maroon" title="c2">X</font>,<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="c3">o</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font></span>)</span>)
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="30_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</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="c10">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">p</font> implies <font color="Maroon" title="c4">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="msafree.html#K7" title="MSAFREE:func.7">FreeSort</a> (<font color="Maroon" title="c2">X</font>,<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="c3">o</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">n</font></span>)</span>) )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c11">s</font> = <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="c3">o</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">n</font>;<br/>
<span class="kw">assume </span><a NAME="E1:30_1_1"/><span class="lab"><font color="Green" title="E21">A6</font></span>: 
<font color="Maroon" title="c10">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">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"> <font color="Maroon" title="c4">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="msafree.html#K7" title="MSAFREE:func.7">FreeSort</a> (<font color="Maroon" title="c2">X</font>,<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="c3">o</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">n</font></span>)</span>)</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">T</font> being   <a href="trees_2.html#NM4" title="TREES_2:NM.4">DecoratedTree</a><span class="kw"> such that </span><br/><a NAME="E3:30_1_1"/><span class="lab"><font color="Green" title="E22">A7</font></span>: 
<font color="Maroon" title="c12">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font>
 <span class="kw">and </span><br/><a NAME="E4:30_1_1"/><span class="lab"><font color="Green" title="E23">A8</font></span>: 
<font color="Maroon" title="c9">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">T</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="trees_3.html#D18" onmouseover="rs('trees_3/D18')" onmouseout="rh()">TREES_3:def 18</a></span>;<br/>
<a NAME="E5:30_1_1"/>
(  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c4">p</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> &amp; <font color="Maroon" title="c4">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">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="c4">p</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1_1"><span class="lab"><font color="Green" title="E21">A6</font></span></a>, <a class="ref" href="finseq_1.html#D4" onmouseover="rs('finseq_1/D4')" onmouseout="rh()">FINSEQ_1:def 4</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="c13">T</font> = <font color="Maroon" title="c12">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="#E3:30_1_1"><span class="lab"><font color="Green" title="E22">A7</font></span></a></span>;<br/>
<a NAME="E7:30_1_1"/><span class="lab"><font color="Green" title="E24">A9</font></span>: 
(  <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c9">r</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"> the <a href="struct_0.html#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> &amp; <font color="Maroon" title="c9">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">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="c9">r</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:30_1"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E1:30_1_1"><span class="lab"><font color="Green" title="E21">A6</font></span></a>, <a class="ref" href="finseq_1.html#D4" onmouseover="rs('finseq_1/D4')" onmouseout="rh()">FINSEQ_1:def 4</a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:30_1_1_1"/>
( <font color="Maroon" title="c9">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"> the <a href="struct_0.html#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> or <font color="Maroon" title="c9">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<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 class="kw">by</span> <span class="lab"><a class="txt" href="#E7:30_1_1"><span class="lab"><font color="Green" title="E24">A9</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:30_1_1_1_1"/><span class="lab"><font color="Green" title="E25">A10</font></span>: 
<font color="Maroon" title="c9">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"> the <a href="struct_0.html#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 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="c4">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="msafree.html#K7" title="MSAFREE:func.7">FreeSort</a> (<font color="Maroon" title="c2">X</font>,<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="c3">o</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">n</font></span>)</span>)</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">o1</font> being   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font>, <font color="Maroon" title="c15">x2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span><span class="kw"> such that </span><br/><a NAME="E3:30_1_1_1_1"/><span class="lab"><font color="Green" title="E26">A11</font></span>: 
<font color="Maroon" title="c9">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</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="c14">o1</font>,<font color="Maroon" title="c15">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="domain_1.html#T1" onmouseover="rs('domain_1/T1')" onmouseout="rh()">DOMAIN_1:1</a></span>;<br/><a NAME="E4:30_1_1_1_1"/><span class="lab"><font color="Green" title="E27">A12</font></span>: 
<font color="Maroon" title="c15">x2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><span class="kw">then </span> <a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c14">o1</font> = 
<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="c3">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30_1"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="txt" href="#E4:30_1"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E1:30_1_1"><span class="lab"><font color="Green" title="E21">A6</font></span></a>, <a class="txt" href="#E1:30_1_1_1_1"><span class="lab"><font color="Green" title="E25">A10</font></span></a>, <a class="txt" href="#E3:30_1_1_1_1"><span class="lab"><font color="Green" title="E26">A11</font></span></a>, <a class="ref" href="msafree.html#T5" target="_self" onmouseover="rs('msafree/T5')" onmouseout="rh()">Th5</a></span>
<br/>.= 
<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="c3">o</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">n</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:30_1"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="txt" href="#E4:30_1"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E5:30_1"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="txt" href="#E1:30_1_1"><span class="lab"><font color="Green" title="E21">A6</font></span></a>, <a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>
;<br/><a NAME="E6:30_1_1_1_1"/><span class="kw">then </span>
(  ex <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  st <br/>( <font color="Olive" title="b1">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> <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="c3">o</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">n</font></span>)</span> &amp; <font color="Maroon" title="c13">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<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="c3">o</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">n</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) or  ex <font color="Olive" title="b1">o</font> being   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font> st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">T</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> &amp;  <a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Olive" title="b1">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c3">o</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">n</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:30_1_1"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="txt" href="#E3:30_1_1_1_1"><span class="lab"><font color="Green" title="E26">A11</font></span></a>, <a class="txt" href="#E4:30_1_1_1_1"><span class="lab"><font color="Green" title="E27">A12</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E7:30_1_1_1_1"/>
<font color="Maroon" title="c4">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="msafree.html#K7" title="MSAFREE:func.7">FreeSort</a> (<font color="Maroon" title="c2">X</font>,<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="c3">o</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">n</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:30_1_1"><span class="lab"><font color="Green" title="E22">A7</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:30_1_1_1_2"/><span class="lab"><font color="Green" title="E25">A13</font></span>: 
<font color="Maroon" title="c9">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="msafree.html#K3" title="MSAFREE:func.3">coprod</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="c4">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="msafree.html#K7" title="MSAFREE:func.7">FreeSort</a> (<font color="Maroon" title="c2">X</font>,<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="c3">o</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">n</font></span>)</span>)</span><br/><div class="add"><a NAME="E2:30_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="msafree.html#K2" title="MSAFREE:func.2">coprod</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="c3">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font></span>)</span>,<font color="Maroon" title="c2">X</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30_1"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="txt" href="#E4:30_1"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E1:30_1_1"><span class="lab"><font color="Green" title="E21">A6</font></span></a>, <a class="ref" href="msafree.html#T5" target="_self" onmouseover="rs('msafree/T5')" onmouseout="rh()">Th5</a></span>;<br/><a NAME="E3:30_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="msafree.html#K2" title="MSAFREE:func.2">coprod</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="c3">o</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">n</font></span>)</span>,<font color="Maroon" title="c2">X</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:30_1"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="txt" href="#E4:30_1"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E5:30_1"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="txt" href="#E1:30_1_1"><span class="lab"><font color="Green" title="E21">A6</font></span></a>, <a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>;<br/><a NAME="E4:30_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A14</font></span>: 
 ex <font color="Olive" title="b1">a</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  st <br/>( <font color="Olive" title="b1">a</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> <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="c3">o</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">n</font></span>)</span> &amp; <font color="Maroon" title="c9">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</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="Olive" title="b1">a</font>,<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="c3">o</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">n</font></span>)</span></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#D2" target="_self" onmouseover="rs('msafree/D2')" onmouseout="rh()">Def2</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c14">t</font> = <font color="Maroon" title="c9">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font> as   <a href="dtconstr.html#NM2" title="DTCONSTR:NM.2">Terminal</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="#E1:30_1_1_1_2"><span class="lab"><font color="Green" title="E25">A13</font></span></a>, <a class="ref" href="msafree.html#T6" target="_self" onmouseover="rs('msafree/T6')" onmouseout="rh()">Th6</a></span>;<br/><a NAME="E6:30_1_1_1_2"/>
<font color="Maroon" title="c13">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dtconstr.html#K8" title="DTCONSTR:func.8">root-tree</a> <font color="Maroon" title="c14">t</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:30_1_1"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="ref" href="dtconstr.html#T9" onmouseover="rs('dtconstr/T9')" onmouseout="rh()">DTCONSTR:9</a></span>;<br/><span class="kw">hence </span><a NAME="E7:30_1_1_1_2"/>
<font color="Maroon" title="c4">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="msafree.html#K7" title="MSAFREE:func.7">FreeSort</a> (<font color="Maroon" title="c2">X</font>,<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="c3">o</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">n</font></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:30_1_1"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="txt" href="#E4:30_1_1_1_2"><span class="lab"><font color="Green" title="E26">A14</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></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E8:30_1"/>
<font color="Maroon" title="c4">p</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="c3">o</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:30_1"><span class="lab"><font color="Green" title="E20">A5</font></span></a>, <a class="txt" href="#E4:30_1"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="txt" href="#E5:30_1"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="ref" href="msafree.html#T9" target="_self" onmouseover="rs('msafree/T9')" onmouseout="rh()">Th9</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="E3:30"/><span class="lab"><font color="Green" title="E17">A15</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c4">p</font></span>)</span> <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> <span class="p2">(<span class="default"><a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c4">p</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></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">r</font> =  <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c4">p</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"> the <a href="struct_0.html#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">reconsider </span><font color="Maroon" title="c10">r</font> = <font color="Maroon" title="c9">r</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="finseq_1.html#D11" onmouseover="rs('finseq_1/D11')" onmouseout="rh()">FINSEQ_1:def 11</a></span>;<br/>
<span class="kw">assume </span><a NAME="E6:30"/><span class="lab"><font color="Green" title="E18">A16</font></span>: 
<font color="Maroon" title="c4">p</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="c3">o</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#K6" title="MSAFREE:func.6">Sym</a> (<font color="Maroon" title="c3">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="c4">p</font></span><br/>

<a NAME="E7:30"/><span class="kw">then </span><span class="lab"><font color="Green" title="E19">A17</font></span>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">p</font> <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="c3">o</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msafree.html#T9" target="_self" onmouseover="rs('msafree/T9')" onmouseout="rh()">Th9</a></span>;<br/>
<a NAME="E8:30"/><span class="lab"><font color="Green" title="E20">A18</font></span>: 
 <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="msafree.html#K3" title="MSAFREE:func.3">coprod</a> <font color="Maroon" title="c2">X</font></span>)</span> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"> the <a href="struct_0.html#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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msafree.html#T4" target="_self" onmouseover="rs('msafree/T4')" onmouseout="rh()">Th4</a></span>;<br/>
<a NAME="E9:30"/><span class="lab"><font color="Green" title="E21">A19</font></span>: 
 for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c10">r</font> holds <br/>( ( <font color="Maroon" title="c10">r</font> <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"><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> implies  for <font color="Olive" title="b2">o1</font> being   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">o1</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> holds <br/> <a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Olive" title="b2">o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c3">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> ) &amp; ( <font color="Maroon" title="c10">r</font> <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>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="msafree.html#K3" title="MSAFREE:func.3">coprod</a> <font color="Maroon" title="c2">X</font></span>)</span> implies <font color="Maroon" title="c10">r</font> <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>  <a href="msafree.html#K2" title="MSAFREE:func.2">coprod</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="c3">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font></span>)</span>,<font color="Maroon" title="c2">X</font>) ) )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="30_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">x</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</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="c11">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c10">r</font> implies ( ( <font color="Maroon" title="c10">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"> the <a href="struct_0.html#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> implies  for <font color="Olive" title="b1">o1</font> being   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">o1</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> holds <br/> <a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Olive" title="b1">o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c3">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> ) &amp; ( <font color="Maroon" title="c10">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="msafree.html#K3" title="MSAFREE:func.3">coprod</a> <font color="Maroon" title="c2">X</font></span>)</span> implies <font color="Maroon" title="c10">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="msafree.html#K2" title="MSAFREE:func.2">coprod</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="c3">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font></span>)</span>,<font color="Maroon" title="c2">X</font>) ) ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:30_2"/><span class="lab"><font color="Green" title="E22">A20</font></span>: 
<font color="Maroon" title="c11">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c10">r</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="c10">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"> the <a href="struct_0.html#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> implies  for <font color="Olive" title="b1">o1</font> being   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">o1</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> holds <br/> <a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Olive" title="b1">o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c3">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> ) &amp; ( <font color="Maroon" title="c10">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="msafree.html#K3" title="MSAFREE:func.3">coprod</a> <font color="Maroon" title="c2">X</font></span>)</span> implies <font color="Maroon" title="c10">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="msafree.html#K2" title="MSAFREE:func.2">coprod</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="c3">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font></span>)</span>,<font color="Maroon" title="c2">X</font>) ) )</span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">n</font> = <font color="Maroon" title="c11">x</font> as   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> ;<br/>
<a NAME="E3:30_2"/><span class="lab"><font color="Green" title="E23">A21</font></span>: 
 ex <font color="Olive" title="b1">T</font> being   <a href="trees_2.html#NM4" title="TREES_2:NM.4">DecoratedTree</a> st <br/>( <font color="Olive" title="b1">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">n</font> &amp; <font color="Maroon" title="c10">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">T</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="txt" href="#E1:30_2"><span class="lab"><font color="Green" title="E22">A20</font></span></a>, <a class="ref" href="trees_3.html#D18" onmouseover="rs('trees_3/D18')" onmouseout="rh()">TREES_3:def 18</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c13">s</font> = <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="c3">o</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">n</font>;<br/>
<a NAME="E4:30_2"/>
<font color="Maroon" title="c4">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">n</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="msafree.html#K7" title="MSAFREE:func.7">FreeSort</a> (<font color="Maroon" title="c2">X</font>,<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="c3">o</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">n</font></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:30"><span class="lab"><font color="Green" title="E18">A16</font></span></a>, <a class="txt" href="#E1:30"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="txt" href="#E1:30_2"><span class="lab"><font color="Green" title="E22">A20</font></span></a>, <a class="ref" href="msafree.html#T9" target="_self" onmouseover="rs('msafree/T9')" onmouseout="rh()">Th9</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">a</font> being    <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"> such that </span><br/><a NAME="E6:30_2"/><span class="lab"><font color="Green" title="E24">A22</font></span>: 
<font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">n</font>
 <span class="kw">and </span><br/><a NAME="E7:30_2"/><span class="lab"><font color="Green" title="E25">A23</font></span>: 
(  ex <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  st <br/>( <font color="Olive" title="b1">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> <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="c3">o</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">n</font></span>)</span> &amp; <font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<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="c3">o</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">n</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> ) or  ex <font color="Olive" title="b1">o</font> being   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font> st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> &amp;  <a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Olive" title="b1">o</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c3">o</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">n</font> ) )
 ;<br/>
<a NAME="E8:30_2"/><span class="lab"><font color="Green" title="E26">A24</font></span>: 
<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="c3">o</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c3">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:30"><span class="lab"><font color="Green" title="E19">A17</font></span></a>, <a class="txt" href="#E1:30"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="txt" href="#E1:30_2"><span class="lab"><font color="Green" title="E22">A20</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">thus </span><a NAME="E9:30_2"/>
( <font color="Maroon" title="c10">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"> the <a href="struct_0.html#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> implies  for <font color="Olive" title="b1">o1</font> being   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">o1</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> holds <br/> <a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Olive" title="b1">o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c3">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</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="c10">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="msafree.html#K3" title="MSAFREE:func.3">coprod</a> <font color="Maroon" title="c2">X</font></span>)</span> implies <font color="Maroon" title="c10">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="msafree.html#K2" title="MSAFREE:func.2">coprod</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="c3">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font></span>)</span>,<font color="Maroon" title="c2">X</font>) )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="30_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:30_2_1"/><span class="lab"><font color="Green" title="E27">A25</font></span>: 
<font color="Maroon" title="c10">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"> the <a href="struct_0.html#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 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">o1</font> being   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font>  st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">o1</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> holds <br/> <a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Olive" title="b1">o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c3">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font></span><br/>

<div><span class="lab"><font color="Green" title="E28">A26</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="30_2_1_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">  for <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   holds <br/>(  not <font color="Olive" title="b1">y</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> <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="c3">o</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">n</font></span>)</span> or  not <font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">y</font>,<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="c3">o</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">n</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )</span></a><div class="add"><a NAME="E1:30_2_1_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="msafree.html#K3" title="MSAFREE:func.3">coprod</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="E2:30_2_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="msafree.html#K3" title="MSAFREE:func.3">coprod</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="c3">o</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">n</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="msafree.html#K3" title="MSAFREE:func.3">coprod</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="E3:30_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A27</font></span>: 
 <a href="msafree.html#K2" title="MSAFREE:func.2">coprod</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="c3">o</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">n</font></span>)</span>,<font color="Maroon" title="c2">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="msafree.html#K3" title="MSAFREE:func.3">coprod</a> <font color="Maroon" title="c2">X</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="msafree.html#D3" target="_self" onmouseover="rs('msafree/D3')" onmouseout="rh()">Def3</a></span>;<br/><span class="kw">given </span><font color="Maroon" title="c15">y</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><a NAME="E5:30_2_1_1"/><span class="lab"><font color="Green" title="E30">A28</font></span>: 
( <font color="Maroon" title="c15">y</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> <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="c3">o</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">n</font></span>)</span> &amp; <font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">y</font>,<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="c3">o</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">n</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.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"> contradiction</span><br/><a NAME="E6:30_2_1_1"/>
( <font color="Maroon" title="c10">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</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="c15">y</font>,<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="c3">o</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">n</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">y</font>,<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="c3">o</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">n</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msafree.html#K2" title="MSAFREE:func.2">coprod</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="c3">o</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">n</font></span>)</span>,<font color="Maroon" title="c2">X</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:30_2"><span class="lab"><font color="Green" title="E24">A22</font></span></a>, <a class="txt" href="#E3:30_2"><span class="lab"><font color="Green" title="E23">A21</font></span></a>, <a class="txt" href="#E5:30_2_1_1"><span class="lab"><font color="Green" title="E30">A28</font></span></a>, <a class="ref" href="msafree.html#D2" target="_self" onmouseover="rs('msafree/D2')" onmouseout="rh()">Def2</a>, <a class="ref" href="trees_4.html#T3" onmouseover="rs('trees_4/T3')" onmouseout="rh()">TREES_4:3</a></span>;<br/><a NAME="E7:30_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <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> <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:30_2_1_1"><span class="lab"><font color="Green" title="E29">A27</font></span></a>, <a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>;<br/><a NAME="E8:30_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="msafree.html#K3" title="MSAFREE:func.3">coprod</a> <font color="Maroon" title="c2">X</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_3.html#D4" onmouseover="rs('card_3/D4')" onmouseout="rh()">CARD_3:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E9:30_2_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:30"><span class="lab"><font color="Green" title="E20">A18</font></span></a>, <a class="txt" href="#E1:30_2_1"><span class="lab"><font color="Green" title="E27">A25</font></span></a>, <a class="ref" href="xboole_0.html#T3" onmouseover="rs('xboole_0/T3')" onmouseout="rh()">XBOOLE_0:3</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">let </span><font color="Maroon" title="c15">o1</font> be   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</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"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">o1</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> implies  <a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c15">o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c3">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> )</span><br/>

<span class="kw">assume </span><a NAME="E3:30_2_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">o1</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</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="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c15">o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c3">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font></span><br/>

<span class="kw">hence </span><a NAME="E4:30_2_1"/>
 <a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c15">o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c3">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:30_2"><span class="lab"><font color="Green" title="E24">A22</font></span></a>, <a class="txt" href="#E7:30_2"><span class="lab"><font color="Green" title="E25">A23</font></span></a>, <a class="txt" href="#E3:30_2"><span class="lab"><font color="Green" title="E23">A21</font></span></a>, <a class="txt" href="#E8:30_2"><span class="lab"><font color="Green" title="E26">A24</font></span></a>, <a class="txt" href="#E2:30_2_1"><span class="lab"><font color="Green" title="E28">A26</font></span></a>, <a class="ref" href="xtuple_0.html#T1" onmouseover="rs('xtuple_0/T1')" onmouseout="rh()">XTUPLE_0:1</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">assume </span><a NAME="E10:30_2"/><span class="lab"><font color="Green" title="E27">A29</font></span>: 
<font color="Maroon" title="c10">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</a> <span class="p1">(<span class="default"><a href="msafree.html#K3" title="MSAFREE:func.3">coprod</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="c10">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="msafree.html#K2" title="MSAFREE:func.2">coprod</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="c3">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font></span>)</span>,<font color="Maroon" title="c2">X</font>)</span><br/>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="30_2_2"><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">  for <font color="Olive" title="b1">o1</font> being   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font>  holds <br/>(  not <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">o1</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> or  not  <a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Olive" title="b1">o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c3">o</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">n</font> )</span></a><div class="add"><span class="kw">given </span><font color="Maroon" title="c15">o1</font> being   <a href="msualg_1.html#NM2" title="MSUALG_1:NM.2">OperSymbol</a> of <font color="Maroon" title="c1">S</font><span class="kw"> such that </span><a NAME="E2:30_2_2"/><span class="lab"><font color="Green" title="E28">A30</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">o1</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">a</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>
 <span class="kw">and </span><br/><a NAME="E3:30_2_2"/>
 <a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c15">o1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c3">o</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">n</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/><a NAME="E4:30_2_2"/>
 the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/><a NAME="E5:30_2_2"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">o1</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> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"> the <a href="struct_0.html#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>
 <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/><span class="kw">hence </span><a NAME="E6:30_2_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:30"><span class="lab"><font color="Green" title="E20">A18</font></span></a>, <a class="txt" href="#E6:30_2"><span class="lab"><font color="Green" title="E24">A22</font></span></a>, <a class="txt" href="#E3:30_2"><span class="lab"><font color="Green" title="E23">A21</font></span></a>, <a class="txt" href="#E10:30_2"><span class="lab"><font color="Green" title="E27">A29</font></span></a>, <a class="txt" href="#E2:30_2_2"><span class="lab"><font color="Green" title="E28">A30</font></span></a>, <a class="ref" href="xboole_0.html#T3" onmouseover="rs('xboole_0/T3')" onmouseout="rh()">XBOOLE_0:3</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">consider </span><font color="Maroon" title="c15">y</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E13:30_2"/><span class="lab"><font color="Green" title="E28">A31</font></span>: 
<font color="Maroon" title="c15">y</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> <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="c3">o</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">n</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E14:30_2"/><span class="lab"><font color="Green" title="E29">A32</font></span>: 
<font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_4.html#K1" title="TREES_4:func.1">root-tree</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">y</font>,<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="c3">o</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">n</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:30_2"><span class="lab"><font color="Green" title="E25">A23</font></span></a></span>;<br/>
<a NAME="E15:30_2"/>
<font color="Maroon" title="c10">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</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="c15">y</font>,<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="c3">o</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c12">n</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:30_2"><span class="lab"><font color="Green" title="E24">A22</font></span></a>, <a class="txt" href="#E3:30_2"><span class="lab"><font color="Green" title="E23">A21</font></span></a>, <a class="txt" href="#E14:30_2"><span class="lab"><font color="Green" title="E29">A32</font></span></a>, <a class="ref" href="trees_4.html#T3" onmouseover="rs('trees_4/T3')" onmouseout="rh()">TREES_4:3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:30_2"/>
<font color="Maroon" title="c10">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="msafree.html#K2" title="MSAFREE:func.2">coprod</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="c3">o</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">x</font></span>)</span>,<font color="Maroon" title="c2">X</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:30_2"><span class="lab"><font color="Green" title="E26">A24</font></span></a>, <a class="txt" href="#E13:30_2"><span class="lab"><font color="Green" title="E28">A31</font></span></a>, <a class="ref" href="msafree.html#D2" target="_self" onmouseover="rs('msafree/D2')" onmouseout="rh()">Def2</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="E10:30"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">r</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="c3">o</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:30"><span class="lab"><font color="Green" title="E19">A17</font></span></a>, <a class="txt" href="#E1:30"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="txt" href="#E3:30"><span class="lab"><font color="Green" title="E17">A15</font></span></a>, <a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E11:30"/><span class="kw">then </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="c3">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>,<font color="Maroon" title="c10">r</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msafree.html#K4" title="MSAFREE:func.4">REL</a> <font color="Maroon" title="c2">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:30"><span class="lab"><font color="Green" title="E21">A19</font></span></a>, <a class="ref" href="msafree.html#T5" target="_self" onmouseover="rs('msafree/T5')" onmouseout="rh()">Th5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:30"/>
 <a href="msafree.html#K6" title="MSAFREE:func.6">Sym</a> (<font color="Maroon" title="c3">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="c4">p</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lang1.html#D1" onmouseover="rs('lang1/D1')" onmouseout="rh()">LANG1:def 1</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>
