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<b>let </b><font color="Maroon">A</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">J</font> be    <a href="valuat_1.html#M1">interpretation</a> of <font color="Maroon">A</font>;<br/>



<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="substut1.html#K38">CQC-Sub-WFF</a> ] <b>means </b>for <font color="Olive">v</font> being   <a href="fraenkel.html#M2">Element</a> of  <a href="valuat_1.html#K2">Valuations_in</a> <font color="Maroon">A</font> holds <br/> ( <font color="Maroon">J</font>,<font color="Olive">v</font> <a href="valuat_1.html#R1">|=</a>  <a href="substut1.html#K39">CQC_Sub</a> <font color="Maroon">a<sub>1</sub></font> iff <font color="Maroon">J</font>,<font color="Olive">v</font> <a href="sublemma.html#K1" target="_self">.</a> <span class="p1">(<span class="default"><a href="sublemma.html#K3" target="_self">Val_S</a> <font color="Olive">v</font>,<font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="sublemma.html#R1" target="_self">|=</a> <font color="Maroon">a<sub>1</sub></font> );<br/>
<a NAME="E1:109"/><i><font color="Green">E55</font></i>: 
for <font color="Olive">S</font>, <font color="Olive">S'</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="substut1.html#K38">CQC-Sub-WFF</a> <br/> for <font color="Olive">x</font> being  <a href="qc_lang1.html#NM3">bound_QC-variable</a><br/> for <font color="Olive">SQ</font> being   <a href="substut1.html#M1">second_Q_comp</a> of <span class="p1"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Olive">S</font>,<font color="Olive">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span><br/> for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">ll</font> being  <a href="cqc_lang.html#NM2">CQC-variable_list</a> of <font color="Olive">k</font><br/> for <font color="Olive">P</font> being  <a href="qc_lang1.html#NM6">QC-pred_symbol</a> of <font color="Olive">k</font><br/> for <font color="Olive">e</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="substut1.html#K1">vSUB</a>  holds <br/> ( <font color="Maroon">S<sub>1</sub></font>[ <a href="sublemma.html#K4" target="_self">Sub_P</a> <font color="Olive">P</font>,<font color="Olive">ll</font>,<font color="Olive">e</font>] &amp; ( <font color="Olive">S</font> is <a href="substut1.html#V2">Sub_VERUM</a> implies <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">S</font>] ) &amp; ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">S</font>] implies <font color="Maroon">S<sub>1</sub></font>[ <a href="sublemma.html#K6" target="_self">Sub_not</a> <font color="Olive">S</font>] ) &amp; ( <font color="Olive">S</font> <a href="substut1.html#K19">`2</a>  <a href="hidden.html#R1">=</a> <font color="Olive">S'</font> <a href="substut1.html#K19">`2</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">S</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">S'</font>] implies <font color="Maroon">S<sub>1</sub></font>[ <a href="sublemma.html#K7" target="_self">CQCSub_&amp;</a> <font color="Olive">S</font>,<font color="Olive">S'</font>] ) &amp; ( <span class="p1"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Olive">S</font>,<font color="Olive">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span> is <a href="substut1.html#V3">quantifiable</a> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">S</font>] implies <font color="Maroon">S<sub>1</sub></font>[ <a href="sublemma.html#K10" target="_self">CQCSub_All</a> <span class="p1"><a href="sublemma.html#K8" target="_self">[</a><span class="default"><font color="Olive">S</font>,<font color="Olive">x</font></span><a href="sublemma.html#K8" target="_self">]</a></span>,<font color="Olive">SQ</font>] ) )
 
<b>by </b><i>, , , , </i>;<br/>
<b>thus </b><a NAME="E2:109"/>
for <font color="Olive">S</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="substut1.html#K38">CQC-Sub-WFF</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">S</font>]
 <b>from </b><i><a class="ref" href="sublemma.html#S1" target="_self">SUBLEMMA:sch 1</a>(<a class="ref" href="sublemma.html#T12" target="_self">Th12</a>);</i><br/>


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