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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <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>means </b> <a href="substut1.html#K9" title="SUBSTUT1:func.9">Bound_Vars</a> <font color="Maroon" title="c1">p</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>;<br/>
<a NAME="E1:16_1_1"/><i><font color="Green" title="E5">A1</font></i>: 
for <font color="Olive" title="b1">p</font> being   <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>  holds <br/> ( ( <font color="Olive" title="b1">p</font> is <a href="qc_lang1.html#V2" title="QC_LANG1:attr.2">atomic</a> implies <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">p</font>] ) &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="cqc_lang.html#K6" title="CQC_LANG:func.6">VERUM</a> ] &amp; ( <font color="Olive" title="b1">p</font> is <a href="qc_lang1.html#V3" title="QC_LANG1:attr.3">negative</a> &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K17" title="QC_LANG1:func.17">the_argument_of</a> <font color="Olive" title="b1">p</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">p</font>] ) &amp; ( <font color="Olive" title="b1">p</font> is <a href="qc_lang1.html#V4" title="QC_LANG1:attr.4">conjunctive</a> &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_left_argument_of</a> <font color="Olive" title="b1">p</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_right_argument_of</a> <font color="Olive" title="b1">p</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">p</font>] ) &amp; ( <font color="Olive" title="b1">p</font> is <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a> &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K21" title="QC_LANG1:func.21">the_scope_of</a> <font color="Olive" title="b1">p</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">p</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" title="c2">p</font> be    <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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( <font color="Maroon" title="c2">p</font> is <a href="qc_lang1.html#V2" title="QC_LANG1:attr.2">atomic</a> implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">p</font>] ) &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="cqc_lang.html#K6" title="CQC_LANG:func.6">VERUM</a> ] &amp; ( <font color="Maroon" title="c2">p</font> is <a href="qc_lang1.html#V3" title="QC_LANG1:attr.3">negative</a> &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K17" title="QC_LANG1:func.17">the_argument_of</a> <font color="Maroon" title="c2">p</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">p</font>] ) &amp; ( <font color="Maroon" title="c2">p</font> is <a href="qc_lang1.html#V4" title="QC_LANG1:attr.4">conjunctive</a> &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_left_argument_of</a> <font color="Maroon" title="c2">p</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_right_argument_of</a> <font color="Maroon" title="c2">p</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">p</font>] ) &amp; ( <font color="Maroon" title="c2">p</font> is <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a> &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K21" title="QC_LANG1:func.21">the_scope_of</a> <font color="Maroon" title="c2">p</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">p</font>] ) )</span><br/>

<b>thus </b><a NAME="E1:16_1_1_1"/>
( <font color="Maroon" title="c2">p</font> is <a href="qc_lang1.html#V2" title="QC_LANG1:attr.2">atomic</a> implies  <a href="substut1.html#K9" title="SUBSTUT1:func.9">Bound_Vars</a> <font color="Maroon" title="c2">p</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[ <a href="cqc_lang.html#K6" title="CQC_LANG:func.6">VERUM</a> ] &amp; ( <font color="Maroon" title="c2">p</font> is <a href="qc_lang1.html#V3" title="QC_LANG1:attr.3">negative</a> &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K17" title="QC_LANG1:func.17">the_argument_of</a> <font color="Maroon" title="c2">p</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">p</font>] ) &amp; ( <font color="Maroon" title="c2">p</font> is <a href="qc_lang1.html#V4" title="QC_LANG1:attr.4">conjunctive</a> &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_left_argument_of</a> <font color="Maroon" title="c2">p</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_right_argument_of</a> <font color="Maroon" title="c2">p</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">p</font>] ) &amp; ( <font color="Maroon" title="c2">p</font> is <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a> &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K21" title="QC_LANG1:func.21">the_scope_of</a> <font color="Maroon" title="c2">p</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">p</font>] ) )</span><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" title="c2">p</font> is <a href="qc_lang1.html#V2" title="QC_LANG1:attr.2">atomic</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="substut1.html#K9" title="SUBSTUT1:func.9">Bound_Vars</a> <font color="Maroon" title="c2">p</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a></span><br/>

<b>then </b><i><font color="Green" title="E6">A2</font></i>:  <a href="substut1.html#K9" title="SUBSTUT1:func.9">Bound_Vars</a> <font color="Maroon" title="c2">p</font> = 
 <a href="substut1.html#K8" title="SUBSTUT1:func.8">Bound_Vars</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K16" title="QC_LANG1:func.16">the_arguments_of</a> <font color="Maroon" title="c2">p</font></span>)</span>
<b>by </b><i><a class="txt" href="substut1.html#E10"><i><font color="Green" title="E4">Lm1</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" title="QC_LANG1:func.16">the_arguments_of</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> where <font color="Olive" title="b1">k</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K16" title="QC_LANG1:func.16">the_arguments_of</a> <font color="Maroon" title="c2">p</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="qc_lang1.html#K16" title="QC_LANG1:func.16">the_arguments_of</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>  ) </span>}</span> 

;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">p</font> &amp; <font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K16" title="QC_LANG1:func.16">the_arguments_of</a> <font color="Maroon" title="c2">p</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="qc_lang1.html#K16" title="QC_LANG1:func.16">the_arguments_of</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>  );<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">p</font> &amp; <font color="Maroon" title="c1">p</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K16" title="QC_LANG1:func.16">the_arguments_of</a> <font color="Maroon" title="c2">p</font></span>)</span> );<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <span class="p1">(<span class="default"><a href="qc_lang1.html#K16" title="QC_LANG1:func.16">the_arguments_of</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">p</font>;<br/>
<a NAME="E3:16_1_1_1_1"/><i><font color="Green" title="E7">A3</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">k</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" title="b1">k</font>) where <font color="Olive" title="b1">k</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font>] </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">n</font>) where <font color="Olive" title="b1">n</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">n</font>] </span>}</span> 
 
<b>from </b><i><a class="ref" href="fraenkel.html#S1" title="FRAENKEL:sch.1">FRAENKEL:sch 1</a>(<a class="txt" href="#E3:16_1_1_1_1"><i><font color="Green" title="E7">A3</font></i></a>);<br/></i>
<a NAME="E5:16_1_1_1_1"/><b>then </b>
(  <a href="substut1.html#K9" title="SUBSTUT1:func.9">Bound_Vars</a> <font color="Maroon" title="c2">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K16" title="QC_LANG1:func.16">the_arguments_of</a> <font color="Maroon" title="c2">p</font></span>)</span> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K16" title="QC_LANG1:func.16">the_arguments_of</a> <font color="Maroon" title="c2">p</font></span>)</span> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )
 
<b>by </b><i><a class="txt" href="#E2:16_1_1_1_1"><i><font color="Green" title="E6">A2</font></i></a>, <a class="ref" href="cqc_sim1.html#T9" title="CQC_SIM1:th.9">CQC_SIM1:9</a></i>;<br/>
<b>hence </b><a NAME="E6:16_1_1_1_1"/>
 <a href="substut1.html#K9" title="SUBSTUT1:func.9">Bound_Vars</a> <font color="Maroon" title="c2">p</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>thus </b><a NAME="E2:16_1_1_1"/>
 <a href="substut1.html#K9" title="SUBSTUT1:func.9">Bound_Vars</a> <a href="cqc_lang.html#K6" title="CQC_LANG:func.6">VERUM</a>  is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="txt" href="substut1.html#E10"><i><font color="Green" title="E4">Lm1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( <font color="Maroon" title="c2">p</font> is <a href="qc_lang1.html#V3" title="QC_LANG1:attr.3">negative</a> &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K17" title="QC_LANG1:func.17">the_argument_of</a> <font color="Maroon" title="c2">p</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">p</font>] ) &amp; ( <font color="Maroon" title="c2">p</font> is <a href="qc_lang1.html#V4" title="QC_LANG1:attr.4">conjunctive</a> &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_left_argument_of</a> <font color="Maroon" title="c2">p</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_right_argument_of</a> <font color="Maroon" title="c2">p</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">p</font>] ) &amp; ( <font color="Maroon" title="c2">p</font> is <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a> &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K21" title="QC_LANG1:func.21">the_scope_of</a> <font color="Maroon" title="c2">p</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">p</font>] ) )</span><br/>

<b>thus </b><a NAME="E3:16_1_1_1"/>
( <font color="Maroon" title="c2">p</font> is <a href="qc_lang1.html#V3" title="QC_LANG1:attr.3">negative</a> &amp;  <a href="substut1.html#K9" title="SUBSTUT1:func.9">Bound_Vars</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K17" title="QC_LANG1:func.17">the_argument_of</a> <font color="Maroon" title="c2">p</font></span>)</span> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> implies  <a href="substut1.html#K9" title="SUBSTUT1:func.9">Bound_Vars</a> <font color="Maroon" title="c2">p</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )
 <b>by </b><i><a class="txt" href="substut1.html#E10"><i><font color="Green" title="E4">Lm1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( <font color="Maroon" title="c2">p</font> is <a href="qc_lang1.html#V4" title="QC_LANG1:attr.4">conjunctive</a> &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_left_argument_of</a> <font color="Maroon" title="c2">p</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_right_argument_of</a> <font color="Maroon" title="c2">p</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">p</font>] ) &amp; ( <font color="Maroon" title="c2">p</font> is <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a> &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K21" title="QC_LANG1:func.21">the_scope_of</a> <font color="Maroon" title="c2">p</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">p</font>] ) )</span><br/>

<b>thus </b><a NAME="E4:16_1_1_1"/>
( <font color="Maroon" title="c2">p</font> is <a href="qc_lang1.html#V4" title="QC_LANG1:attr.4">conjunctive</a> &amp;  <a href="substut1.html#K9" title="SUBSTUT1:func.9">Bound_Vars</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_left_argument_of</a> <font color="Maroon" title="c2">p</font></span>)</span> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp;  <a href="substut1.html#K9" title="SUBSTUT1:func.9">Bound_Vars</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_right_argument_of</a> <font color="Maroon" title="c2">p</font></span>)</span> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> implies  <a href="substut1.html#K9" title="SUBSTUT1:func.9">Bound_Vars</a> <font color="Maroon" title="c2">p</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">p</font> is <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a> &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K21" title="QC_LANG1:func.21">the_scope_of</a> <font color="Maroon" title="c2">p</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">p</font>] )</span><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" title="E6">A4</font></i>: 
<font color="Maroon" title="c2">p</font> is <a href="qc_lang1.html#V4" title="QC_LANG1:attr.4">conjunctive</a>
 <b>and </b><br/><a NAME="E2:16_1_1_1_2"/><i><font color="Green" title="E7">A5</font></i>: 
(  <a href="substut1.html#K9" title="SUBSTUT1:func.9">Bound_Vars</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_left_argument_of</a> <font color="Maroon" title="c2">p</font></span>)</span> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> &amp;  <a href="substut1.html#K9" title="SUBSTUT1:func.9">Bound_Vars</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_right_argument_of</a> <font color="Maroon" title="c2">p</font></span>)</span> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="substut1.html#K9" title="SUBSTUT1:func.9">Bound_Vars</a> <font color="Maroon" title="c2">p</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a></span><br/>

<a NAME="E3:16_1_1_1_2"/>
 <a href="substut1.html#K9" title="SUBSTUT1:func.9">Bound_Vars</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="substut1.html#K9" title="SUBSTUT1:func.9">Bound_Vars</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_left_argument_of</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="substut1.html#K9" title="SUBSTUT1:func.9">Bound_Vars</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_right_argument_of</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:16_1_1_1_2"><i><font color="Green" title="E6">A4</font></i></a>, <a class="txt" href="substut1.html#E10"><i><font color="Green" title="E4">Lm1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:16_1_1_1_2"/>
 <a href="substut1.html#K9" title="SUBSTUT1:func.9">Bound_Vars</a> <font color="Maroon" title="c2">p</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="txt" href="#E2:16_1_1_1_2"><i><font color="Green" title="E7">A5</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/>


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

<b>assume </b><b>that </b><br/><a NAME="E5:16_1_1_1"/><i><font color="Green" title="E6">A6</font></i>: 
<font color="Maroon" title="c2">p</font> is <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a>
 <b>and </b><br/><a NAME="E6:16_1_1_1"/><i><font color="Green" title="E7">A7</font></i>: 
 <a href="substut1.html#K9" title="SUBSTUT1:func.9">Bound_Vars</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K21" title="QC_LANG1:func.21">the_scope_of</a> <font color="Maroon" title="c2">p</font></span>)</span> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">p</font>]</span><br/>

<a NAME="E7:16_1_1_1"/>
 <a href="substut1.html#K9" title="SUBSTUT1:func.9">Bound_Vars</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="substut1.html#K9" title="SUBSTUT1:func.9">Bound_Vars</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K21" title="QC_LANG1:func.21">the_scope_of</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">bound_in</a> <font color="Maroon" title="c2">p</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="txt" href="#E5:16_1_1_1"><i><font color="Green" title="E6">A6</font></i></a>, <a class="txt" href="substut1.html#E10"><i><font color="Green" title="E4">Lm1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:16_1_1_1"/>
 <a href="substut1.html#K9" title="SUBSTUT1:func.9">Bound_Vars</a> <font color="Maroon" title="c2">p</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 <b>by </b><i><a class="txt" href="#E6:16_1_1_1"><i><font color="Green" title="E7">A7</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/>


</div><b>end;</b></div>
<a NAME="E2:16_1_1"/>
for <font color="Olive" title="b1">p</font> being   <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>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">p</font>]
 
<b>from </b><i><a class="ref" href="qc_lang1.html#S2" title="QC_LANG1:sch.2">QC_LANG1:sch 2</a>(<a class="txt" href="#E1:16_1_1"><i><font color="Green" title="E5">A1</font></i></a>);<br/></i>
<b>hence </b><a NAME="E3:16_1_1"/>
 <a href="substut1.html#K9" title="SUBSTUT1:func.9">Bound_Vars</a> <font color="Maroon" title="c1">p</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
