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<span class="kw">let </span><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#K3" title="CQC_LANG:func.3">CQC-WFF</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">p</font> <a href="cqc_lang.html#K10" title="CQC_LANG:func.10">&lt;=&gt;</a> <font color="Maroon" title="c2">q</font> is  <a href="cqc_the1.html#V2" title="CQC_THE1:attr.2">valid</a>  implies ( <font color="Maroon" title="c1">p</font> is  <a href="cqc_the1.html#V2" title="CQC_THE1:attr.2">valid</a>  iff <font color="Maroon" title="c2">q</font> is  <a href="cqc_the1.html#V2" title="CQC_THE1:attr.2">valid</a>  ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:19"/><span class="lab"><font color="Green" title="E20">A1</font></span>: 
<font color="Maroon" title="c1">p</font> <a href="cqc_lang.html#K10" title="CQC_LANG:func.10">&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> 
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="cqc_the1.html#D10" title="CQC_THE1:def.10">CQC_THE1:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">p</font> is  <a href="cqc_the1.html#V2" title="CQC_THE1:attr.2">valid</a>  iff <font color="Maroon" title="c2">q</font> is  <a href="cqc_the1.html#V2" title="CQC_THE1:attr.2">valid</a>  )</span><br/>

<a NAME="E2:19"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="cqc_lang.html#K10" title="CQC_LANG:func.10">&lt;=&gt;</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">=&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">=&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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="procal_1.html#T23" title="PROCAL_1:th.23">PROCAL_1:23</a></span>;<br/>
<a NAME="E3:19"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A2</font></span>: 
<font color="Maroon" title="c1">p</font> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">=&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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="ref" href="cqc_the1.html#T46" title="CQC_THE1:th.46">CQC_THE1:46</a></span>;<br/>
<span class="kw">thus </span><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>  )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">q</font> is  <a href="cqc_the1.html#V2" title="CQC_THE1:attr.2">valid</a>  implies <font color="Maroon" title="c1">p</font> is  <a href="cqc_the1.html#V2" title="CQC_THE1:attr.2">valid</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><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> 
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="cqc_the1.html#D10" title="CQC_THE1:def.10">CQC_THE1:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">q</font> is  <a href="cqc_the1.html#V2" title="CQC_THE1:attr.2">valid</a> </span><br/>

<span class="kw">hence </span><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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:19"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="ref" href="cqc_the1.html#T46" title="CQC_THE1:th.46">CQC_THE1:46</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="cqc_the1.html#D10" title="CQC_THE1:def.10">CQC_THE1:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E5:19"/><span class="lab"><font color="Green" title="E22">A3</font></span>: 
<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> 
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="cqc_the1.html#D10" title="CQC_THE1:def.10">CQC_THE1:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">p</font> is  <a href="cqc_the1.html#V2" title="CQC_THE1:attr.2">valid</a> </span><br/>

<a NAME="E6:19"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="cqc_lang.html#K10" title="CQC_LANG:func.10">&lt;=&gt;</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">=&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">q</font> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">=&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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="procal_1.html#T24" title="PROCAL_1:th.24">PROCAL_1:24</a></span>;<br/>
<a NAME="E7:19"/><span class="kw">then </span>
<font color="Maroon" title="c2">q</font> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">=&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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:19"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="ref" href="cqc_the1.html#T46" title="CQC_THE1:th.46">CQC_THE1:46</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:19"/>
<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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:19"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="ref" href="cqc_the1.html#T46" title="CQC_THE1:th.46">CQC_THE1:46</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="cqc_the1.html#D10" title="CQC_THE1:def.10">CQC_THE1:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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