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

<b>let </b><font color="Maroon" title="c1">n</font> be    <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> ;<br/><b>let </b><font color="Maroon" title="c2">p</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> ;<br/>



<b>assume </b><a NAME="E1:43"/><i><font color="Green" title="E24">A1</font></i>: 
for <font color="Olive" title="b1">q</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a>   st <font color="Olive" title="b1">q</font> <a href="qc_lang2.html#R2" title="QC_LANG2:pred.2">is_subformula_of</a> <font color="Maroon" title="c2">p</font> holds <br/> <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Olive" title="b1">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> ] <b>means </b>( $1 <a href="qc_lang2.html#R2" title="QC_LANG2:pred.2">is_subformula_of</a> <font color="Maroon" title="c2">p</font> implies  <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 );<br/>
<a NAME="E2:43"/><i><font color="Green" title="E25">A2</font></i>: 
for <font color="Olive" title="b1">r</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">r</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="cqc_lang.html#K7" title="CQC_LANG:func.7">'not'</a> <font color="Olive" title="b1">r</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">r</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> ;<br/>

<b>assume </b><a NAME="E1:43_1"/><i><font color="Green" title="E26">A3</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">r</font>]
 ;<br/>

<b>assume </b><a NAME="E2:43_1"/><i><font color="Green" title="E27">A4</font></i>: 
 <a href="cqc_lang.html#K7" title="CQC_LANG:func.7">'not'</a> <font color="Maroon" title="c3">r</font> <a href="qc_lang2.html#R2" title="QC_LANG2:pred.2">is_subformula_of</a> <font color="Maroon" title="c2">p</font>
 ;<br/>

<a NAME="E3:43_1"/>
<font color="Maroon" title="c3">r</font> <a href="qc_lang2.html#R1" title="QC_LANG2:pred.1">is_immediate_constituent_of</a>  <a href="cqc_lang.html#K7" title="CQC_LANG:func.7">'not'</a> <font color="Maroon" title="c3">r</font>
 
<b>by </b><i><a class="ref" href="qc_lang2.html#T60" title="QC_LANG2:th.60">QC_LANG2:60</a></i>;<br/>
<a NAME="E4:43_1"/><b>then </b>
<font color="Maroon" title="c3">r</font> <a href="qc_lang2.html#R3" title="QC_LANG2:pred.3">is_proper_subformula_of</a> <font color="Maroon" title="c2">p</font>
 
<b>by </b><i><a class="txt" href="substut2.html#E2:43_1"><i><font color="Green" title="E27">A4</font></i></a>, <a class="ref" href="qc_lang2.html#T83" title="QC_LANG2:th.83">QC_LANG2:83</a></i>;<br/>
<b>hence </b><a NAME="E5:43_1"/>
 <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K7" title="CQC_LANG:func.7">'not'</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="txt" href="substut2.html#E1:43_1"><i><font color="Green" title="E26">A3</font></i></a>, <a class="ref" href="cqc_sim1.html#T16" title="CQC_SIM1:th.16">CQC_SIM1:16</a>, <a class="ref" href="qc_lang2.html#D22" title="QC_LANG2:def.22">QC_LANG2:def 22</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:43"/><i><font color="Green" title="E26">A5</font></i>: 
for <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">r</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">s</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">r</font> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">'&amp;'</a> <font color="Olive" title="b2">s</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">r</font>, <font color="Maroon" title="c4">s</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> ;<br/>



<b>assume </b><a NAME="E1:43_2"/><i><font color="Green" title="E27">A6</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">r</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">s</font>] )
 ;<br/>

<b>assume </b><a NAME="E2:43_2"/><i><font color="Green" title="E28">A7</font></i>: 
<font color="Maroon" title="c3">r</font> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">'&amp;'</a> <font color="Maroon" title="c4">s</font> <a href="qc_lang2.html#R2" title="QC_LANG2:pred.2">is_subformula_of</a> <font color="Maroon" title="c2">p</font>
 ;<br/>

<a NAME="E3:43_2"/>
( <font color="Maroon" title="c3">r</font> <a href="qc_lang2.html#R1" title="QC_LANG2:pred.1">is_immediate_constituent_of</a> <font color="Maroon" title="c3">r</font> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">'&amp;'</a> <font color="Maroon" title="c4">s</font> &amp; <font color="Maroon" title="c4">s</font> <a href="qc_lang2.html#R1" title="QC_LANG2:pred.1">is_immediate_constituent_of</a> <font color="Maroon" title="c3">r</font> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">'&amp;'</a> <font color="Maroon" title="c4">s</font> )
 
<b>by </b><i><a class="ref" href="qc_lang2.html#T62" title="QC_LANG2:th.62">QC_LANG2:62</a></i>;<br/>
<a NAME="E4:43_2"/><b>then </b>
( <font color="Maroon" title="c3">r</font> <a href="qc_lang2.html#R3" title="QC_LANG2:pred.3">is_proper_subformula_of</a> <font color="Maroon" title="c2">p</font> &amp; <font color="Maroon" title="c4">s</font> <a href="qc_lang2.html#R3" title="QC_LANG2:pred.3">is_proper_subformula_of</a> <font color="Maroon" title="c2">p</font> )
 
<b>by </b><i><a class="txt" href="substut2.html#E2:43_2"><i><font color="Green" title="E28">A7</font></i></a>, <a class="ref" href="qc_lang2.html#T83" title="QC_LANG2:th.83">QC_LANG2:83</a></i>;<br/>
<a NAME="E5:43_2"/><b>then </b>
 <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">'&amp;'</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 0
 
<b>by </b><i><a class="txt" href="substut2.html#E1:43_2"><i><font color="Green" title="E27">A6</font></i></a>, <a class="ref" href="cqc_sim1.html#T17" title="CQC_SIM1:th.17">CQC_SIM1:17</a>, <a class="ref" href="qc_lang2.html#D22" title="QC_LANG2:def.22">QC_LANG2:def 22</a></i>;<br/>
<b>hence </b><a NAME="E6:43_2"/>
 <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">'&amp;'</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:43"/>
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">r</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">r</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">r</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">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="c4">r</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> ;<br/>



<b>assume </b><a NAME="E1:43_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">r</font>]
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:43_3_1"/><i><font color="Green" title="E27">A8</font></i>: 
 <a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">r</font> <a href="qc_lang2.html#R2" title="QC_LANG2:pred.2">is_subformula_of</a> <font color="Maroon" title="c2">p</font>
 ;<br/><a NAME="E2:43_3_1"/>
<font color="Maroon" title="c4">r</font> <a href="qc_lang2.html#R1" title="QC_LANG2:pred.1">is_immediate_constituent_of</a>  <a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">r</font>
 <b>by </b><i><a class="ref" href="qc_lang2.html#T63" title="QC_LANG2:th.63">QC_LANG2:63</a></i>;<br/><a NAME="E3:43_3_1"/><b>then </b>
<font color="Maroon" title="c4">r</font> <a href="qc_lang2.html#R3" title="QC_LANG2:pred.3">is_proper_subformula_of</a> <font color="Maroon" title="c2">p</font>
 <b>by </b><i><a class="txt" href="substut2.html#E1:43_3_1"><i><font color="Green" title="E27">A8</font></i></a>, <a class="ref" href="qc_lang2.html#T83" title="QC_LANG2:th.83">QC_LANG2:83</a></i>;<br/><a NAME="E4:43_3_1"/><b>then </b>
<font color="Maroon" title="c4">r</font> <a href="qc_lang2.html#R2" title="QC_LANG2:pred.2">is_subformula_of</a> <font color="Maroon" title="c2">p</font>
 <b>by </b><i><a class="ref" href="qc_lang2.html#D22" title="QC_LANG2:def.22">QC_LANG2:def 22</a></i>;<br/><a NAME="E5:43_3_1"/><b>then </b>
(  <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Maroon" title="c4">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp;  <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> )
 <b>by </b><i><a class="txt" href="substut2.html#E1:43"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="substut2.html#E1:43_3_1"><i><font color="Green" title="E27">A8</font></i></a></i>;<br/><a NAME="E6:43_3_1"/><b>then </b>
<font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c1">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="cqc_sim1.html#T18" title="CQC_SIM1:th.18">CQC_SIM1:18</a></i>;<br/><b>hence </b><a NAME="E7:43_3_1"/>
contradiction
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:43_3"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">r</font>]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E5:43"/><b>then </b><i><font color="Green" title="E27">A9</font></i>: 
for <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> <br/> for <font color="Olive" title="b3">x</font> being  <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a><br/> for <font color="Olive" title="b4">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> <br/> for <font color="Olive" title="b5">l</font> being  <a href="cqc_lang.html#NM2" title="CQC_LANG:NM.2">CQC-variable_list</a> of <font color="Olive" title="b4">k</font><br/> for <font color="Olive" title="b6">P</font> being  <a href="qc_lang1.html#NM6" title="QC_LANG1:NM.6">QC-pred_symbol</a> of <font color="Olive" title="b4">k</font> holds <br/> ( <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">S<sub>1</sub></font>[<font color="Olive" title="b6">P</font> <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">!</a> <font color="Olive" title="b5">l</font>] &amp; ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">r</font>] implies <font color="Maroon">S<sub>1</sub></font>[ <a href="cqc_lang.html#K7" title="CQC_LANG:func.7">'not'</a> <font color="Olive" title="b1">r</font>] ) &amp; ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">r</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">s</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">r</font> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">'&amp;'</a> <font color="Olive" title="b2">s</font>] ) &amp; ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">r</font>] implies <font color="Maroon">S<sub>1</sub></font>[ <a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Olive" title="b3">x</font>,<font color="Olive" title="b1">r</font>] ) )
 
<b>by </b><i><a class="txt" href="substut2.html#E2:43"><i><font color="Green" title="E25">A2</font></i></a>, <a class="txt" href="substut2.html#E3:43"><i><font color="Green" title="E26">A5</font></i></a>, <a class="ref" href="cqc_sim1.html#T14" title="CQC_SIM1:th.14">CQC_SIM1:14</a>, <a class="ref" href="cqc_sim1.html#T15" title="CQC_SIM1:th.15">CQC_SIM1:15</a></i>;<br/>
<a NAME="E6:43"/><i><font color="Green" title="E28">A10</font></i>: 
for <font color="Olive" title="b1">r</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">r</font>]
 
<b>from </b><i><a class="ref" href="cqc_lang.html#S1" title="CQC_LANG:sch.1">CQC_LANG:sch 1</a>(<a class="txt" href="substut2.html#E5:43"><i><font color="Green" title="E27">A9</font></i></a>);<br/></i>
<a NAME="E7:43"/>
 <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="txt" href="substut2.html#E1:43"><i><font color="Green" title="E24">A1</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:43"/>
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 <b>by </b><i><a class="txt" href="substut2.html#E6:43"><i><font color="Green" title="E28">A10</font></i></a></i>;<br/>


</div>
