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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">Y</font> being  <a href="relat_1.html#V2" title="RELAT_1:attr.2">V8</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="msaterm.html#NM1" title="MSATERM:NM.1">Term</a> of <font color="Maroon" title="c1">S</font>,<font color="Olive" title="b1">Y</font><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   <a href="msafree3.html#K4" title="MSAFREE3:func.4">variables_in</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">t</font> <a href="msaterm.html#K8" title="MSATERM:func.8">|</a> <font color="Olive" title="b3">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="Olive" title="b2">t</font></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">V8</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="msaterm.html#NM1" title="MSATERM:NM.1">Term</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">Y</font><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   <a href="msafree3.html#K4" title="MSAFREE3:func.4">variables_in</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">t</font> <a href="msaterm.html#K8" title="MSATERM:func.8">|</a> <font color="Olive" title="b2">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="Olive" title="b1">t</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">t</font> be   <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 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   <a href="msafree3.html#K4" title="MSAFREE3:func.4">variables_in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">t</font> <a href="msaterm.html#K8" title="MSATERM:func.8">|</a> <font color="Olive" title="b1">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="c3">t</font></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">  <a href="msafree3.html#K4" title="MSAFREE3:func.4">variables_in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">t</font> <a href="msaterm.html#K8" title="MSATERM:func.8">|</a> <font color="Maroon" title="c4">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="c3">t</font></span><br/>







<span class="kw">reconsider </span><font color="Maroon" title="c5">q</font> = <font color="Maroon" title="c3">t</font> <a href="msaterm.html#K8" title="MSATERM:func.8">|</a> <font color="Maroon" title="c4">p</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> ;<br/>
<span class="kw">let </span><font color="Maroon" title="c6">s</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="pboole.html#D2" onmouseover="rs('pboole/D2')" onmouseout="rh()">PBOOLE:def 2</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c6">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font> 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"><font color="Maroon" title="c3">t</font> <a href="msaterm.html#K8" title="MSATERM:func.8">|</a> <font color="Maroon" title="c4">p</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="msafree3.html#K4" title="MSAFREE3:func.4">variables_in</a> <font color="Maroon" title="c3">t</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">s</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:43"/><span class="lab"><font color="Green" title="E32">A1</font></span>: 
<font color="Maroon" title="c6">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="msafree3.html#K4" title="MSAFREE3:func.4">variables_in</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">t</font> <a href="msaterm.html#K8" title="MSATERM:func.8">|</a> <font color="Maroon" title="c4">p</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">s</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="msafree3.html#K4" title="MSAFREE3:func.4">variables_in</a> <font color="Maroon" title="c3">t</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">s</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c7">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"><font color="Maroon" title="c3">t</font> <a href="msaterm.html#K8" title="MSATERM:func.8">|</a> <font color="Maroon" title="c4">p</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">s</font> or <font color="Maroon" title="c7">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> <font color="Maroon" title="c3">t</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">s</font> )</span><br/>

<span class="kw">assume </span><a NAME="E3:43"/>
<font color="Maroon" title="c7">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"><font color="Maroon" title="c3">t</font> <a href="msaterm.html#K8" title="MSATERM:func.8">|</a> <font color="Maroon" title="c4">p</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">s</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">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> <font color="Maroon" title="c3">t</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">s</font></span><br/>

<a NAME="E4:43"/><span class="kw">then </span>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">a</font> <a href="xtuple_0.html#K1" title="XTUPLE_0:func.1">`1</a></span>)</span> where <font color="Olive" title="b1">a</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c5">q</font> : <font color="Olive" title="b1">a</font> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">s</font> </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="ref" href="msafree3.html#D2" target="_self" onmouseover="rs('msafree3/D2')" onmouseout="rh()">Def2</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">a</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">t</font> <a href="msaterm.html#K8" title="MSATERM:func.8">|</a> <font color="Maroon" title="c4">p</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E6:43"/><span class="lab"><font color="Green" title="E33">A2</font></span>: 
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a</font> <a href="xtuple_0.html#K1" title="XTUPLE_0:func.1">`1</a>  &amp; <font color="Maroon" title="c8">a</font> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">s</font> )
 ;<br/>
<a NAME="E7:43"/>
(  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">t</font> <a href="msaterm.html#K8" title="MSATERM:func.8">|</a> <font color="Maroon" title="c4">p</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">t</font> &amp; <font color="Maroon" title="c8">a</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">t</font> <a href="msaterm.html#K8" title="MSATERM:func.8">|</a> <font color="Maroon" title="c4">p</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="trees_2.html#T32" onmouseover="rs('trees_2/T32')" onmouseout="rh()">TREES_2:32</a></span>;<br/>
<a NAME="E8:43"/><span class="kw">then </span>
<font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">b</font> <a href="xtuple_0.html#K1" title="XTUPLE_0:func.1">`1</a></span>)</span> where <font color="Olive" title="b1">b</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c3">t</font> : <font color="Olive" title="b1">b</font> <a href="xtuple_0.html#K2" title="XTUPLE_0:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">s</font> </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:43"><span class="lab"><font color="Green" title="E33">A2</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:43"/>
<font color="Maroon" title="c7">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> <font color="Maroon" title="c3">t</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">s</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:43"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="ref" href="msafree3.html#D2" target="_self" onmouseover="rs('msafree3/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/>


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