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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a> ] means  <a href="substut1.html#K9" target="_self">Bound_Vars</a> <font color="Maroon">a<sub>1</sub></font> is <a href="finset_1.html#V1">finite</a>;<br/>
<a NAME="E1:16_1_1"/><i><font color="Green">E5</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a>  holds <br/>( ( <font color="Olive">b<sub>1</sub></font> is <a href="qc_lang1.html#V2">atomic</a> implies <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] ) &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="cqc_lang.html#K9">VERUM</a> ] &amp; ( <font color="Olive">b<sub>1</sub></font> is <a href="qc_lang1.html#V3">negative</a> &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K17">the_argument_of</a> <font color="Olive">b<sub>1</sub></font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] ) &amp; ( <font color="Olive">b<sub>1</sub></font> is <a href="qc_lang1.html#V4">conjunctive</a> &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K18">the_left_argument_of</a> <font color="Olive">b<sub>1</sub></font>] &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K19">the_right_argument_of</a> <font color="Olive">b<sub>1</sub></font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] ) &amp; ( <font color="Olive">b<sub>1</sub></font> is <a href="qc_lang1.html#V5">universal</a> &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K21">the_scope_of</a> <font color="Olive">b<sub>1</sub></font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a> ;<br/>

<b>thus </b><a NAME="E1:16_1_1_1"/>
( <font color="Maroon">c<sub>2</sub></font> is <a href="qc_lang1.html#V2">atomic</a> implies  <a href="substut1.html#K9" target="_self">Bound_Vars</a> <font color="Maroon">c<sub>2</sub></font> is <a href="finset_1.html#V1">finite</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:16_1_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="qc_lang1.html#V2">atomic</a>
 ;<br/>

<b>then </b><i><font color="Green">E6</font></i>:  <a href="substut1.html#K9" target="_self">Bound_Vars</a> <font color="Maroon">c<sub>2</sub></font> = 
 <a href="substut1.html#K8" target="_self">Bound_Vars</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K16">the_arguments_of</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="substut1.html#E10"><i><font color="Green">Lemma4</font></i></a></i>
<br/>.= 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang1.html#K16">the_arguments_of</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K16">the_arguments_of</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="qc_lang1.html#K16">the_arguments_of</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K2">bound_QC-variables</a>  ) </span>}</span> 

;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K16">the_arguments_of</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="qc_lang1.html#K16">the_arguments_of</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K2">bound_QC-variables</a>  );<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K16">the_arguments_of</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> );<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <span class="p1">(<span class="default"><a href="qc_lang1.html#K16">the_arguments_of</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E3:16_1_1_1_1"/><i><font color="Green">E7</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
;<br/>
<a NAME="E4:16_1_1_1_1"/>
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span>  <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span> 
 
<b>from </b><i><a class="ref" href="fraenkel.html#S1">FRAENKEL:sch 1</a>(<a class="txt" href="substut1.html#E3:16_1_1_1_1"><i><font color="Green">E7</font></i></a>);<br/></i>
<a NAME="E5:16_1_1_1_1"/><b>then </b>
(  <a href="substut1.html#K9" target="_self">Bound_Vars</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K16">the_arguments_of</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp;  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K16">the_arguments_of</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> is <a href="finset_1.html#V1">finite</a> )
 
<b>by </b><i><a class="txt" href="substut1.html#E2:16_1_1_1_1"><i><font color="Green">E6</font></i></a>, <a class="ref" href="cqc_sim1.html#T9">CQC_SIM1:9</a></i>;<br/>
<b>hence </b><a NAME="E6:16_1_1_1_1"/>
 <a href="substut1.html#K9" target="_self">Bound_Vars</a> <font color="Maroon">c<sub>2</sub></font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:16_1_1_1"/>
 <a href="substut1.html#K9" target="_self">Bound_Vars</a> <a href="cqc_lang.html#K9">VERUM</a>  is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="txt" href="substut1.html#E10"><i><font color="Green">Lemma4</font></i></a></i>;<br/>

<b>thus </b><a NAME="E3:16_1_1_1"/>
( <font color="Maroon">c<sub>2</sub></font> is <a href="qc_lang1.html#V3">negative</a> &amp;  <a href="substut1.html#K9" target="_self">Bound_Vars</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K17">the_argument_of</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> is <a href="finset_1.html#V1">finite</a> implies  <a href="substut1.html#K9" target="_self">Bound_Vars</a> <font color="Maroon">c<sub>2</sub></font> is <a href="finset_1.html#V1">finite</a> )
 <b>by </b><i><a class="txt" href="substut1.html#E10"><i><font color="Green">Lemma4</font></i></a></i>;<br/>

<b>thus </b><a NAME="E4:16_1_1_1"/>
( <font color="Maroon">c<sub>2</sub></font> is <a href="qc_lang1.html#V4">conjunctive</a> &amp;  <a href="substut1.html#K9" target="_self">Bound_Vars</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K18">the_left_argument_of</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> is <a href="finset_1.html#V1">finite</a> &amp;  <a href="substut1.html#K9" target="_self">Bound_Vars</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K19">the_right_argument_of</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> is <a href="finset_1.html#V1">finite</a> implies  <a href="substut1.html#K9" target="_self">Bound_Vars</a> <font color="Maroon">c<sub>2</sub></font> is <a href="finset_1.html#V1">finite</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_1_2"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:16_1_1_1_2"/><i><font color="Green">E6</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="qc_lang1.html#V4">conjunctive</a>
 <b>and </b><br/><a NAME="E2:16_1_1_1_2"/><i><font color="Green">E7</font></i>: 
(  <a href="substut1.html#K9" target="_self">Bound_Vars</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K18">the_left_argument_of</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> is <a href="finset_1.html#V1">finite</a> &amp;  <a href="substut1.html#K9" target="_self">Bound_Vars</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K19">the_right_argument_of</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> is <a href="finset_1.html#V1">finite</a> )
 ;<br/>

<a NAME="E3:16_1_1_1_2"/>
 <a href="substut1.html#K9" target="_self">Bound_Vars</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="substut1.html#K9" target="_self">Bound_Vars</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K18">the_left_argument_of</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="substut1.html#K9" target="_self">Bound_Vars</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K19">the_right_argument_of</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="substut1.html#E1:16_1_1_1_2"><i><font color="Green">E6</font></i></a>, <a class="txt" href="substut1.html#E10"><i><font color="Green">Lemma4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:16_1_1_1_2"/>
 <a href="substut1.html#K9" target="_self">Bound_Vars</a> <font color="Maroon">c<sub>2</sub></font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="txt" href="substut1.html#E2:16_1_1_1_2"><i><font color="Green">E7</font></i></a>, <a class="ref" href="finset_1.html#T14">FINSET_1:14</a></i>;<br/>


</div><b>end;</b></div>

<b>assume </b><b>that </b><br/><a NAME="E5:16_1_1_1"/><i><font color="Green">E6</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="qc_lang1.html#V5">universal</a>
 <b>and </b><br/><a NAME="E6:16_1_1_1"/><i><font color="Green">E7</font></i>: 
 <a href="substut1.html#K9" target="_self">Bound_Vars</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K21">the_scope_of</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> is <a href="finset_1.html#V1">finite</a>
 ;<br/>

<a NAME="E7:16_1_1_1"/>
 <a href="substut1.html#K9" target="_self">Bound_Vars</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="substut1.html#K9" target="_self">Bound_Vars</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K21">the_scope_of</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K20">bound_in</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="txt" href="substut1.html#E5:16_1_1_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="substut1.html#E10"><i><font color="Green">Lemma4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:16_1_1_1"/>
 <a href="substut1.html#K9" target="_self">Bound_Vars</a> <font color="Maroon">c<sub>2</sub></font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="txt" href="substut1.html#E6:16_1_1_1"><i><font color="Green">E7</font></i></a>, <a class="ref" href="finset_1.html#T14">FINSET_1:14</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:16_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a>  holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="qc_lang1.html#S2">QC_LANG1:sch 2</a>(<a class="txt" href="substut1.html#E1:16_1_1"><i><font color="Green">E5</font></i></a>);<br/></i>
<b>hence </b><a NAME="E3:16_1_1"/>
 <a href="substut1.html#K9" target="_self">Bound_Vars</a> <font color="Maroon">c<sub>1</sub></font> is <a href="finset_1.html#V1">finite</a>
 ;<br/>


</div>
