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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#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> ;<br/>



<div><i><font color="Green">E46</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:57_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="cqc_the3.html#R7" target="_self">&lt;==&gt;</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><a NAME="E2:57_1"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K14">&lt;=&gt;</a> <font color="Maroon">c<sub>2</sub></font> is <a href="cqc_the1.html#V2">valid</a>
 <b>by </b><i><a class="ref" href="cqc_the3.html#D7" target="_self">Def7</a></i>;<br/><a NAME="E3:57_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="cqc_lang.html#K11">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> is <a href="cqc_the1.html#V2">valid</a>
 <b>by </b><i><a class="ref" href="qc_lang2.html#D4">QC_LANG2:def 4</a></i>;<br/><a NAME="E4:57_1"/><b>then </b>
 <a href="subset_1.html#K1">{}</a> <a href="cqc_lang.html#K7">CQC-WFF</a>  <a href="cqc_the1.html#R3">|-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="cqc_lang.html#K11">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="cqc_the1.html#D10">CQC_THE1:def 10</a></i>;<br/><a NAME="E5:57_1"/><b>then </b>
(  <a href="subset_1.html#K1">{}</a> <a href="cqc_lang.html#K7">CQC-WFF</a>  <a href="cqc_the1.html#R3">|-</a> <font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="subset_1.html#K1">{}</a> <a href="cqc_lang.html#K7">CQC-WFF</a>  <a href="cqc_the1.html#R3">|-</a> <font color="Maroon">c<sub>2</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="cqc_the3.html#E40"><i><font color="Green">Lemma32</font></i></a></i>;<br/><b>hence </b><a NAME="E6:57_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">c<sub>2</sub></font> is <a href="cqc_the1.html#V2">valid</a> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">c<sub>1</sub></font> is <a href="cqc_the1.html#V2">valid</a> )
 <b>by </b><i><a class="ref" href="cqc_the1.html#D10">CQC_THE1:def 10</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:57_2"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">c<sub>2</sub></font> is <a href="cqc_the1.html#V2">valid</a> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">c<sub>1</sub></font> is <a href="cqc_the1.html#V2">valid</a> )
 ;<br/><a NAME="E2:57_2"/><b>then </b>
(  <a href="subset_1.html#K1">{}</a> <a href="cqc_lang.html#K7">CQC-WFF</a>  <a href="cqc_the1.html#R3">|-</a> <font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="subset_1.html#K1">{}</a> <a href="cqc_lang.html#K7">CQC-WFF</a>  <a href="cqc_the1.html#R3">|-</a> <font color="Maroon">c<sub>2</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="ref" href="cqc_the1.html#D10">CQC_THE1:def 10</a></i>;<br/><a NAME="E3:57_2"/><b>then </b>
 <a href="subset_1.html#K1">{}</a> <a href="cqc_lang.html#K7">CQC-WFF</a>  <a href="cqc_the1.html#R3">|-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="cqc_lang.html#K11">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="cqc_the3.html#E39"><i><font color="Green">Lemma31</font></i></a></i>;<br/><a NAME="E4:57_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="cqc_lang.html#K11">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> is <a href="cqc_the1.html#V2">valid</a>
 <b>by </b><i><a class="ref" href="cqc_the1.html#D10">CQC_THE1:def 10</a></i>;<br/><a NAME="E5:57_2"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K14">&lt;=&gt;</a> <font color="Maroon">c<sub>2</sub></font> is <a href="cqc_the1.html#V2">valid</a>
 <b>by </b><i><a class="ref" href="qc_lang2.html#D4">QC_LANG2:def 4</a></i>;<br/><b>hence </b><a NAME="E6:57_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="cqc_the3.html#R7" target="_self">&lt;==&gt;</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="cqc_the3.html#D7" target="_self">Def7</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:57"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="cqc_the3.html#R7" target="_self">&lt;==&gt;</a> <font color="Maroon">c<sub>2</sub></font> iff ( <font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">c<sub>2</sub></font> is <a href="cqc_the1.html#V2">valid</a> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">c<sub>1</sub></font> is <a href="cqc_the1.html#V2">valid</a> ) )
 <b>by </b><i><a class="txt" href="cqc_the3.html#E1:57"><i><font color="Green">E46</font></i></a></i>;<br/>


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