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<b>let </b><font color="Maroon" title="c1">F</font>, <font color="Maroon" title="c2">G</font>, <font color="Maroon" title="c3">H</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K8" title="QC_LANG1:func.8">QC-WFF</a> ;<br/>





<b>thus </b><a NAME="E1:42"/>
(  not <font color="Maroon" title="c1">F</font> <a href="qc_lang2.html#R1" title="QC_LANG2:pred.1">is_immediate_constituent_of</a> <font color="Maroon" title="c2">G</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <font color="Maroon" title="c3">H</font> or <font color="Maroon" title="c1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> or <font color="Maroon" title="c1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">H</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:42_1"/><i><font color="Green" title="E25">A1</font></i>: 
( <font color="Maroon" title="c2">G</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <font color="Maroon" title="c3">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K12" title="QC_LANG1:func.12">'not'</a> <font color="Maroon" title="c1">F</font> or  ex <font color="Olive" title="b1">H1</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K8" title="QC_LANG1:func.8">QC-WFF</a>  st <br/>( <font color="Maroon" title="c2">G</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <font color="Maroon" title="c3">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">F</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <font color="Olive" title="b1">H1</font> or <font color="Maroon" title="c2">G</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <font color="Maroon" title="c3">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">H1</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <font color="Maroon" title="c1">F</font> ) or  ex <font color="Olive" title="b1">x</font> being  <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a> st <font color="Maroon" title="c2">G</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <font color="Maroon" title="c3">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> <font color="Olive" title="b1">x</font>,<font color="Maroon" title="c1">F</font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="qc_lang2.html#D20" target="_self" title="QC_LANG2:def.20">QC_LANG2:def 20</a><br/>

<a NAME="E2:42_1"/>
<font color="Maroon" title="c2">G</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <font color="Maroon" title="c3">H</font> is <a href="qc_lang1.html#V4" title="QC_LANG1:attr.4">conjunctive</a>
 
<b>by </b><i><a class="ref" href="qc_lang1.html#D19" title="QC_LANG1:def.19">QC_LANG1:def 19</a></i>;<br/>
<a NAME="E3:42_1"/><b>then </b><i><font color="Green" title="E26">A2</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">G</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <font color="Maroon" title="c3">H</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 
<b>by </b><i><a class="ref" href="qc_lang1.html#T49" title="QC_LANG1:th.49">QC_LANG1:49</a></i>;<br/>
<div><i><font color="Green" title="E27">A3</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:42_1_1"/><i><font color="Green" title="E28">A4</font></i>: 
<font color="Maroon" title="c2">G</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <font color="Maroon" title="c3">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K12" title="QC_LANG1:func.12">'not'</a> <font color="Maroon" title="c1">F</font>
 ;<br/><a NAME="E2:42_1_1"/>
 <a href="qc_lang1.html#K12" title="QC_LANG1:func.12">'not'</a> <font color="Maroon" title="c1">F</font> is <a href="qc_lang1.html#V3" title="QC_LANG1:attr.3">negative</a>
 <b>by </b><i><a class="ref" href="qc_lang1.html#D18" title="QC_LANG1:def.18">QC_LANG1:def 18</a></i>;<br/><b>hence </b><a NAME="E3:42_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="qc_lang2.html#E3:42_1"><i><font color="Green" title="E26">A2</font></i></a>, <a class="txt" href="qc_lang2.html#E1:42_1_1"><i><font color="Green" title="E28">A4</font></i></a>, <a class="ref" href="qc_lang1.html#T49" title="QC_LANG1:th.49">QC_LANG1:49</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_2"><b>now </b></a><div class="add"><b>given </b><font color="Maroon" title="c4">x</font> being   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a><b> such that </b><a NAME="E1:42_1_2"/><i><font color="Green" title="E28">A5</font></i>: 
<font color="Maroon" title="c2">G</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <font color="Maroon" title="c3">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c1">F</font>
 ;<br/><a NAME="E2:42_1_2"/>
 <a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c1">F</font> is <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a>
 <b>by </b><i><a class="ref" href="qc_lang1.html#D20" title="QC_LANG1:def.20">QC_LANG1:def 20</a></i>;<br/><b>hence </b><a NAME="E3:42_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="qc_lang2.html#E3:42_1"><i><font color="Green" title="E26">A2</font></i></a>, <a class="txt" href="qc_lang2.html#E1:42_1_2"><i><font color="Green" title="E28">A5</font></i></a>, <a class="ref" href="qc_lang1.html#T49" title="QC_LANG1:th.49">QC_LANG1:49</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:42_1"/>
( <font color="Maroon" title="c1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> or <font color="Maroon" title="c1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">H</font> )
 <b>by </b><i><a class="txt" href="qc_lang2.html#E1:42_1"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="qc_lang2.html#E4:42_1"><i><font color="Green" title="E27">A3</font></i></a>, <a class="ref" href="qc_lang2.html#T3" target="_self" title="QC_LANG2:th.3">Th3</a></i>;<br/>


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

<b>assume </b><a NAME="E2:42"/>
( <font color="Maroon" title="c1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> or <font color="Maroon" title="c1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">H</font> )
 ;<br/>

<b>hence </b><a NAME="E3:42"/>
<font color="Maroon" title="c1">F</font> <a href="qc_lang2.html#R1" title="QC_LANG2:pred.1">is_immediate_constituent_of</a> <font color="Maroon" title="c2">G</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <font color="Maroon" title="c3">H</font>
 <b>by </b><i><a class="ref" href="qc_lang2.html#D20" target="_self" title="QC_LANG2:def.20">Def20</a></i>;<br/>


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