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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="qc_lang1.html#NM8">QC-formula</a>;<br/>

<a NAME="E1:33"/>
<font color="Maroon">c<sub>1</sub></font> <a href="qc_lang2.html#K4">&lt;=&gt;</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="qc_lang2.html#K2">=&gt;</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="qc_lang1.html#K13">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="qc_lang2.html#K2">=&gt;</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="qc_lang2.html#D4">QC_LANG2:def 4</a></i>;<br/>
<a NAME="E2:33"/><b>then </b>
( ( <font color="Maroon">c<sub>1</sub></font> <a href="qc_lang2.html#K4">&lt;=&gt;</a> <font color="Maroon">c<sub>2</sub></font> is <a href="qc_lang1.html#V6">closed</a> implies ( <font color="Maroon">c<sub>1</sub></font> <a href="qc_lang2.html#K2">=&gt;</a> <font color="Maroon">c<sub>2</sub></font> is <a href="qc_lang1.html#V6">closed</a> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="qc_lang2.html#K2">=&gt;</a> <font color="Maroon">c<sub>1</sub></font> is <a href="qc_lang1.html#V6">closed</a> ) ) &amp; ( <font color="Maroon">c<sub>1</sub></font> <a href="qc_lang2.html#K2">=&gt;</a> <font color="Maroon">c<sub>2</sub></font> is <a href="qc_lang1.html#V6">closed</a> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="qc_lang2.html#K2">=&gt;</a> <font color="Maroon">c<sub>1</sub></font> is <a href="qc_lang1.html#V6">closed</a> implies <font color="Maroon">c<sub>1</sub></font> <a href="qc_lang2.html#K4">&lt;=&gt;</a> <font color="Maroon">c<sub>2</sub></font> is <a href="qc_lang1.html#V6">closed</a> ) &amp; ( <font color="Maroon">c<sub>2</sub></font> <a href="qc_lang2.html#K2">=&gt;</a> <font color="Maroon">c<sub>1</sub></font> is <a href="qc_lang1.html#V6">closed</a> implies ( <font color="Maroon">c<sub>1</sub></font> is <a href="qc_lang1.html#V6">closed</a> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="qc_lang1.html#V6">closed</a> ) ) &amp; ( <font color="Maroon">c<sub>1</sub></font> is <a href="qc_lang1.html#V6">closed</a> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="qc_lang1.html#V6">closed</a> implies <font color="Maroon">c<sub>2</sub></font> <a href="qc_lang2.html#K2">=&gt;</a> <font color="Maroon">c<sub>1</sub></font> is <a href="qc_lang1.html#V6">closed</a> ) &amp; ( <font color="Maroon">c<sub>1</sub></font> <a href="qc_lang2.html#K2">=&gt;</a> <font color="Maroon">c<sub>2</sub></font> is <a href="qc_lang1.html#V6">closed</a> implies ( <font color="Maroon">c<sub>1</sub></font> is <a href="qc_lang1.html#V6">closed</a> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="qc_lang1.html#V6">closed</a> ) ) &amp; ( <font color="Maroon">c<sub>1</sub></font> is <a href="qc_lang1.html#V6">closed</a> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="qc_lang1.html#V6">closed</a> implies <font color="Maroon">c<sub>1</sub></font> <a href="qc_lang2.html#K2">=&gt;</a> <font color="Maroon">c<sub>2</sub></font> is <a href="qc_lang1.html#V6">closed</a> ) )
 
<b>by </b><i><a class="ref" href="qc_lang3.html#T26" target="_self">Th26</a>, <a class="ref" href="qc_lang3.html#T30" target="_self">Th30</a></i>;<br/>
<b>hence </b><a NAME="E3:33"/>
( ( <font color="Maroon">c<sub>1</sub></font> is <a href="qc_lang1.html#V6">closed</a> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="qc_lang1.html#V6">closed</a> ) iff <font color="Maroon">c<sub>1</sub></font> <a href="qc_lang2.html#K4">&lt;=&gt;</a> <font color="Maroon">c<sub>2</sub></font> is <a href="qc_lang1.html#V6">closed</a> )
 ;<br/>


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