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

<span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">p</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">CQC-WFF</a>   st (  for <font color="Olive" title="b2">q</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">CQC-WFF</a>   st <font color="Olive" title="b2">q</font> <a href="qc_lang2.html#R2" title="QC_LANG2:pred.2">is_subformula_of</a> <font color="Olive" title="b1">p</font> holds <br/> <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Olive" title="b2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> ) holds <br/><font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/><span class="kw">let </span><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#K3" title="CQC_LANG:func.3">CQC-WFF</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  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#K3" title="CQC_LANG:func.3">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> ) implies <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/>



<span class="kw">assume </span><a NAME="E1:44"/><span class="lab"><font color="Green" title="E24">A1</font></span>: 
 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#K3" title="CQC_LANG:func.3">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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/>

<span class="kw">defpred </span><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#K3" title="CQC_LANG:func.3">CQC-WFF</a> ] <span class="kw">means </span>( $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>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  );<br/>
<a NAME="E2:44"/><span class="lab"><font color="Green" title="E25">A2</font></span>: 
 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#K3" title="CQC_LANG:func.3">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#K11" title="CQC_LANG:func.11">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="44_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">x</font> be   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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#K3" title="CQC_LANG:func.3">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#K11" title="CQC_LANG:func.11">All</a> (<font color="Maroon" title="c3">x</font>,<font color="Olive" title="b1">r</font>)]</span><br/><span class="kw">let </span><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#K3" title="CQC_LANG:func.3">CQC-WFF</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">r</font>] implies <font color="Maroon">S<sub>1</sub></font>[ <a href="cqc_lang.html#K11" title="CQC_LANG:func.11">All</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">r</font>)] )</span><br/>



<span class="kw">assume </span><a NAME="E1:44_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">r</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[ <a href="cqc_lang.html#K11" title="CQC_LANG:func.11">All</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">r</font>)]</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:44_1_1"/><span class="lab"><font color="Green" title="E26">A3</font></span>: 
 <a href="cqc_lang.html#K11" title="CQC_LANG:func.11">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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:44_1_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#K11" title="CQC_LANG:func.11">All</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">r</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="qc_lang2.html#T46" title="QC_LANG2:th.46">QC_LANG2:46</a></span>;<br/><a NAME="E3:44_1_1"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:44_1_1"><span class="lab"><font color="Green" title="E26">A3</font></span></a>, <a class="ref" href="qc_lang2.html#T63" title="QC_LANG2:th.63">QC_LANG2:63</a></span>;<br/><a NAME="E4:44_1_1"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="qc_lang2.html#D21" title="QC_LANG2:def.21">QC_LANG2:def 21</a></span>;<br/><a NAME="E5:44_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A4</font></span>: 
 <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:44"><span class="lab"><font color="Green" title="E24">A1</font></span></a></span>;<br/><a NAME="E6:44_1_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#K11" title="CQC_LANG:func.11">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:44"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="txt" href="#E1:44_1_1"><span class="lab"><font color="Green" title="E26">A3</font></span></a></span>;<br/><a NAME="E7:44_1_1"/><span class="kw">then </span>
<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#K2" title="NAT_1:func.2">+</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:44_1_1"><span class="lab"><font color="Green" title="E27">A4</font></span></a>, <a class="ref" href="cqc_sim1.html#T18" title="CQC_SIM1:th.18">CQC_SIM1:18</a></span>;<br/><span class="kw">hence </span><a NAME="E8:44_1_1"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:44_1"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="cqc_lang.html#K11" title="CQC_LANG:func.11">All</a> (<font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">r</font>)]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:44"/><span class="lab"><font color="Green" title="E26">A5</font></span>: 
 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#K3" title="CQC_LANG:func.3">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#K7" title="CQC_LANG:func.7">'&amp;'</a> <font color="Olive" title="b2">s</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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#K3" title="CQC_LANG:func.3">CQC-WFF</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">r</font> <a href="cqc_lang.html#K7" title="CQC_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c4">s</font>] )</span><br/>



<span class="kw">assume </span><a NAME="E1:44_2"/><span class="lab"><font color="Green" title="E27">A6</font></span>: 
( <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>] )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">r</font> <a href="cqc_lang.html#K7" title="CQC_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c4">s</font>]</span><br/>

<span class="kw">assume </span><a NAME="E2:44_2"/><span class="lab"><font color="Green" title="E28">A7</font></span>: 
<font color="Maroon" title="c3">r</font> <a href="cqc_lang.html#K7" title="CQC_LANG:func.7">'&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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <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#K7" title="CQC_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/>

<a NAME="E3:44_2"/>
<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#K7" title="CQC_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c4">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="qc_lang2.html#T45" title="QC_LANG2:th.45">QC_LANG2:45</a></span>;<br/>
<a NAME="E4:44_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A8</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:44_2"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="ref" href="qc_lang2.html#T63" title="QC_LANG2:th.63">QC_LANG2:63</a></span>;<br/>
<a NAME="E5:44_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#K7" title="CQC_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c4">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="qc_lang2.html#T45" title="QC_LANG2:th.45">QC_LANG2:45</a></span>;<br/>
<a NAME="E6:44_2"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:44_2"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="ref" href="qc_lang2.html#T63" title="QC_LANG2:th.63">QC_LANG2:63</a></span>;<br/>
<a NAME="E7:44_2"/><span class="kw">then </span>
 <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#K7" title="CQC_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:44_2"><span class="lab"><font color="Green" title="E27">A6</font></span></a>, <a class="txt" href="#E4:44_2"><span class="lab"><font color="Green" title="E29">A8</font></span></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#D21" title="QC_LANG2:def.21">QC_LANG2:def 21</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:44_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#K7" title="CQC_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E4:44"/>
 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#K3" title="CQC_LANG:func.3">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#K6" title="CQC_LANG:func.6">'not'</a> <font color="Olive" title="b1">r</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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#K3" title="CQC_LANG:func.3">CQC-WFF</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">r</font>] implies <font color="Maroon">S<sub>1</sub></font>[ <a href="cqc_lang.html#K6" title="CQC_LANG:func.6">'not'</a> <font color="Maroon" title="c3">r</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:44_3"/><span class="lab"><font color="Green" title="E27">A9</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">r</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[ <a href="cqc_lang.html#K6" title="CQC_LANG:func.6">'not'</a> <font color="Maroon" title="c3">r</font>]</span><br/>

<a NAME="E2:44_3"/><span class="lab"><font color="Green" title="E28">A10</font></span>: 
<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#K6" title="CQC_LANG:func.6">'not'</a> <font color="Maroon" title="c3">r</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="qc_lang2.html#T43" title="QC_LANG2:th.43">QC_LANG2:43</a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:44_3"/>
 <a href="cqc_lang.html#K6" title="CQC_LANG:func.6">'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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K6" title="CQC_LANG:func.6">'not'</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/>

<a NAME="E4:44_3"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:44_3"><span class="lab"><font color="Green" title="E28">A10</font></span></a>, <a class="ref" href="qc_lang2.html#T63" title="QC_LANG2:th.63">QC_LANG2:63</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:44_3"/>
 <a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K6" title="CQC_LANG:func.6">'not'</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:44_3"><span class="lab"><font color="Green" title="E27">A9</font></span></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#D21" title="QC_LANG2:def.21">QC_LANG2:def 21</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E5:44"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A11</font></span>: 
 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#K3" title="CQC_LANG:func.3">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#K5" title="CQC_LANG:func.5">VERUM</a> ] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b6">P</font> <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">!</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#K6" title="CQC_LANG:func.6">'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#K7" title="CQC_LANG:func.7">'&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#K11" title="CQC_LANG:func.11">All</a> (<font color="Olive" title="b3">x</font>,<font color="Olive" title="b1">r</font>)] ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:44"><span class="lab"><font color="Green" title="E26">A5</font></span></a>, <a class="txt" href="#E2:44"><span class="lab"><font color="Green" title="E25">A2</font></span></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></span>;<br/>
<a NAME="E6:44"/><span class="lab"><font color="Green" title="E28">A12</font></span>: 
 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#K3" title="CQC_LANG:func.3">CQC-WFF</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">r</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="cqc_lang.html#S1" title="CQC_LANG:sch.1">CQC_LANG:sch 1</a>(<a class="txt" href="#E5:44"><span class="lab"><font color="Green" title="E27">A11</font></span></a>);<br/></span>
<a NAME="E7:44"/>
 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:44"><span class="lab"><font color="Green" title="E24">A1</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:44"/>
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:44"><span class="lab"><font color="Green" title="E28">A12</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
