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

<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><font color="Maroon">F<sub>2</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1;<br/>
<a NAME="E1:1_1"/><i><font color="Green" title="E3">A3</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/> for <font color="Olive" title="b2">P</font> being  <a href="qc_lang1.html#NM6" title="QC_LANG1:NM.6">QC-pred_symbol</a> of <font color="Olive" title="b1">k</font><br/> for <font color="Olive" title="b3">l</font> being   <a href="qc_lang1.html#M1" title="QC_LANG1:mode.1">QC-variable_list</a> of <font color="Olive" title="b1">k</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">P</font> <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">!</a> <font color="Olive" title="b3">l</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c1">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c2">P</font> be   <a href="qc_lang1.html#NM6" title="QC_LANG1:NM.6">QC-pred_symbol</a> of <font color="Maroon" title="c1">k</font>;<br/><b>let </b><font color="Maroon" title="c3">l</font> be    <a href="qc_lang1.html#M1" title="QC_LANG1:mode.1">QC-variable_list</a> of <font color="Maroon" title="c1">k</font>;<br/>





<a NAME="E1:1_1_1"/><i><font color="Green" title="E4">A4</font></i>: 
<font color="Maroon" title="c2">P</font> <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">!</a> <font color="Maroon" title="c3">l</font> is <a href="qc_lang1.html#V2" title="QC_LANG1:attr.2">atomic</a>
 
<b>by </b><i><a class="ref" href="qc_lang1.html#D17" title="QC_LANG1:def.17">QC_LANG1:def 17</a></i>;<br/>
<b>hence </b><font color="Maroon">F<sub>2</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">!</a> <font color="Maroon" title="c3">l</font></span>)</span> = 
<font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">!</a> <font color="Maroon" title="c3">l</font></span>)</span>)
<b>by </b><i><a class="txt" href="qc_lang3.html#E1:1"><i><font color="Green" title="E1">A1</font></i></a></i>
<br/>.= 
<font color="Maroon">F<sub>3</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">P</font> <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">!</a> <font color="Maroon" title="c3">l</font></span>)</span>
<b>by </b><i><a class="txt" href="qc_lang3.html#E2:1"><i><font color="Green" title="E2">A2</font></i></a>, <a class="txt" href="qc_lang3.html#E1:1_1_1"><i><font color="Green" title="E4">A4</font></i></a></i>
;<br/><br/>


</div><b>end;</b></div>
<a NAME="E2:1_1"/>
( <font color="Maroon">F<sub>2</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="qc_lang1.html#K11" title="QC_LANG1:func.11">VERUM</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>() &amp; <font color="Maroon">F<sub>3</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="qc_lang1.html#K11" title="QC_LANG1:func.11">VERUM</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>4</sub></font>() )
 
<b>by </b><i><a class="txt" href="qc_lang3.html#E1:1"><i><font color="Green" title="E1">A1</font></i></a>, <a class="txt" href="qc_lang3.html#E2:1"><i><font color="Green" title="E2">A2</font></i></a></i>;<br/>
<a NAME="E3:1_1"/><b>then </b><i><font color="Green" title="E4">A5</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K11" title="QC_LANG1:func.11">VERUM</a> ]
 
;<br/>
<a NAME="E4:1_1"/><i><font color="Green" title="E5">A6</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>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">p</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K12" title="QC_LANG1:func.12">'not'</a> <font color="Olive" title="b1">p</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c1">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> ;<br/>

<b>assume </b><a NAME="E1:1_1_2"/><i><font color="Green" title="E6">A7</font></i>: 
<font color="Maroon">F<sub>2</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p</font>
 ;<br/>

<a NAME="E2:1_1_2"/><i><font color="Green" title="E7">A8</font></i>: 
 <a href="qc_lang1.html#K12" title="QC_LANG1:func.12">'not'</a> <font color="Maroon" title="c1">p</font> is <a href="qc_lang1.html#V3" title="QC_LANG1:attr.3">negative</a>
 
<b>by </b><i><a class="ref" href="qc_lang1.html#D18" title="QC_LANG1:def.18">QC_LANG1:def 18</a></i>;<br/>
<a NAME="E3:1_1_2"/><b>then </b><i><font color="Green" title="E8">A9</font></i>: 
 <a href="qc_lang1.html#K17" title="QC_LANG1:func.17">the_argument_of</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K12" title="QC_LANG1:func.12">'not'</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font>
 
<b>by </b><i><a class="ref" href="qc_lang1.html#D23" title="QC_LANG1:def.23">QC_LANG1:def 23</a></i>;<br/>
<b>hence </b><font color="Maroon">F<sub>2</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K12" title="QC_LANG1:func.12">'not'</a> <font color="Maroon" title="c1">p</font></span>)</span> = 
<font color="Maroon">F<sub>6</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p</font></span>)</span>)
<b>by </b><i><a class="txt" href="qc_lang3.html#E1:1"><i><font color="Green" title="E1">A1</font></i></a>, <a class="txt" href="qc_lang3.html#E1:1_1_2"><i><font color="Green" title="E6">A7</font></i></a>, <a class="txt" href="qc_lang3.html#E2:1_1_2"><i><font color="Green" title="E7">A8</font></i></a></i>
<br/>.= 
<font color="Maroon">F<sub>3</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K12" title="QC_LANG1:func.12">'not'</a> <font color="Maroon" title="c1">p</font></span>)</span>
<b>by </b><i><a class="txt" href="qc_lang3.html#E2:1"><i><font color="Green" title="E2">A2</font></i></a>, <a class="txt" href="qc_lang3.html#E2:1_1_2"><i><font color="Green" title="E7">A8</font></i></a>, <a class="txt" href="qc_lang3.html#E3:1_1_2"><i><font color="Green" title="E8">A9</font></i></a></i>
;<br/><br/>


</div><b>end;</b></div>
<a NAME="E5:1_1"/><i><font color="Green" title="E6">A10</font></i>: 
for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</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>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">p</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">q</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">p</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <font color="Olive" title="b2">q</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c1">p</font>, <font color="Maroon" title="c2">q</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> ;<br/>



<b>assume </b><a NAME="E1:1_1_3"/><i><font color="Green" title="E7">A11</font></i>: 
( <font color="Maroon">F<sub>2</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p</font> &amp; <font color="Maroon">F<sub>2</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">q</font> )
 ;<br/>

<a NAME="E2:1_1_3"/><i><font color="Green" title="E8">A12</font></i>: 
<font color="Maroon" title="c1">p</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <font color="Maroon" title="c2">q</font> is <a href="qc_lang1.html#V4" title="QC_LANG1:attr.4">conjunctive</a>
 
<b>by </b><i><a class="ref" href="qc_lang1.html#D19" title="QC_LANG1:def.19">QC_LANG1:def 19</a></i>;<br/>
<a NAME="E3:1_1_3"/><b>then </b><i><font color="Green" title="E9">A13</font></i>: 
(  <a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_left_argument_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">p</font> &amp;  <a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_right_argument_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">q</font> )
 
<b>by </b><i><a class="ref" href="qc_lang1.html#D24" title="QC_LANG1:def.24">QC_LANG1:def 24</a>, <a class="ref" href="qc_lang1.html#D25" title="QC_LANG1:def.25">QC_LANG1:def 25</a></i>;<br/>
<b>hence </b><font color="Maroon">F<sub>2</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <font color="Maroon" title="c2">q</font></span>)</span> = 
<font color="Maroon">F<sub>7</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">q</font></span>)</span>)
<b>by </b><i><a class="txt" href="qc_lang3.html#E1:1"><i><font color="Green" title="E1">A1</font></i></a>, <a class="txt" href="qc_lang3.html#E1:1_1_3"><i><font color="Green" title="E7">A11</font></i></a>, <a class="txt" href="qc_lang3.html#E2:1_1_3"><i><font color="Green" title="E8">A12</font></i></a></i>
<br/>.= 
<font color="Maroon">F<sub>3</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <font color="Maroon" title="c2">q</font></span>)</span>
<b>by </b><i><a class="txt" href="qc_lang3.html#E2:1"><i><font color="Green" title="E2">A2</font></i></a>, <a class="txt" href="qc_lang3.html#E2:1_1_3"><i><font color="Green" title="E8">A12</font></i></a>, <a class="txt" href="qc_lang3.html#E3:1_1_3"><i><font color="Green" title="E9">A13</font></i></a></i>
;<br/><br/>


</div><b>end;</b></div>
<a NAME="E6:1_1"/><i><font color="Green" title="E7">A14</font></i>: 
for <font color="Olive" title="b1">x</font> being  <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a><br/> for <font color="Olive" title="b2">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>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">p</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">p</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c1">x</font> be   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a>;<br/><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> ;<br/>



<b>assume </b><a NAME="E1:1_1_4"/><i><font color="Green" title="E8">A15</font></i>: 
<font color="Maroon">F<sub>2</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">F<sub>3</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p</font>
 ;<br/>

<a NAME="E2:1_1_4"/><i><font color="Green" title="E9">A16</font></i>: 
 <a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">p</font> is <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a>
 
<b>by </b><i><a class="ref" href="qc_lang1.html#D20" title="QC_LANG1:def.20">QC_LANG1:def 20</a></i>;<br/>
<a NAME="E3:1_1_4"/><b>then </b>
(  <a href="qc_lang1.html#K21" title="QC_LANG1:func.21">the_scope_of</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p</font> &amp;  <a href="qc_lang1.html#K20" title="QC_LANG1:func.20">bound_in</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> )
 
<b>by </b><i><a class="ref" href="qc_lang1.html#D26" title="QC_LANG1:def.26">QC_LANG1:def 26</a>, <a class="ref" href="qc_lang1.html#D27" title="QC_LANG1:def.27">QC_LANG1:def 27</a></i>;<br/>
<b>hence </b><font color="Maroon">F<sub>2</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">p</font></span>)</span> = 
<font color="Maroon">F<sub>8</sub></font>(<span class="p1">(<span class="default"><a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">p</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K21" title="QC_LANG1:func.21">the_scope_of</a> <span class="p3">(<span class="default"><a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">p</font></span>)</span></span>)</span></span>)</span>)
<b>by </b><i><a class="txt" href="qc_lang3.html#E1:1"><i><font color="Green" title="E1">A1</font></i></a>, <a class="txt" href="qc_lang3.html#E1:1_1_4"><i><font color="Green" title="E8">A15</font></i></a>, <a class="txt" href="qc_lang3.html#E2:1_1_4"><i><font color="Green" title="E9">A16</font></i></a></i>
<br/>.= 
<font color="Maroon">F<sub>3</sub></font>() <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">p</font></span>)</span>
<b>by </b><i><a class="txt" href="qc_lang3.html#E2:1"><i><font color="Green" title="E2">A2</font></i></a>, <a class="txt" href="qc_lang3.html#E2:1_1_4"><i><font color="Green" title="E9">A16</font></i></a></i>
;<br/><br/>


</div><b>end;</b></div>
<a NAME="E7:1_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#S1" title="QC_LANG1:sch.1">QC_LANG1:sch 1</a>(<a class="txt" href="qc_lang3.html#E1:1_1"><i><font color="Green" title="E3">A3</font></i></a>, <a class="txt" href="qc_lang3.html#E3:1_1"><i><font color="Green" title="E4">A5</font></i></a>, <a class="txt" href="qc_lang3.html#E4:1_1"><i><font color="Green" title="E5">A6</font></i></a>, <a class="txt" href="qc_lang3.html#E5:1_1"><i><font color="Green" title="E6">A10</font></i></a>, <a class="txt" href="qc_lang3.html#E6:1_1"><i><font color="Green" title="E7">A14</font></i></a>);<br/></i>
<b>hence </b><a NAME="E8:1_1"/>
<font color="Maroon">F<sub>2</sub></font>() <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon">F<sub>3</sub></font>()
 <b>by </b><i><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>;<br/>


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