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<b>let </b><font color="Maroon" title="c1">S</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="substut1.html#K16" title="SUBSTUT1:func.16">QC-Sub-WFF</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">p</font> being  <a href="qc_lang1.html#NM8" title="QC_LANG1:NM.8">QC-formula</a> ex <font color="Olive" title="b2">e</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="substut1.html#K1" title="SUBSTUT1:func.1">vSUB</a>  st <font color="Maroon" title="c1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Olive" title="b1">p</font>,<font color="Olive" title="b2">e</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><br/>

<a NAME="E1:31"/>
<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="qc_lang1.html#K8" title="QC_LANG1:func.8">QC-WFF</a> ,<a href="substut1.html#K1" title="SUBSTUT1:func.1">vSUB</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> is <a href="substut1.html#V1" title="SUBSTUT1:attr.1">QC-Sub-closed</a>
 
<b>by </b><i><a class="ref" href="substut1.html#D16" target="_self" title="SUBSTUT1:def.16">Def16</a>, <a class="ref" href="substut1.html#T7" target="_self" title="SUBSTUT1:th.7">Th7</a></i>;<br/>
<a NAME="E2:31"/><b>then </b>
 <a href="substut1.html#K16" title="SUBSTUT1:func.16">QC-Sub-WFF</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="qc_lang1.html#K8" title="QC_LANG1:func.8">QC-WFF</a> ,<a href="substut1.html#K1" title="SUBSTUT1:func.1">vSUB</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="substut1.html#D17" target="_self" title="SUBSTUT1:def.17">Def17</a></i>;<br/>
<a NAME="E3:31"/><b>then </b>
<font color="Maroon" title="c1">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="qc_lang1.html#K8" title="QC_LANG1:func.8">QC-WFF</a> ,<a href="substut1.html#K1" title="SUBSTUT1:func.1">vSUB</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:31"/><i><font color="Green" title="E13">A1</font></i>: 
( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K8" title="QC_LANG1:func.8">QC-WFF</a>  &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="substut1.html#K1" title="SUBSTUT1:func.1">vSUB</a>  &amp; <font color="Maroon" title="c1">S</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="c2">a</font>,<font color="Maroon" title="c3">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#D2" title="ZFMISC_1:def.2">ZFMISC_1:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">p</font> = <font color="Maroon" title="c2">a</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K8" title="QC_LANG1:func.8">QC-WFF</a>  <b>by </b><i><a class="txt" href="substut1.html#E5:31"><i><font color="Green" title="E13">A1</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">e</font> = <font color="Maroon" title="c3">b</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="substut1.html#K1" title="SUBSTUT1:func.1">vSUB</a>  <b>by </b><i><a class="txt" href="substut1.html#E5:31"><i><font color="Green" title="E13">A1</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c4">p</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">e</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="substut1.html#K1" title="SUBSTUT1:func.1">vSUB</a>  st <font color="Maroon" title="c1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">p</font>,<font color="Olive" title="b1">e</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><br/>

<b>take </b>
<font color="Maroon" title="c5">e</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">e</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span></span><br/>

<b>thus </b><a NAME="E8:31"/>
<font color="Maroon" title="c1">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c5">e</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>
 <b>by </b><i><a class="txt" href="substut1.html#E5:31"><i><font color="Green" title="E13">A1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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