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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" 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="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">X</font> being   <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="b3">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="b4">p</font> being    <a href="msaterm.html#M1" title="MSATERM:mode.1">ArgumentSeq</a> of  <a href="msaterm.html#K2" title="MSATERM:func.2">Sym</a> (<font color="Olive" title="b3">o</font>,<font color="Olive" title="b1">Y</font>) holds <br/> ( <span class="p1">(<span class="default"><a href="msaterm.html#K2" title="MSATERM:func.2">Sym</a> (<font color="Olive" title="b3">o</font>,<font color="Olive" title="b1">Y</font>)</span>)</span> <a href="msaterm.html#K6" title="MSATERM:func.6">-tree</a> <font color="Olive" title="b4">p</font> <a href="hidden.html#R2" title="HIDDEN: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="Olive" title="b2">X</font>,<font color="Olive" title="b1">Y</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Olive" title="b3">o</font></span>)</span> iff  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b4">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</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="Olive" title="b2">X</font>,<font color="Olive" title="b1">Y</font>)</span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">Y</font> be  <a href="relat_1.html#V2" title="RELAT_1:attr.2">V5</a>() <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">X</font> being   <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="msaterm.html#M1" title="MSATERM:mode.1">ArgumentSeq</a> of  <a href="msaterm.html#K2" title="MSATERM:func.2">Sym</a> (<font color="Olive" title="b2">o</font>,<font color="Maroon" title="c2">Y</font>) holds <br/> ( <span class="p1">(<span class="default"><a href="msaterm.html#K2" title="MSATERM:func.2">Sym</a> (<font color="Olive" title="b2">o</font>,<font color="Maroon" title="c2">Y</font>)</span>)</span> <a href="msaterm.html#K6" title="MSATERM:func.6">-tree</a> <font color="Olive" title="b3">p</font> <a href="hidden.html#R2" title="HIDDEN: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="Olive" title="b1">X</font>,<font color="Maroon" title="c2">Y</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Olive" title="b2">o</font></span>)</span> iff  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b3">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</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="Olive" title="b1">X</font>,<font color="Maroon" title="c2">Y</font>)</span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">X</font> be   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="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="msaterm.html#M1" title="MSATERM:mode.1">ArgumentSeq</a> of  <a href="msaterm.html#K2" title="MSATERM:func.2">Sym</a> (<font color="Olive" title="b1">o</font>,<font color="Maroon" title="c2">Y</font>) holds <br/> ( <span class="p1">(<span class="default"><a href="msaterm.html#K2" title="MSATERM:func.2">Sym</a> (<font color="Olive" title="b1">o</font>,<font color="Maroon" title="c2">Y</font>)</span>)</span> <a href="msaterm.html#K6" title="MSATERM:func.6">-tree</a> <font color="Olive" title="b2">p</font> <a href="hidden.html#R2" title="HIDDEN: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="c3">X</font>,<font color="Maroon" title="c2">Y</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Olive" title="b1">o</font></span>)</span> iff  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b2">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</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="c3">X</font>,<font color="Maroon" title="c2">Y</font>)</span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">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" 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="msaterm.html#M1" title="MSATERM:mode.1">ArgumentSeq</a> of  <a href="msaterm.html#K2" title="MSATERM:func.2">Sym</a> (<font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c2">Y</font>) holds <br/> ( <span class="p1">(<span class="default"><a href="msaterm.html#K2" title="MSATERM:func.2">Sym</a> (<font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c2">Y</font>)</span>)</span> <a href="msaterm.html#K6" title="MSATERM:func.6">-tree</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN: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="c3">X</font>,<font color="Maroon" title="c2">Y</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c4">o</font></span>)</span> iff  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b1">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</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="c3">X</font>,<font color="Maroon" title="c2">Y</font>)</span>)</span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">p</font> be    <a href="msaterm.html#M1" title="MSATERM:mode.1">ArgumentSeq</a> of  <a href="msaterm.html#K2" title="MSATERM:func.2">Sym</a> (<font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c2">Y</font>); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="msaterm.html#K2" title="MSATERM:func.2">Sym</a> (<font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c2">Y</font>)</span>)</span> <a href="msaterm.html#K6" title="MSATERM:func.6">-tree</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN: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="c3">X</font>,<font color="Maroon" title="c2">Y</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c4">o</font></span>)</span> iff  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c5">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</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="c3">X</font>,<font color="Maroon" title="c2">Y</font>)</span>)</span> )</span><br/>









<span class="kw">set </span><font color="Maroon" title="c6">s</font> =  <a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c4">o</font>;<br/>
<a NAME="E1:31"/><span class="lab"><font color="Green" title="E21">A1</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="msafree3.html#K5" title="MSAFREE3:func.5">-Terms</a> (<font color="Maroon" title="c3">X</font>,<font color="Maroon" title="c2">Y</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" title="PARTFUN1:def.2">PARTFUN1:def 2</a></span>;<br/>
<a NAME="E2:31"/><span class="lab"><font color="Green" title="E22">A2</font></span>: 
 <a href="msaterm.html#K2" title="MSATERM:func.2">Sym</a> (<font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c2">Y</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="c4">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="kw">by</span> <span class="lab"><a class="ref" href="msafree.html#D9" title="MSAFREE:def.9">MSAFREE:def 9</a></span>;<br/>
<a NAME="E3:31"/><span class="lab"><font color="Green" title="E23">A3</font></span>: 
<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="c3">X</font>,<font color="Maroon" title="c2">Y</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c4">o</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">t</font> where <font color="Olive" title="b1">t</font> is   <a href="msaterm.html#NM1" title="MSATERM:NM.1">Term</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">Y</font> : (  <a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c4">o</font> &amp;  <a href="msafree3.html#K4" title="MSAFREE3:func.4">variables_in</a> <font color="Olive" title="b1">t</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c3">X</font> ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msafree3.html#D5" target="_self" title="MSAFREE3:def.5">Def6</a></span>;<br/>
<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c5">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</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="c3">X</font>,<font color="Maroon" title="c2">Y</font>)</span>)</span> implies <span class="p1">(<span class="default"><a href="msaterm.html#K2" title="MSATERM:func.2">Sym</a> (<font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c2">Y</font>)</span>)</span> <a href="msaterm.html#K6" title="MSATERM:func.6">-tree</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN: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="c3">X</font>,<font color="Maroon" title="c2">Y</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c4">o</font></span>)</span> )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:31_1"/>
<span class="p1">(<span class="default"><a href="msaterm.html#K2" title="MSATERM:func.2">Sym</a> (<font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c2">Y</font>)</span>)</span> <a href="msaterm.html#K6" title="MSATERM:func.6">-tree</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN: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="c3">X</font>,<font color="Maroon" title="c2">Y</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c4">o</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c5">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</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="c3">X</font>,<font color="Maroon" title="c2">Y</font>)</span>)</span></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">t</font> being   <a href="msaterm.html#NM1" title="MSATERM:NM.1">Term</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">Y</font><span class="kw"> such that </span><br/><a NAME="E3:31_1"/><span class="lab"><font color="Green" title="E24">A4</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">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="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">t</font>
 <span class="kw">and </span><br/><a NAME="E4:31_1"/>
 <a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Maroon" title="c7">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c4">o</font>
 <span class="kw">and </span><br/><a NAME="E5:31_1"/><span class="lab"><font color="Green" title="E25">A5</font></span>: 
 <a href="msafree3.html#K4" title="MSAFREE3:func.4">variables_in</a> <font color="Maroon" title="c7">t</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c3">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:31"><span class="lab"><font color="Green" title="E23">A3</font></span></a>, <a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E22">A2</font></span></a></span>;<br/><span class="kw">thus </span><a NAME="E6:31_1"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c5">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</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="c3">X</font>,<font color="Maroon" title="c2">Y</font>)</span>)</span>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c5">p</font> or <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</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="c3">X</font>,<font color="Maroon" title="c2">Y</font>)</span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:31_1_1"/><span class="lab"><font color="Green" title="E26">A6</font></span>: 
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c5">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</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="c3">X</font>,<font color="Maroon" title="c2">Y</font>)</span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:31_1_1"/><span class="lab"><font color="Green" title="E27">A7</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">p</font>
 <span class="kw">and </span><br/><a NAME="E4:31_1_1"/><span class="lab"><font color="Green" title="E28">A8</font></span>: 
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">x</font> = <font color="Maroon" title="c9">x</font> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:31_1_1"><span class="lab"><font color="Green" title="E27">A7</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">q</font> = <font color="Maroon" title="c5">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">x</font> as   <a href="msaterm.html#NM1" title="MSATERM:NM.1">Term</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">Y</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:31_1_1"><span class="lab"><font color="Green" title="E27">A7</font></span></a>, <a class="ref" href="msaterm.html#T22" title="MSATERM:th.22">MSATERM:22</a></span>;<br/>
<a NAME="E7:31_1_1"/><span class="lab"><font color="Green" title="E29">A9</font></span>: 
 <a href="msafree3.html#K4" title="MSAFREE3:func.4">variables_in</a> <font color="Maroon" title="c11">q</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c3">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="pboole.html#D2" title="PBOOLE:def.2">PBOOLE:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c12">z</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> or <span class="p1">(<span class="default"><a href="msafree3.html#K4" title="MSAFREE3:func.4">variables_in</a> <font color="Maroon" title="c11">q</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">z</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:31_1_1_1"/><span class="lab"><font color="Green" title="E30">A10</font></span>: 
<font color="Maroon" title="c12">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="msafree3.html#K4" title="MSAFREE3:func.4">variables_in</a> <font color="Maroon" title="c11">q</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">z</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c13">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c13">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="msafree3.html#K4" title="MSAFREE3:func.4">variables_in</a> <font color="Maroon" title="c11">q</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">z</font> or <font color="Maroon" title="c13">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">z</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:31_1_1_1"/>
<font color="Maroon" title="c13">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="msafree3.html#K4" title="MSAFREE3:func.4">variables_in</a> <font color="Maroon" title="c11">q</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">z</font></span><br/>

<a NAME="E3:31_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A11</font></span>: 
<font color="Maroon" title="c13">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="msafree3.html#K4" title="MSAFREE3:func.4">variables_in</a> <font color="Maroon" title="c7">t</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">z</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:31_1"><span class="lab"><font color="Green" title="E24">A4</font></span></a>, <a class="txt" href="#E1:31_1_1"><span class="lab"><font color="Green" title="E26">A6</font></span></a>, <a class="txt" href="#E4:31_1_1"><span class="lab"><font color="Green" title="E28">A8</font></span></a>, <a class="txt" href="#E1:31_1_1_1"><span class="lab"><font color="Green" title="E30">A10</font></span></a>, <a class="ref" href="msafree3.html#T11" target="_self" title="MSAFREE3:th.11">Th12</a></span>;<br/>
<a NAME="E4:31_1_1_1"/>
<span class="p1">(<span class="default"><a href="msafree3.html#K4" title="MSAFREE3:func.4">variables_in</a> <font color="Maroon" title="c7">t</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">z</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:31_1"><span class="lab"><font color="Green" title="E25">A5</font></span></a>, <a class="txt" href="#E1:31_1_1_1"><span class="lab"><font color="Green" title="E30">A10</font></span></a>, <a class="ref" href="pboole.html#D2" title="PBOOLE:def.2">PBOOLE:def 2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:31_1_1_1"/>
<font color="Maroon" title="c13">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:31_1_1_1"><span class="lab"><font color="Green" title="E31">A11</font></span></a></span>; <a class="txt" 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">set </span><font color="Maroon" title="c12">sq</font> =  <a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Maroon" title="c11">q</font>;<br/>
<a NAME="E8:31_1_1"/>
<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="c3">X</font>,<font color="Maroon" title="c2">Y</font>)</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> <font color="Maroon" title="c11">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">t9</font> where <font color="Olive" title="b1">t9</font> is   <a href="msaterm.html#NM1" title="MSATERM:NM.1">Term</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">Y</font> : (  <a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Olive" title="b1">t9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Maroon" title="c11">q</font> &amp;  <a href="msafree3.html#K4" title="MSAFREE3:func.4">variables_in</a> <font color="Olive" title="b1">t9</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c3">X</font> ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msafree3.html#D5" target="_self" title="MSAFREE3:def.5">Def6</a></span>;<br/>
<a NAME="E9:31_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">q</font> <a href="hidden.html#R2" title="HIDDEN: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="c3">X</font>,<font color="Maroon" title="c2">Y</font>)</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> <font color="Maroon" title="c11">q</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:31_1_1"><span class="lab"><font color="Green" title="E29">A9</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:31_1_1"/>
<font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</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="c3">X</font>,<font color="Maroon" title="c2">Y</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E21">A1</font></span></a>, <a class="txt" href="#E4:31_1_1"><span class="lab"><font color="Green" title="E28">A8</font></span></a>, <a class="ref" href="card_5.html#T2" title="CARD_5:th.2">CARD_5:2</a></span>; <a class="txt" 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>

<span class="kw">set </span><font color="Maroon" title="c7">t</font> = <span class="p1">(<span class="default"><a href="msaterm.html#K2" title="MSATERM:func.2">Sym</a> (<font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c2">Y</font>)</span>)</span> <a href="msaterm.html#K6" title="MSATERM:func.6">-tree</a> <font color="Maroon" title="c5">p</font>;<br/>
<span class="kw">assume </span><a NAME="E5:31"/><span class="lab"><font color="Green" title="E24">A12</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c5">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="card_3.html#K3" title="CARD_3:func.3">Union</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="c3">X</font>,<font color="Maroon" title="c2">Y</font>)</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="msaterm.html#K2" title="MSATERM:func.2">Sym</a> (<font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c2">Y</font>)</span>)</span> <a href="msaterm.html#K6" title="MSATERM:func.6">-tree</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN: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="c3">X</font>,<font color="Maroon" title="c2">Y</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c4">o</font></span>)</span></span><br/>

<a NAME="E6:31"/><span class="lab"><font color="Green" title="E25">A13</font></span>: 
 <a href="msafree3.html#K4" title="MSAFREE3:func.4">variables_in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msaterm.html#K2" title="MSATERM:func.2">Sym</a> (<font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c2">Y</font>)</span>)</span> <a href="msaterm.html#K6" title="MSATERM:func.6">-tree</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c3">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="pboole.html#D2" title="PBOOLE:def.2">PBOOLE:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c8">z</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> or <span class="p1">(<span class="default"><a href="msafree3.html#K4" title="MSAFREE3:func.4">variables_in</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msaterm.html#K2" title="MSATERM:func.2">Sym</a> (<font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c2">Y</font>)</span>)</span> <a href="msaterm.html#K6" title="MSATERM:func.6">-tree</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:31_2"/><span class="lab"><font color="Green" title="E26">A14</font></span>: 
<font color="Maroon" title="c8">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="msafree3.html#K4" title="MSAFREE3:func.4">variables_in</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msaterm.html#K2" title="MSATERM:func.2">Sym</a> (<font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c2">Y</font>)</span>)</span> <a href="msaterm.html#K6" title="MSATERM:func.6">-tree</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="msafree3.html#K4" title="MSAFREE3:func.4">variables_in</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msaterm.html#K2" title="MSATERM:func.2">Sym</a> (<font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c2">Y</font>)</span>)</span> <a href="msaterm.html#K6" title="MSATERM:func.6">-tree</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font> or <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:31_2"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="msafree3.html#K4" title="MSAFREE3:func.4">variables_in</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="msaterm.html#K2" title="MSATERM:func.2">Sym</a> (<font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c2">Y</font>)</span>)</span> <a href="msaterm.html#K6" title="MSATERM:func.6">-tree</a> <font color="Maroon" title="c5">p</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">q</font> being   <a href="trees_2.html#NM4" title="TREES_2:NM.4">DecoratedTree</a><span class="kw"> such that </span><br/><a NAME="E4:31_2"/><span class="lab"><font color="Green" title="E27">A15</font></span>: 
<font color="Maroon" title="c10">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c5">p</font>
 <span class="kw">and </span><br/><a NAME="E5:31_2"/><span class="lab"><font color="Green" title="E28">A16</font></span>: 
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">S</font> <a href="msafree3.html#K2" title="MSAFREE3:func.2">variables_in</a> <font color="Maroon" title="c10">q</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:31"><span class="lab"><font color="Green" title="E22">A2</font></span></a>, <a class="txt" href="#E1:31_2"><span class="lab"><font color="Green" title="E26">A14</font></span></a>, <a class="ref" href="msafree3.html#T11" target="_self" title="MSAFREE3:th.11">Th12</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c11">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E7:31_2"/><span class="lab"><font color="Green" title="E29">A17</font></span>: 
<font color="Maroon" title="c11">y</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>
 <span class="kw">and </span><br/><a NAME="E8:31_2"/><span class="lab"><font color="Green" title="E30">A18</font></span>: 
<font color="Maroon" title="c10">q</font> <a href="hidden.html#R2" title="HIDDEN: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="c3">X</font>,<font color="Maroon" title="c2">Y</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">y</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31"><span class="lab"><font color="Green" title="E21">A1</font></span></a>, <a class="txt" href="#E5:31"><span class="lab"><font color="Green" title="E24">A12</font></span></a>, <a class="txt" href="#E4:31_2"><span class="lab"><font color="Green" title="E27">A15</font></span></a>, <a class="ref" href="card_5.html#T2" title="CARD_5:th.2">CARD_5:2</a></span>;<br/>
<a NAME="E9:31_2"/>
<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="c3">X</font>,<font color="Maroon" title="c2">Y</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">t9</font> where <font color="Olive" title="b1">t9</font> is   <a href="msaterm.html#NM1" title="MSATERM:NM.1">Term</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">Y</font> : (  <a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Olive" title="b1">t9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">y</font> &amp;  <a href="msafree3.html#K4" title="MSAFREE3:func.4">variables_in</a> <font color="Olive" title="b1">t9</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c3">X</font> ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:31_2"><span class="lab"><font color="Green" title="E29">A17</font></span></a>, <a class="ref" href="msafree3.html#D5" target="_self" title="MSAFREE3:def.5">Def6</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">t9</font> being   <a href="msaterm.html#NM1" title="MSATERM:NM.1">Term</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">Y</font><span class="kw"> such that </span><br/><a NAME="E11:31_2"/><span class="lab"><font color="Green" title="E31">A19</font></span>: 
<font color="Maroon" title="c10">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">t9</font>
 <span class="kw">and </span><br/><a NAME="E12:31_2"/>
 <a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <font color="Maroon" title="c12">t9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">y</font>
 <span class="kw">and </span><br/><a NAME="E13:31_2"/><span class="lab"><font color="Green" title="E32">A20</font></span>: 
 <a href="msafree3.html#K4" title="MSAFREE3:func.4">variables_in</a> <font color="Maroon" title="c12">t9</font> <a href="pboole.html#R2" title="PBOOLE:pred.2">c=</a> <font color="Maroon" title="c3">X</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:31_2"><span class="lab"><font color="Green" title="E30">A18</font></span></a></span>;<br/>
<a NAME="E14:31_2"/>
<span class="p1">(<span class="default"><a href="msafree3.html#K4" title="MSAFREE3:func.4">variables_in</a> <font color="Maroon" title="c12">t9</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:31_2"><span class="lab"><font color="Green" title="E26">A14</font></span></a>, <a class="txt" href="#E13:31_2"><span class="lab"><font color="Green" title="E32">A20</font></span></a>, <a class="ref" href="pboole.html#D2" title="PBOOLE:def.2">PBOOLE:def 2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:31_2"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">X</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">z</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:31_2"><span class="lab"><font color="Green" title="E28">A16</font></span></a>, <a class="txt" href="#E11:31_2"><span class="lab"><font color="Green" title="E31">A19</font></span></a></span>; <a class="txt" 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="E7:31"/>
 <a href="msaterm.html#K7" title="MSATERM:func.7">the_sort_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msaterm.html#K2" title="MSATERM:func.2">Sym</a> (<font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c2">Y</font>)</span>)</span> <a href="msaterm.html#K6" title="MSATERM:func.6">-tree</a> <font color="Maroon" title="c5">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c4">o</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="msaterm.html#T20" title="MSATERM:th.20">MSATERM:20</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:31"/>
<span class="p1">(<span class="default"><a href="msaterm.html#K2" title="MSATERM:func.2">Sym</a> (<font color="Maroon" title="c4">o</font>,<font color="Maroon" title="c2">Y</font>)</span>)</span> <a href="msaterm.html#K6" title="MSATERM:func.6">-tree</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN: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="c3">X</font>,<font color="Maroon" title="c2">Y</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K2" title="MSUALG_1:func.2">the_result_sort_of</a> <font color="Maroon" title="c4">o</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:31"><span class="lab"><font color="Green" title="E23">A3</font></span></a>, <a class="txt" href="#E6:31"><span class="lab"><font color="Green" title="E25">A13</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
