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

<b>let </b><font color="Maroon">c<sub>1</sub></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="E1:45"/><i><font color="Green">E25</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  st <font color="Olive">b<sub>1</sub></font> <a href="qc_lang2.html#R2">is_subformula_of</a> <font color="Maroon">c<sub>1</sub></font> holds <br/> <a href="cqc_sim1.html#K6">QuantNbr</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<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> ] means ( <font color="Maroon">a<sub>1</sub></font> <a href="qc_lang2.html#R2">is_subformula_of</a> <font color="Maroon">c<sub>1</sub></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="E2:45"/><i><font color="Green">E26</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a><br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="qc_lang1.html#NM6">QC-pred_symbol</a> of <font color="Olive">b<sub>1</sub></font><br/>for <font color="Olive">b<sub>3</sub></font> being  <a href="cqc_lang.html#NM2">CQC-variable_list</a> of <font color="Olive">b<sub>1</sub></font> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font> <a href="cqc_lang.html#K8">!</a> <font color="Olive">b<sub>3</sub></font>]
 
<b>by </b><i><a class="ref" href="cqc_sim1.html#T15">CQC_SIM1:15</a></i>;<br/>
<a NAME="E3:45"/><i><font color="Green">E27</font></i>: 
for <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="cqc_lang.html#K10">'not'</a> <font color="Olive">b<sub>1</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></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="E1:45_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>]
 ;<br/>

<b>assume </b><a NAME="E2:45_1"/><i><font color="Green">E29</font></i>: 
 <a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">c<sub>2</sub></font> <a href="qc_lang2.html#R2">is_subformula_of</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E3:45_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="qc_lang2.html#R1">is_immediate_constituent_of</a>  <a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="qc_lang2.html#T60">QC_LANG2:60</a></i>;<br/>
<a NAME="E4:45_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="qc_lang2.html#R3">is_proper_subformula_of</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="substut2.html#E2:45_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="qc_lang2.html#T83">QC_LANG2:83</a></i>;<br/>
<b>hence </b><a NAME="E5:45_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">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="txt" href="substut2.html#E1:45_1"><i><font color="Green">E28</font></i></a>, <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="E4:45"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></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">b<sub>1</sub></font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Olive">b<sub>2</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></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="E1:45_2"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>2</sub></font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>3</sub></font>] )
 ;<br/>

<b>assume </b><a NAME="E2:45_2"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">c<sub>3</sub></font> <a href="qc_lang2.html#R2">is_subformula_of</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E3:45_2"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="qc_lang2.html#R1">is_immediate_constituent_of</a> <font color="Maroon">c<sub>2</sub></font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="qc_lang2.html#R1">is_immediate_constituent_of</a> <font color="Maroon">c<sub>2</sub></font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="ref" href="qc_lang2.html#T62">QC_LANG2:62</a></i>;<br/>
<a NAME="E4:45_2"/><b>then </b>
( <font color="Maroon">c<sub>2</sub></font> <a href="qc_lang2.html#R3">is_proper_subformula_of</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="qc_lang2.html#R3">is_proper_subformula_of</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="substut2.html#E2:45_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="qc_lang2.html#T83">QC_LANG2:83</a></i>;<br/>
<a NAME="E5:45_2"/><b>then </b>
 <a href="cqc_sim1.html#K6">QuantNbr</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 0 <a href="nat_1.html#K1">+</a> 0
 
<b>by </b><i><a class="txt" href="substut2.html#E1:45_2"><i><font color="Green">E29</font></i></a>, <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="E6:45_2"/>
 <a href="cqc_sim1.html#K6">QuantNbr</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 0
 ;<br/>


</div><b>end;</b></div>
<a NAME="E5:45"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="qc_lang1.html#NM3">bound_QC-variable</a><br/>for <font color="Olive">b<sub>2</sub></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">b<sub>2</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="cqc_lang.html#K15">All</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="qc_lang1.html#NM3">bound_QC-variable</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></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="E1:45_3"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>3</sub></font>]
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:45_3_1"/><i><font color="Green">E31</font></i>: 
 <a href="cqc_lang.html#K15">All</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="qc_lang2.html#R2">is_subformula_of</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><a NAME="E2:45_3_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="qc_lang2.html#R1">is_immediate_constituent_of</a>  <a href="cqc_lang.html#K15">All</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="qc_lang2.html#T63">QC_LANG2:63</a></i>;<br/><a NAME="E3:45_3_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="qc_lang2.html#R3">is_proper_subformula_of</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="substut2.html#E1:45_3_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="qc_lang2.html#T83">QC_LANG2:83</a></i>;<br/><a NAME="E4:45_3_1"/><b>then </b>
 <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">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 0 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="substut2.html#E1:45_3"><i><font color="Green">E30</font></i></a>, <a class="ref" href="cqc_sim1.html#T18">CQC_SIM1:18</a>, <a class="ref" href="qc_lang2.html#D22">QC_LANG2:def 22</a></i>;<br/><b>hence </b><a NAME="E5:45_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="substut2.html#E1:45"><i><font color="Green">E25</font></i></a>, <a class="txt" href="substut2.html#E1:45_3_1"><i><font color="Green">E31</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:45_3"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="cqc_lang.html#K15">All</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E6:45"/><i><font color="Green">E30</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></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">b<sub>3</sub></font> being  <a href="qc_lang1.html#NM3">bound_QC-variable</a><br/>for <font color="Olive">b<sub>4</sub></font> being  <a href="nat_1.html#NM1">Nat</a><br/>for <font color="Olive">b<sub>5</sub></font> being  <a href="cqc_lang.html#NM2">CQC-variable_list</a> of <font color="Olive">b<sub>4</sub></font><br/>for <font color="Olive">b<sub>6</sub></font> being  <a href="qc_lang1.html#NM6">QC-pred_symbol</a> of <font color="Olive">b<sub>4</sub></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">b<sub>6</sub></font> <a href="cqc_lang.html#K8">!</a> <font color="Olive">b<sub>5</sub></font>] &amp; ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] implies <font color="Maroon">S<sub>1</sub></font>[ <a href="cqc_lang.html#K10">'not'</a> <font color="Olive">b<sub>1</sub></font>] ) &amp; ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Olive">b<sub>2</sub></font>] ) &amp; ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] implies <font color="Maroon">S<sub>1</sub></font>[ <a href="cqc_lang.html#K15">All</a> <font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] ) )
 
<b>by </b><i><a class="txt" href="substut2.html#E2:45"><i><font color="Green">E26</font></i></a>, <a class="txt" href="substut2.html#E3:45"><i><font color="Green">E27</font></i></a>, <a class="txt" href="substut2.html#E4:45"><i><font color="Green">E28</font></i></a>, <a class="txt" href="substut2.html#E5:45"><i><font color="Green">E29</font></i></a>, <a class="ref" href="cqc_sim1.html#T14">CQC_SIM1:14</a></i>;<br/>
<a NAME="E7:45"/>
for <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="cqc_lang.html#S1">CQC_LANG:sch 1</a>(<a class="txt" href="substut2.html#E6:45"><i><font color="Green">E30</font></i></a>);<br/></i>
<b>hence </b><a NAME="E8:45"/>
 <a href="cqc_sim1.html#K6">QuantNbr</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/>


</div>
