<?xml version="1.0"?>
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<span class="kw">let </span><font color="Maroon" title="c1">S</font> be   non  <a href="struct_0.html#V11" title="STRUCT_0:attr.11">void</a>  <a href="catalg_1.html#NM1" title="CATALG_1:NM.1">Signature</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#V3" title="RELAT_1:attr.3">V9</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">t</font> being   <a href="msafree3.html#NM1" title="MSAFREE3:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> (<font color="Maroon" title="c1">S</font>,<font color="Olive" title="b1">X</font>)</span>)</span><br/>  for <font color="Olive" title="b3">p</font> being    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Olive" title="b2">t</font> holds  <font color="Olive" title="b2">t</font> <a href="trees_2.html#K5" title="TREES_2:func.5">|</a> <font color="Olive" title="b3">p</font> is   <a href="msafree3.html#NM1" title="MSAFREE3:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> (<font color="Maroon" title="c1">S</font>,<font color="Olive" title="b1">X</font>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">X</font> be  <a href="relat_1.html#V3" title="RELAT_1:attr.3">V9</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">t</font> being   <a href="msafree3.html#NM1" title="MSAFREE3:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> (<font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">X</font>)</span>)</span><br/>  for <font color="Olive" title="b2">p</font> being    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Olive" title="b1">t</font> holds  <font color="Olive" title="b1">t</font> <a href="trees_2.html#K5" title="TREES_2:func.5">|</a> <font color="Olive" title="b2">p</font> is   <a href="msafree3.html#NM1" title="MSAFREE3:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> (<font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">X</font>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">t</font> be   <a href="msafree3.html#NM1" title="MSAFREE3:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> (<font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">X</font>)</span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">p</font> being    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c3">t</font> holds  <font color="Maroon" title="c3">t</font> <a href="trees_2.html#K5" title="TREES_2:func.5">|</a> <font color="Olive" title="b1">p</font> is   <a href="msafree3.html#NM1" title="MSAFREE3:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> (<font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">X</font>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c4">p</font> be    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c3">t</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">t</font> <a href="trees_2.html#K5" title="TREES_2:func.5">|</a> <font color="Maroon" title="c4">p</font> is   <a href="msafree3.html#NM1" title="MSAFREE3:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> (<font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">X</font>)</span>)</span></span><br/>







<span class="kw">set </span><font color="Maroon" title="c5">Y</font> = <font color="Maroon" title="c2">X</font> <a href="pboole.html#K2" title="PBOOLE:func.2">(\/)</a> <span class="p1">(<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> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">t</font> = <font color="Maroon" title="c3">t</font> as   <a href="msaterm.html#NM1" title="MSATERM:NM.1">Term</a> of <font color="Maroon" title="c1">S</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="pboole.html#K2" title="PBOOLE:func.2">(\/)</a> <span class="p2">(<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> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="msafree3.html#T8" target="_self" onmouseover="rs('msafree3/T8')" onmouseout="rh()">Th8</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">p</font> = <font color="Maroon" title="c4">p</font> as    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c6">t</font> ;<br/>
<a NAME="E3:44"/><span class="lab"><font color="Green" title="E33">A1</font></span>: 
 <a href="msafree3.html#K4" title="MSAFREE3:func.4">variables_in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="msaterm.html#K8" title="MSATERM:func.8">|</a> <font color="Maroon" title="c7">p</font></span>)</span> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a>  <a href="msafree3.html#K4" title="MSAFREE3:func.4">variables_in</a> <font color="Maroon" title="c6">t</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msafree3.html#T32" target="_self" onmouseover="rs('msafree3/T32')" onmouseout="rh()">Th32</a></span>;<br/>
<a NAME="E4:44"/><span class="lab"><font color="Green" title="E34">A2</font></span>: 
(  the <a href="msualg_1.html#U3" title="MSUALG_1:sel.3">Sorts</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> (<font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">X</font>)</span>)</span> <a href="pboole.html#R8" title="PBOOLE:pred.8">=</a> <font color="Maroon" title="c1">S</font> <a href="msafree3.html#K5" title="MSAFREE3:func.5">-Terms</a> (<font color="Maroon" title="c2">X</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="pboole.html#K2" title="PBOOLE:func.2">(\/)</a> <span class="p2">(<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> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>) &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="msafree3.html#K5" title="MSAFREE3:func.5">-Terms</a> (<font color="Maroon" title="c2">X</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="pboole.html#K2" title="PBOOLE:func.2">(\/)</a> <span class="p3">(<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> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>)</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="msafree3.html#T24" target="_self" onmouseover="rs('msafree3/T24')" onmouseout="rh()">Th24</a>, <a class="ref" href="partfun1.html#D2" onmouseover="rs('partfun1/D2')" onmouseout="rh()">PARTFUN1:def 2</a></span>;<br/>
<a NAME="E5:44"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> &amp; <font color="Maroon" title="c6">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="msafree3.html#K5" title="MSAFREE3:func.5">-Terms</a> (<font color="Maroon" title="c2">X</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="pboole.html#K2" title="PBOOLE:func.2">(\/)</a> <span class="p3">(<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> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_5.html#T2" onmouseover="rs('card_5/T2')" onmouseout="rh()">CARD_5:2</a></span>;<br/>
<a NAME="E6:44"/><span class="kw">then </span>
 <a href="msafree3.html#K4" title="MSAFREE3:func.4">variables_in</a> <font color="Maroon" title="c6">t</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c2">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msafree3.html#T17" target="_self" onmouseover="rs('msafree3/T17')" onmouseout="rh()">Th17</a></span>;<br/>
<a NAME="E7:44"/><span class="kw">then </span>
 <a href="msafree3.html#K4" title="MSAFREE3:func.4">variables_in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="msaterm.html#K8" title="MSATERM:func.8">|</a> <font color="Maroon" title="c7">p</font></span>)</span> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c2">X</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:44"><span class="lab"><font color="Green" title="E33">A1</font></span></a>, <a class="ref" href="pboole.html#T13" onmouseover="rs('pboole/T13')" onmouseout="rh()">PBOOLE:13</a></span>;<br/>
<a NAME="E8:44"/><span class="kw">then </span>
<font color="Maroon" title="c6">t</font> <a href="msaterm.html#K8" title="MSATERM:func.8">|</a> <font color="Maroon" title="c7">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">q</font> where <font color="Olive" title="b1">q</font> is   <a href="msaterm.html#NM1" title="MSATERM:NM.1">Term</a> of <font color="Maroon" title="c1">S</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="pboole.html#K2" title="PBOOLE:func.2">(\/)</a> <span class="p2">(<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> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p3"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span> : (  <a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="msaterm.html#K8" title="MSATERM:func.8">|</a> <font color="Maroon" title="c7">p</font></span>)</span> &amp;  <a href="msafree3.html#K4" title="MSAFREE3:func.4">variables_in</a> <font color="Olive" title="b1">q</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c2">X</font> ) </span> } </span> 
 
;<br/>
<a NAME="E9:44"/><span class="kw">then </span>
<font color="Maroon" title="c6">t</font> <a href="msaterm.html#K8" title="MSATERM:func.8">|</a> <font color="Maroon" title="c7">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="msafree3.html#K5" title="MSAFREE3:func.5">-Terms</a> (<font color="Maroon" title="c2">X</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="pboole.html#K2" title="PBOOLE:func.2">(\/)</a> <span class="p3">(<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> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p4"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span></span>)</span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">t</font> <a href="msaterm.html#K8" title="MSATERM:func.8">|</a> <font color="Maroon" title="c7">p</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msafree3.html#D5" target="_self" onmouseover="rs('msafree3/D5')" onmouseout="rh()">Def5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:44"/>
<font color="Maroon" title="c3">t</font> <a href="trees_2.html#K5" title="TREES_2:func.5">|</a> <font color="Maroon" title="c4">p</font> is   <a href="msafree3.html#NM1" title="MSAFREE3:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="msafree3.html#K1" title="MSAFREE3:func.1">Free</a> (<font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">X</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:44"><span class="lab"><font color="Green" title="E34">A2</font></span></a>, <a class="ref" href="card_5.html#T2" onmouseover="rs('card_5/T2')" onmouseout="rh()">CARD_5:2</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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