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<b>let </b><font color="Maroon">p1</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">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">q1</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>assume </b><b>that </b><br/><a NAME="E1:54"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">p1</font> <a href="cqc_the3.html#R6" target="_self">is_an_universal_closure_of</a> <font color="Maroon">p</font>
 <b>and </b><br/><a NAME="E2:54"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">q1</font> <a href="cqc_the3.html#R6" target="_self">is_an_universal_closure_of</a> <font color="Maroon">q</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:54_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">p</font> <a href="cqc_the3.html#R1" target="_self">|-</a> <font color="Maroon">q</font>
 ;<br/><a NAME="E2:54_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">p1</font> is <a href="qc_lang1.html#V6">closed</a>
 <b>by </b><i><a class="ref" href="cqc_the3.html#T4" target="_self">Th4</a>, </i>;<br/><a NAME="E3:54_1"/>
<font color="Maroon">p1</font> <a href="cqc_the3.html#R5" target="_self">|-|</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="cqc_the3.html#T4" target="_self">Th4</a>, </i>;<br/><a NAME="E4:54_1"/><b>then </b>
<font color="Maroon">p1</font> <a href="cqc_the3.html#R1" target="_self">|-</a> <font color="Maroon">p</font>
 <b>by </b><i/>;<br/><a NAME="E5:54_1"/><b>then </b><i><font color="Green">E61</font></i>: 
<font color="Maroon">p1</font> <a href="cqc_the3.html#R1" target="_self">|-</a> <font color="Maroon">q</font>
 <b>by </b><i><a class="ref" href="cqc_the3.html#T5" target="_self">Th5</a>, </i>;<br/><a NAME="E6:54_1"/>
<font color="Maroon">q</font> <a href="cqc_the3.html#R5" target="_self">|-|</a> <font color="Maroon">q1</font>
 <b>by </b><i><a class="ref" href="cqc_the3.html#D1" target="_self">Def1</a>, </i>;<br/><a NAME="E7:54_1"/><b>then </b>
<font color="Maroon">q</font> <a href="cqc_the3.html#R1" target="_self">|-</a> <font color="Maroon">q1</font>
 <b>by </b><i/>;<br/><a NAME="E8:54_1"/><b>then </b>
<font color="Maroon">p1</font> <a href="cqc_the3.html#R1" target="_self">|-</a> <font color="Maroon">q1</font>
 <b>by </b><i><a class="ref" href="cqc_the3.html#D2" target="_self">Def2</a>, </i>;<br/><b>hence </b><a NAME="E9:54_1"/>
 <a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">q1</font> <a href="cqc_the3.html#R1" target="_self">|-</a>  <a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p1</font>
 <b>by </b><i><a class="ref" href="cqc_the3.html#T6" target="_self">Th6</a>, <a class="ref" href="cqc_the3.html#T3" target="_self">Th3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:54"/>
( <font color="Maroon">p</font> <a href="cqc_the3.html#R1" target="_self">|-</a> <font color="Maroon">q</font> implies  <a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">q1</font> <a href="cqc_the3.html#R1" target="_self">|-</a>  <a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p1</font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:54_2"/><i><font color="Green">E77</font></i>: 
 <a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">q1</font> <a href="cqc_the3.html#R1" target="_self">|-</a>  <a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p1</font>
 ;<br/><a NAME="E2:54_2"/>
<font color="Maroon">q1</font> is <a href="qc_lang1.html#V6">closed</a>
 <b>by </b><i><a class="ref" href="cqc_the3.html#D1" target="_self">Def1</a>, </i>;<br/><a NAME="E3:54_2"/><b>then </b><i><font color="Green">E85</font></i>: 
<font color="Maroon">p1</font> <a href="cqc_the3.html#R1" target="_self">|-</a> <font color="Maroon">q1</font>
 <b>by </b><i><a class="ref" href="cqc_the3.html#T5" target="_self">Th5</a>, <a class="txt" href="cqc_the3.html#E2:54"><i><font color="Green">E42</font></i></a></i>;<br/><a NAME="E4:54_2"/>
<font color="Maroon">p1</font> <a href="cqc_the3.html#R5" target="_self">|-|</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="cqc_the3.html#T4" target="_self">Th4</a>, </i>;<br/><a NAME="E5:54_2"/><b>then </b>
<font color="Maroon">p</font> <a href="cqc_the3.html#R1" target="_self">|-</a> <font color="Maroon">p1</font>
 <b>by </b><i/>;<br/><a NAME="E6:54_2"/><b>then </b><i><font color="Green">E86</font></i>: 
<font color="Maroon">p</font> <a href="cqc_the3.html#R1" target="_self">|-</a> <font color="Maroon">q1</font>
 <b>by </b><i><a class="ref" href="cqc_the3.html#T6" target="_self">Th6</a>, </i>;<br/><a NAME="E7:54_2"/>
<font color="Maroon">q1</font> <a href="cqc_the3.html#R5" target="_self">|-|</a> <font color="Maroon">q</font>
 <b>by </b><i><a class="ref" href="cqc_the3.html#D1" target="_self">Def1</a>, </i>;<br/><a NAME="E8:54_2"/><b>then </b>
<font color="Maroon">q1</font> <a href="cqc_the3.html#R1" target="_self">|-</a> <font color="Maroon">q</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E9:54_2"/>
<font color="Maroon">p</font> <a href="cqc_the3.html#R1" target="_self">|-</a> <font color="Maroon">q</font>
 <b>by </b><i><a class="ref" href="cqc_the3.html#D2" target="_self">Def2</a>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:54"/>
(  <a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">q1</font> <a href="cqc_the3.html#R1" target="_self">|-</a>  <a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">p1</font> implies <font color="Maroon">p</font> <a href="cqc_the3.html#R1" target="_self">|-</a> <font color="Maroon">q</font> )
 ;<br/>


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