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<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <a href="qc_lang1.html#K1">QC-variables</a> ;<br/>



<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="qc_lang2.html#K12">the_antecedent_of</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="qc_lang2.html#NK13" target="_self">the_consequent_of</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>assume </b><a NAME="E1:53"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="qc_lang2.html#V2">conditional</a>
 ;<br/>

<a NAME="E2:53"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="qc_lang2.html#K12">the_antecedent_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="qc_lang2.html#K2">=&gt;</a> <span class="p1">(<span class="default"><a href="qc_lang2.html#NK13" target="_self">the_consequent_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="qc_lang2.html#T54">QC_LANG2:54</a></i>;<br/>
<a NAME="E3:53"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K12">'not'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang2.html#K12">the_antecedent_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="qc_lang1.html#K13">'&amp;'</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K12">'not'</a> <span class="p3">(<span class="default"><a href="qc_lang2.html#NK13" target="_self">the_consequent_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="qc_lang2.html#D2">QC_LANG2:def 2</a></i>;<br/>
<b>hence </b> <a href="qc_lang3.html#K4" target="_self">Vars</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> = 
 <a href="qc_lang3.html#K4" target="_self">Vars</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang2.html#K12">the_antecedent_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="qc_lang1.html#K13">'&amp;'</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K12">'not'</a> <span class="p3">(<span class="default"><a href="qc_lang2.html#NK13" target="_self">the_consequent_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>,<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="qc_lang3.html#T50" target="_self">Th50</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="qc_lang3.html#K4" target="_self">Vars</a> <span class="p2">(<span class="default"><a href="qc_lang2.html#K12">the_antecedent_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="qc_lang3.html#K4" target="_self">Vars</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K12">'not'</a> <span class="p3">(<span class="default"><a href="qc_lang2.html#NK13" target="_self">the_consequent_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="qc_lang3.html#T53" target="_self">Th53</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="qc_lang3.html#K4" target="_self">Vars</a> <span class="p2">(<span class="default"><a href="qc_lang2.html#K12">the_antecedent_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="qc_lang3.html#K4" target="_self">Vars</a> <span class="p2">(<span class="default"><a href="qc_lang2.html#NK13" target="_self">the_consequent_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="qc_lang3.html#T50" target="_self">Th50</a></i>
;<br/><br/>


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