<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">C</font> be   <a href="abcmiz_1.html#V3" title="ABCMIZ_1:attr.3">initialized</a>  <a href="abcmiz_1.html#NM5" title="ABCMIZ_1:NM.5">ConstructorSignature</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">q</font> being   <a href="abcmiz_1.html#V9" title="ABCMIZ_1:attr.9">pure</a>   <a href="abcmiz_1.html#M1" title="ABCMIZ_1:mode.1">expression</a> of <font color="Maroon" title="c1">C</font>, <a href="abcmiz_1.html#K12" title="ABCMIZ_1:func.12">a_Type</a> <font color="Maroon" title="c1">C</font> holds   <a href="abcmiz_1.html#K47" title="ABCMIZ_1:func.47">variables_in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <span class="p3">(<span class="default"><a href="abcmiz_1.html#K38" title="ABCMIZ_1:func.38">QuasiAdjs</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="abcmiz_1.html#K40" title="ABCMIZ_1:func.40">ast</a> <font color="Olive" title="b1">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="abcmiz_1.html#K45" title="ABCMIZ_1:func.45">variables_in</a> <font color="Olive" title="b1">q</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">q</font> be   <a href="abcmiz_1.html#V9" title="ABCMIZ_1:attr.9">pure</a>   <a href="abcmiz_1.html#M1" title="ABCMIZ_1:mode.1">expression</a> of <font color="Maroon" title="c1">C</font>, <a href="abcmiz_1.html#K12" title="ABCMIZ_1:func.12">a_Type</a> <font color="Maroon" title="c1">C</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="abcmiz_1.html#K47" title="ABCMIZ_1:func.47">variables_in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <span class="p3">(<span class="default"><a href="abcmiz_1.html#K38" title="ABCMIZ_1:func.38">QuasiAdjs</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="abcmiz_1.html#K40" title="ABCMIZ_1:func.40">ast</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="abcmiz_1.html#K45" title="ABCMIZ_1:func.45">variables_in</a> <font color="Maroon" title="c2">q</font></span><br/>



<span class="kw">set </span><font color="Maroon" title="c3">A</font> =  <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <span class="p1">(<span class="default"><a href="abcmiz_1.html#K38" title="ABCMIZ_1:func.38">QuasiAdjs</a> <font color="Maroon" title="c1">C</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">AA</font> = <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="abcmiz_1.html#K45" title="ABCMIZ_1:func.45">variables_in</a> <font color="Olive" title="b1">a</font></span>)</span> where <font color="Olive" title="b1">a</font> is   <a href="abcmiz_1.html#NM10" title="ABCMIZ_1:NM.10">quasi-adjective</a> of <font color="Maroon" title="c1">C</font> : <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <span class="p1">(<span class="default"><a href="abcmiz_1.html#K38" title="ABCMIZ_1:func.38">QuasiAdjs</a> <font color="Maroon" title="c1">C</font></span>)</span> </span> } </span> ;<br/>
<a NAME="E1:189"/>
<span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="abcmiz_1.html#K45" title="ABCMIZ_1:func.45">variables_in</a> <font color="Olive" title="b1">a</font></span>)</span> where <font color="Olive" title="b1">a</font> is   <a href="abcmiz_1.html#NM10" title="ABCMIZ_1:NM.10">quasi-adjective</a> of <font color="Maroon" title="c1">C</font> : <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <span class="p1">(<span class="default"><a href="abcmiz_1.html#K38" title="ABCMIZ_1:func.38">QuasiAdjs</a> <font color="Maroon" title="c1">C</font></span>)</span> </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="189_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">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="c5">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"><a href="abcmiz_1.html#K45" title="ABCMIZ_1:func.45">variables_in</a> <font color="Olive" title="b1">a</font></span>)</span> where <font color="Olive" title="b1">a</font> is   <a href="abcmiz_1.html#NM10" title="ABCMIZ_1:NM.10">quasi-adjective</a> of <font color="Maroon" title="c1">C</font> : <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <span class="p1">(<span class="default"><a href="abcmiz_1.html#K38" title="ABCMIZ_1:func.38">QuasiAdjs</a> <font color="Maroon" title="c1">C</font></span>)</span> </span> } </span>  or <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:189_1"/>
<font color="Maroon" title="c5">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"><a href="abcmiz_1.html#K45" title="ABCMIZ_1:func.45">variables_in</a> <font color="Olive" title="b1">a</font></span>)</span> where <font color="Olive" title="b1">a</font> is   <a href="abcmiz_1.html#NM10" title="ABCMIZ_1:NM.10">quasi-adjective</a> of <font color="Maroon" title="c1">C</font> : <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <span class="p1">(<span class="default"><a href="abcmiz_1.html#K38" title="ABCMIZ_1:func.38">QuasiAdjs</a> <font color="Maroon" title="c1">C</font></span>)</span> </span> } </span> 
 ; <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="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><br/>

<a NAME="E2:189_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">a</font> being   <a href="abcmiz_1.html#NM10" title="ABCMIZ_1:NM.10">quasi-adjective</a> of <font color="Maroon" title="c1">C</font> st <br/>( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="abcmiz_1.html#K45" title="ABCMIZ_1:func.45">variables_in</a> <font color="Olive" title="b1">a</font> &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <span class="p1">(<span class="default"><a href="abcmiz_1.html#K38" title="ABCMIZ_1:func.38">QuasiAdjs</a> <font color="Maroon" title="c1">C</font></span>)</span> )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:189_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <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="E2:189"/><span class="kw">then </span><span class="lab"><font color="Green" title="E93">A1</font></span>: 
<span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="abcmiz_1.html#K45" title="ABCMIZ_1:func.45">variables_in</a> <font color="Olive" title="b1">a</font></span>)</span> where <font color="Olive" title="b1">a</font> is   <a href="abcmiz_1.html#NM10" title="ABCMIZ_1:NM.10">quasi-adjective</a> of <font color="Maroon" title="c1">C</font> : <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <span class="p1">(<span class="default"><a href="abcmiz_1.html#K38" title="ABCMIZ_1:func.38">QuasiAdjs</a> <font color="Maroon" title="c1">C</font></span>)</span> </span> } </span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
;<br/>
<a NAME="E3:189"/>
 <a href="abcmiz_1.html#K47" title="ABCMIZ_1:func.47">variables_in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <span class="p3">(<span class="default"><a href="abcmiz_1.html#K38" title="ABCMIZ_1:func.38">QuasiAdjs</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="abcmiz_1.html#K40" title="ABCMIZ_1:func.40">ast</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p2"> { <span class="default"> <span class="p3">(<span class="default"><a href="abcmiz_1.html#K45" title="ABCMIZ_1:func.45">variables_in</a> <font color="Olive" title="b1">a</font></span>)</span> where <font color="Olive" title="b1">a</font> is   <a href="abcmiz_1.html#NM10" title="ABCMIZ_1:NM.10">quasi-adjective</a> of <font color="Maroon" title="c1">C</font> : <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <span class="p1">(<span class="default"><a href="abcmiz_1.html#K38" title="ABCMIZ_1:func.38">QuasiAdjs</a> <font color="Maroon" title="c1">C</font></span>)</span> </span> } </span> </span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="abcmiz_1.html#K45" title="ABCMIZ_1:func.45">variables_in</a> <font color="Maroon" title="c2">q</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="abcmiz_1.html#T104" target="_self" title="ABCMIZ_1:th.104">Th104</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:189"/>
 <a href="abcmiz_1.html#K47" title="ABCMIZ_1:func.47">variables_in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <span class="p3">(<span class="default"><a href="abcmiz_1.html#K38" title="ABCMIZ_1:func.38">QuasiAdjs</a> <font color="Maroon" title="c1">C</font></span>)</span></span>)</span> <a href="abcmiz_1.html#K40" title="ABCMIZ_1:func.40">ast</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="abcmiz_1.html#K45" title="ABCMIZ_1:func.45">variables_in</a> <font color="Maroon" title="c2">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:189"><span class="lab"><font color="Green" title="E93">A1</font></span></a>, <a class="ref" href="zfmisc_1.html#T2" title="ZFMISC_1:th.2">ZFMISC_1:2</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>
