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<b>let </b><font color="Maroon" title="c1">p</font>, <font color="Maroon" title="c2">q</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> ;<br/>



<b>assume </b><a NAME="E1:19"/><i><font color="Green" title="E20">A1</font></i>: 
<font color="Maroon" title="c1">p</font> <a href="cqc_lang.html#K11" title="CQC_LANG:func.11">&lt;=&gt;</a> <font color="Maroon" title="c2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cqc_the1.html#K4" title="CQC_THE1:func.4">TAUT</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="cqc_the1.html#D11" title="CQC_THE1:def.11">CQC_THE1:def 11</a><br/>

<a NAME="E2:19"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="cqc_lang.html#K11" title="CQC_LANG:func.11">&lt;=&gt;</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="cqc_lang.html#K9" title="CQC_LANG:func.9">=&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="cqc_lang.html#K9" title="CQC_LANG:func.9">=&gt;</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cqc_the1.html#K4" title="CQC_THE1:func.4">TAUT</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="cqc_lang.html#K11" title="CQC_LANG:func.11">&lt;=&gt;</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="cqc_lang.html#K9" title="CQC_LANG:func.9">=&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">q</font> <a href="cqc_lang.html#K9" title="CQC_LANG:func.9">=&gt;</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cqc_the1.html#K4" title="CQC_THE1:func.4">TAUT</a>  )
 
<b>by </b><i><a class="ref" href="procal_1.html#T23" title="PROCAL_1:th.23">PROCAL_1:23</a>, <a class="ref" href="procal_1.html#T24" title="PROCAL_1:th.24">PROCAL_1:24</a></i>;<br/>
<a NAME="E3:19"/><b>then </b><i><font color="Green" title="E21">A2</font></i>: 
( <font color="Maroon" title="c1">p</font> <a href="cqc_lang.html#K9" title="CQC_LANG:func.9">=&gt;</a> <font color="Maroon" title="c2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cqc_the1.html#K4" title="CQC_THE1:func.4">TAUT</a>  &amp; <font color="Maroon" title="c2">q</font> <a href="cqc_lang.html#K9" title="CQC_LANG:func.9">=&gt;</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cqc_the1.html#K4" title="CQC_THE1:func.4">TAUT</a>  )
 
<b>by </b><i><a class="txt" href="cqc_the2.html#E1:19"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="cqc_the1.html#T82" title="CQC_THE1:th.82">CQC_THE1:82</a></i>;<br/>
<b>thus </b><a NAME="E4:19"/>
( <font color="Maroon" title="c1">p</font> is <a href="cqc_the1.html#V2" title="CQC_THE1:attr.2">valid</a> implies <font color="Maroon" title="c2">q</font> is <a href="cqc_the1.html#V2" title="CQC_THE1:attr.2">valid</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:19_1"/>
<font color="Maroon" title="c1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cqc_the1.html#K4" title="CQC_THE1:func.4">TAUT</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="cqc_the1.html#D11" title="CQC_THE1:def.11">CQC_THE1:def 11</a><br/>

<b>hence </b><a NAME="E2:19_1"/>
<font color="Maroon" title="c2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cqc_the1.html#K4" title="CQC_THE1:func.4">TAUT</a> 
 <b>by </b><i><a class="txt" href="cqc_the2.html#E3:19"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="cqc_the1.html#T82" title="CQC_THE1:th.82">CQC_THE1:82</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="cqc_the1.html#D11" title="CQC_THE1:def.11">CQC_THE1:def 11</a><br/>


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

<b>assume </b><a NAME="E5:19"/>
<font color="Maroon" title="c2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cqc_the1.html#K4" title="CQC_THE1:func.4">TAUT</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="cqc_the1.html#D11" title="CQC_THE1:def.11">CQC_THE1:def 11</a><br/>

<b>hence </b><a NAME="E6:19"/>
<font color="Maroon" title="c1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cqc_the1.html#K4" title="CQC_THE1:func.4">TAUT</a> 
 <b>by </b><i><a class="txt" href="cqc_the2.html#E3:19"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="cqc_the1.html#T82" title="CQC_THE1:th.82">CQC_THE1:82</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="cqc_the1.html#D11" title="CQC_THE1:def.11">CQC_THE1:def 11</a><br/>


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