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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a> ;<br/>



<b>thus </b><a NAME="E1:41"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="qc_lang2.html#R1" target="_self">is_immediate_constituent_of</a>  <a href="qc_lang1.html#K12">'not'</a> <font color="Maroon">c<sub>2</sub></font> implies <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:41_1"/><i><font color="Green">E24</font></i>: 
(  <a href="qc_lang1.html#K12">'not'</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K12">'not'</a> <font color="Maroon">c<sub>1</sub></font> or ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a>  st <br/>(  <a href="qc_lang1.html#K12">'not'</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="qc_lang1.html#K13">'&amp;'</a> <font color="Olive">b<sub>1</sub></font> or  <a href="qc_lang1.html#K12">'not'</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="qc_lang1.html#K13">'&amp;'</a> <font color="Maroon">c<sub>1</sub></font> ) or ex <font color="Olive">b<sub>1</sub></font> being  <a href="qc_lang1.html#NM3">bound_QC-variable</a> st  <a href="qc_lang1.html#K12">'not'</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K14">All</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="qc_lang2.html#D20" target="_self">QC_LANG2:def 20</a><br/>

<a NAME="E2:41_1"/>
 <a href="qc_lang1.html#K12">'not'</a> <font color="Maroon">c<sub>2</sub></font> is <a href="qc_lang1.html#V3">negative</a>
 
<b>by </b><i><a class="ref" href="qc_lang1.html#D18">QC_LANG1:def 18</a></i>;<br/>
<a NAME="E3:41_1"/><b>then </b><i><font color="Green">E25</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p3">(<span class="default"><a href="qc_lang1.html#K12">'not'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> 1</span>)</span> <a href="mcart_1.html#K1">`1</a>  <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="qc_lang1.html#T49">QC_LANG1:49</a></i>;<br/>
<div><i><font color="Green">E26</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">c<sub>3</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a> <b> such that </b><a NAME="E1:41_1_1"/><i><font color="Green">E27</font></i>: 
(  <a href="qc_lang1.html#K12">'not'</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="qc_lang1.html#K13">'&amp;'</a> <font color="Maroon">c<sub>3</sub></font> or  <a href="qc_lang1.html#K12">'not'</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="qc_lang1.html#K13">'&amp;'</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/><a NAME="E2:41_1_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="qc_lang1.html#K13">'&amp;'</a> <font color="Maroon">c<sub>3</sub></font> is <a href="qc_lang1.html#V4">conjunctive</a> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="qc_lang1.html#K13">'&amp;'</a> <font color="Maroon">c<sub>1</sub></font> is <a href="qc_lang1.html#V4">conjunctive</a> )
 <b>by </b><i><a class="ref" href="qc_lang1.html#D19">QC_LANG1:def 19</a></i>;<br/><b>hence </b><a NAME="E3:41_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="qc_lang2.html#E3:41_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="qc_lang2.html#E1:41_1_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="qc_lang1.html#T49">QC_LANG1:49</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_2"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">c<sub>3</sub></font> being   <a href="qc_lang1.html#NM3">bound_QC-variable</a><b> such that </b><a NAME="E1:41_1_2"/><i><font color="Green">E27</font></i>: 
 <a href="qc_lang1.html#K12">'not'</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K14">All</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 ;<br/><a NAME="E2:41_1_2"/>
 <a href="qc_lang1.html#K14">All</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font> is <a href="qc_lang1.html#V5">universal</a>
 <b>by </b><i><a class="ref" href="qc_lang1.html#D20">QC_LANG1:def 20</a></i>;<br/><b>hence </b><a NAME="E3:41_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="qc_lang2.html#E3:41_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="qc_lang2.html#E1:41_1_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="qc_lang1.html#T49">QC_LANG1:49</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:41_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="qc_lang2.html#E1:41_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="qc_lang2.html#E4:41_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="qc_lang2.html#T1" target="_self">Th1</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:41"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> implies <font color="Maroon">c<sub>1</sub></font> <a href="qc_lang2.html#R1" target="_self">is_immediate_constituent_of</a>  <a href="qc_lang1.html#K12">'not'</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="qc_lang2.html#D20" target="_self">Def20</a></i>;<br/>


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