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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="E1:70"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="qc_lang1.html#K13">'&amp;'</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K12">'not'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="qc_lang2.html#R3" target="_self">is_proper_subformula_of</a> <font color="Maroon">c<sub>1</sub></font> <a href="qc_lang2.html#K2" target="_self">=&gt;</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="qc_lang2.html#T86" target="_self">Th86</a></i>;<br/>

<a NAME="E2:70"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="qc_lang2.html#R3" target="_self">is_proper_subformula_of</a> <font color="Maroon">c<sub>1</sub></font> <a href="qc_lang1.html#K13">'&amp;'</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K12">'not'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> &amp;  <a href="qc_lang1.html#K12">'not'</a> <font color="Maroon">c<sub>2</sub></font> <a href="qc_lang2.html#R3" target="_self">is_proper_subformula_of</a> <font color="Maroon">c<sub>1</sub></font> <a href="qc_lang1.html#K13">'&amp;'</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K12">'not'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="qc_lang2.html#T89" target="_self">Th89</a></i>;<br/>
<b>hence </b><a NAME="E3:70"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="qc_lang2.html#R3" target="_self">is_proper_subformula_of</a> <font color="Maroon">c<sub>1</sub></font> <a href="qc_lang2.html#K2" target="_self">=&gt;</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="qc_lang1.html#K12">'not'</a> <font color="Maroon">c<sub>2</sub></font> <a href="qc_lang2.html#R3" target="_self">is_proper_subformula_of</a> <font color="Maroon">c<sub>1</sub></font> <a href="qc_lang2.html#K2" target="_self">=&gt;</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="qc_lang2.html#E1:70"><i><font color="Green">E48</font></i></a>, <a class="ref" href="qc_lang2.html#T76" target="_self">Th76</a></i>;<br/>

<a NAME="E4:70"/>
<font color="Maroon">c<sub>2</sub></font> <a href="qc_lang2.html#R3" target="_self">is_proper_subformula_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#T86" target="_self">Th86</a></i>;<br/>
<b>hence </b><a NAME="E5:70"/>
<font color="Maroon">c<sub>2</sub></font> <a href="qc_lang2.html#R3" target="_self">is_proper_subformula_of</a> <font color="Maroon">c<sub>1</sub></font> <a href="qc_lang2.html#K2" target="_self">=&gt;</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="qc_lang2.html#E3:70"><i><font color="Green">E49</font></i></a>, <a class="ref" href="qc_lang2.html#T76" target="_self">Th76</a></i>;<br/>


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