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<div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">p</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> ;<br/>



<b>assume </b><a NAME="E1:43"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">q</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>   st <font color="Olive">q</font> <a href="qc_lang2.html#R2">is_subformula_of</a> <font color="Maroon">p</font> holds <br/> <a href="cqc_sim1.html#K6">QuantNbr</a> <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="qc_lang2.html#R2">is_subformula_of</a> <font color="Maroon">p</font> implies  <a href="cqc_sim1.html#K6">QuantNbr</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 );<br/>
<a NAME="E2:43"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">r</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">r</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="cqc_lang.html#K10">'not'</a> <font color="Olive">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">r</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> ;<br/>

<b>assume </b><a NAME="E1:43_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">r</font>]
 ;<br/>

<b>assume </b><a NAME="E2:43_1"/><i><font color="Green">E58</font></i>: 
 <a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">r</font> <a href="qc_lang2.html#R2">is_subformula_of</a> <font color="Maroon">p</font>
 ;<br/>

<a NAME="E3:43_1"/>
<font color="Maroon">r</font> <a href="qc_lang2.html#R1">is_immediate_constituent_of</a>  <a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="qc_lang2.html#T60">QC_LANG2:60</a></i>;<br/>
<a NAME="E4:43_1"/><b>then </b>
<font color="Maroon">r</font> <a href="qc_lang2.html#R3">is_proper_subformula_of</a> <font color="Maroon">p</font>
 
<b>by </b><i>, <a class="ref" href="qc_lang2.html#T83">QC_LANG2:83</a></i>;<br/>
<b>hence </b><a NAME="E5:43_1"/>
 <a href="cqc_sim1.html#K6">QuantNbr</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i>, <a class="ref" href="cqc_sim1.html#T16">CQC_SIM1:16</a>, <a class="ref" href="qc_lang2.html#D22">QC_LANG2:def 22</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:43"/><i><font color="Green">E59</font></i>: 
for <font color="Olive">r</font>, <font color="Olive">s</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">r</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">s</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">r</font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Olive">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">r</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> ;<br/><b>let </b><font color="Maroon">s</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> ;<br/>



<b>assume </b><a NAME="E1:43_2"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">r</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">s</font>] )
 ;<br/>

<b>assume </b><a NAME="E2:43_2"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">r</font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">s</font> <a href="qc_lang2.html#R2">is_subformula_of</a> <font color="Maroon">p</font>
 ;<br/>

<a NAME="E3:43_2"/>
( <font color="Maroon">r</font> <a href="qc_lang2.html#R1">is_immediate_constituent_of</a> <font color="Maroon">r</font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">s</font> &amp; <font color="Maroon">s</font> <a href="qc_lang2.html#R1">is_immediate_constituent_of</a> <font color="Maroon">r</font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">s</font> )
 
<b>by </b><i><a class="ref" href="qc_lang2.html#T62">QC_LANG2:62</a></i>;<br/>
<a NAME="E4:43_2"/><b>then </b>
( <font color="Maroon">r</font> <a href="qc_lang2.html#R3">is_proper_subformula_of</a> <font color="Maroon">p</font> &amp; <font color="Maroon">s</font> <a href="qc_lang2.html#R3">is_proper_subformula_of</a> <font color="Maroon">p</font> )
 
<b>by </b><i>, <a class="ref" href="qc_lang2.html#T83">QC_LANG2:83</a></i>;<br/>
<a NAME="E5:43_2"/><b>then </b>
 <a href="cqc_sim1.html#K6">QuantNbr</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a> 0 <a href="nat_1.html#K1">+</a> 0
 
<b>by </b><i>, <a class="ref" href="cqc_sim1.html#T17">CQC_SIM1:17</a>, <a class="ref" href="qc_lang2.html#D22">QC_LANG2:def 22</a></i>;<br/>
<b>hence </b><a NAME="E6:43_2"/>
 <a href="cqc_sim1.html#K6">QuantNbr</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a> 0
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:43"/><i><font color="Green">E62</font></i>: 
for <font color="Olive">x</font> being  <a href="qc_lang1.html#NM3">bound_QC-variable</a><br/> for <font color="Olive">r</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">r</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="cqc_lang.html#K15">All</a> <font color="Olive">x</font>,<font color="Olive">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">x</font> be   <a href="qc_lang1.html#NM3">bound_QC-variable</a>;<br/><b>let </b><font color="Maroon">r</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> ;<br/>



<b>assume </b><a NAME="E1:43_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">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">E63</font></i>: 
 <a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">r</font> <a href="qc_lang2.html#R2">is_subformula_of</a> <font color="Maroon">p</font>
 ;<br/><a NAME="E2:43_3_1"/>
<font color="Maroon">r</font> <a href="qc_lang2.html#R1">is_immediate_constituent_of</a>  <a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="qc_lang2.html#T63">QC_LANG2:63</a></i>;<br/><a NAME="E3:43_3_1"/><b>then </b>
<font color="Maroon">r</font> <a href="qc_lang2.html#R3">is_proper_subformula_of</a> <font color="Maroon">p</font>
 <b>by </b><i>, <a class="ref" href="qc_lang2.html#T83">QC_LANG2:83</a></i>;<br/><a NAME="E4:43_3_1"/><b>then </b>
<font color="Maroon">r</font> <a href="qc_lang2.html#R2">is_subformula_of</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="qc_lang2.html#D22">QC_LANG2:def 22</a></i>;<br/><a NAME="E5:43_3_1"/><b>then </b>
(  <a href="cqc_sim1.html#K6">QuantNbr</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="cqc_sim1.html#K6">QuantNbr</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E6:43_3_1"/><b>then </b>
<font color="Maroon">n</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">n</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="cqc_sim1.html#T18">CQC_SIM1:18</a></i>;<br/><a NAME="E7:43_3_1"/><b>then </b>
<font color="Maroon">n</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><a NAME="E8:43_3_1"/><b>then </b>
0 <a href="hidden.html#R1">=</a> 1
 ;<br/><b>hence </b><a NAME="E9:43_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#K15">All</a> <font color="Maroon">x</font>,<font color="Maroon">r</font>]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E5:43"/><i><font color="Green">E75</font></i>: 
for <font color="Olive">r</font>, <font color="Olive">s</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-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">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">l</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> holds <br/> ( <font color="Maroon">S<sub>1</sub></font>[ <a href="cqc_lang.html#K9">VERUM</a> ] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">P</font> <a href="cqc_lang.html#K8">!</a> <font color="Olive">l</font>] &amp; ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">r</font>] implies <font color="Maroon">S<sub>1</sub></font>[ <a href="cqc_lang.html#K10">'not'</a> <font color="Olive">r</font>] ) &amp; ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">r</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">s</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">r</font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Olive">s</font>] ) &amp; ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">r</font>] implies <font color="Maroon">S<sub>1</sub></font>[ <a href="cqc_lang.html#K15">All</a> <font color="Olive">x</font>,<font color="Olive">r</font>] ) )
 
<b>by </b><i>, , , <a class="ref" href="cqc_sim1.html#T14">CQC_SIM1:14</a>, <a class="ref" href="cqc_sim1.html#T15">CQC_SIM1:15</a></i>;<br/>
<a NAME="E6:43"/><i><font color="Green">E76</font></i>: 
for <font color="Olive">r</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">r</font>]
 
<b>from </b><i><a class="ref" href="cqc_lang.html#S1">CQC_LANG:sch 1</a>();<br/></i>
<a NAME="E7:43"/>
 <a href="cqc_sim1.html#K6">QuantNbr</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E8:43"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i/>;<br/>


</div>
