<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">p</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> ;<br/><b>let </b><font color="Maroon">q</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> ;<br/>



<b>thus </b><a NAME="E1:18"/>
( <font color="Maroon">p</font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">q</font> is <a href="cqc_the1.html#V2">valid</a> &amp; <font color="Maroon">q</font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">p</font> is <a href="cqc_the1.html#V2">valid</a> implies <font color="Maroon">p</font> <a href="cqc_lang.html#K14">&lt;=&gt;</a> <font color="Maroon">q</font> is <a href="cqc_the1.html#V2">valid</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_1"/>
( <font color="Maroon">p</font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">q</font> is <a href="cqc_the1.html#V2">valid</a> &amp; <font color="Maroon">q</font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">p</font> is <a href="cqc_the1.html#V2">valid</a> )
 ;<br/>

<a NAME="E2:18_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">q</font></span>)</span> <a href="cqc_lang.html#K11">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">p</font></span>)</span> is <a href="cqc_the1.html#V2">valid</a>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E3:18_1"/>
<font color="Maroon">p</font> <a href="cqc_lang.html#K14">&lt;=&gt;</a> <font color="Maroon">q</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/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:18"/>
<font color="Maroon">p</font> <a href="cqc_lang.html#K14">&lt;=&gt;</a> <font color="Maroon">q</font> is <a href="cqc_the1.html#V2">valid</a>
 ;<br/>

<a NAME="E3:18"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">q</font></span>)</span> <a href="cqc_lang.html#K11">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">p</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/>
<b>hence </b><a NAME="E4:18"/>
( <font color="Maroon">p</font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">q</font> is <a href="cqc_the1.html#V2">valid</a> &amp; <font color="Maroon">q</font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">p</font> is <a href="cqc_the1.html#V2">valid</a> )
 <b>by </b><i/>;<br/>


</div>
