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<b>let </b><font color="Maroon" title="c1">F</font>, <font color="Maroon" title="c2">G</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="E1:69"/><i><font color="Green" title="E45">A1</font></i>: 
<font color="Maroon" title="c1">F</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K12" title="QC_LANG1:func.12">'not'</a> <font color="Maroon" title="c2">G</font></span>)</span> <a href="qc_lang2.html#R3" title="QC_LANG2:pred.3">is_proper_subformula_of</a> <font color="Maroon" title="c1">F</font> <a href="qc_lang2.html#K2" title="QC_LANG2:func.2">=&gt;</a> <font color="Maroon" title="c2">G</font>
 <b>by </b><i><a class="ref" href="qc_lang2.html#T86" target="_self" title="QC_LANG2:th.86">Th86</a></i>;<br/>

<a NAME="E2:69"/>
( <font color="Maroon" title="c1">F</font> <a href="qc_lang2.html#R3" title="QC_LANG2:pred.3">is_proper_subformula_of</a> <font color="Maroon" title="c1">F</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K12" title="QC_LANG1:func.12">'not'</a> <font color="Maroon" title="c2">G</font></span>)</span> &amp;  <a href="qc_lang1.html#K12" title="QC_LANG1:func.12">'not'</a> <font color="Maroon" title="c2">G</font> <a href="qc_lang2.html#R3" title="QC_LANG2:pred.3">is_proper_subformula_of</a> <font color="Maroon" title="c1">F</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K12" title="QC_LANG1:func.12">'not'</a> <font color="Maroon" title="c2">G</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="qc_lang2.html#T89" target="_self" title="QC_LANG2:th.89">Th89</a></i>;<br/>
<b>hence </b><a NAME="E3:69"/><i><font color="Green" title="E46">A2</font></i>: 
( <font color="Maroon" title="c1">F</font> <a href="qc_lang2.html#R3" title="QC_LANG2:pred.3">is_proper_subformula_of</a> <font color="Maroon" title="c1">F</font> <a href="qc_lang2.html#K2" title="QC_LANG2:func.2">=&gt;</a> <font color="Maroon" title="c2">G</font> &amp;  <a href="qc_lang1.html#K12" title="QC_LANG1:func.12">'not'</a> <font color="Maroon" title="c2">G</font> <a href="qc_lang2.html#R3" title="QC_LANG2:pred.3">is_proper_subformula_of</a> <font color="Maroon" title="c1">F</font> <a href="qc_lang2.html#K2" title="QC_LANG2:func.2">=&gt;</a> <font color="Maroon" title="c2">G</font> )
 <b>by </b><i><a class="txt" href="qc_lang2.html#E1:69"><i><font color="Green" title="E45">A1</font></i></a>, <a class="ref" href="qc_lang2.html#T76" target="_self" title="QC_LANG2:th.76">Th76</a></i>;<br/>

<a NAME="E4:69"/>
<font color="Maroon" title="c2">G</font> <a href="qc_lang2.html#R3" title="QC_LANG2:pred.3">is_proper_subformula_of</a>  <a href="qc_lang1.html#K12" title="QC_LANG1:func.12">'not'</a> <font color="Maroon" title="c2">G</font>
 
<b>by </b><i><a class="ref" href="qc_lang2.html#T86" target="_self" title="QC_LANG2:th.86">Th86</a></i>;<br/>
<b>hence </b><a NAME="E5:69"/>
<font color="Maroon" title="c2">G</font> <a href="qc_lang2.html#R3" title="QC_LANG2:pred.3">is_proper_subformula_of</a> <font color="Maroon" title="c1">F</font> <a href="qc_lang2.html#K2" title="QC_LANG2:func.2">=&gt;</a> <font color="Maroon" title="c2">G</font>
 <b>by </b><i><a class="txt" href="qc_lang2.html#E3:69"><i><font color="Green" title="E46">A2</font></i></a>, <a class="ref" href="qc_lang2.html#T76" target="_self" title="QC_LANG2:th.76">Th76</a></i>;<br/>


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