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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></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">E44</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="cqc_the3.html#R6" target="_self">is_an_universal_closure_of</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E2:54"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="cqc_the3.html#R6" target="_self">is_an_universal_closure_of</a> <font color="Maroon">c<sub>4</sub></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">E46</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="cqc_the3.html#R1" target="_self">|-</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><a NAME="E2:54_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="qc_lang1.html#V6">closed</a>
 <b>by </b><i><a class="txt" href="cqc_the3.html#E1:54"><i><font color="Green">E44</font></i></a>, <a class="ref" href="cqc_the3.html#D6" target="_self">Def6</a></i>;<br/><a NAME="E3:54_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="cqc_the3.html#R5" target="_self">|-|</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="cqc_the3.html#E1:54"><i><font color="Green">E44</font></i></a>, <a class="ref" href="cqc_the3.html#T38" target="_self">Th38</a></i>;<br/><a NAME="E4:54_1"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="cqc_the3.html#R1" target="_self">|-</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="cqc_the3.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E5:54_1"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="cqc_the3.html#R1" target="_self">|-</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="cqc_the3.html#E1:54_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="cqc_the3.html#T6" target="_self">Th6</a></i>;<br/><a NAME="E6:54_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="cqc_the3.html#R5" target="_self">|-|</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="cqc_the3.html#E2:54"><i><font color="Green">E45</font></i></a>, <a class="ref" href="cqc_the3.html#T38" target="_self">Th38</a></i>;<br/><a NAME="E7:54_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="cqc_the3.html#R1" target="_self">|-</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="cqc_the3.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E8:54_1"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="cqc_the3.html#R1" target="_self">|-</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="cqc_the3.html#E5:54_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="cqc_the3.html#T6" target="_self">Th6</a></i>;<br/><b>hence </b><a NAME="E9:54_1"/>
 <a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">c<sub>3</sub></font> <a href="cqc_the3.html#R1" target="_self">|-</a>  <a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="cqc_the3.html#E2:54_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="cqc_the3.html#T43" target="_self">Th43</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:54"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="cqc_the3.html#R1" target="_self">|-</a> <font color="Maroon">c<sub>4</sub></font> implies  <a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">c<sub>3</sub></font> <a href="cqc_the3.html#R1" target="_self">|-</a>  <a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">c<sub>1</sub></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">E46</font></i>: 
 <a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">c<sub>3</sub></font> <a href="cqc_the3.html#R1" target="_self">|-</a>  <a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><a NAME="E2:54_2"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="qc_lang1.html#V6">closed</a>
 <b>by </b><i><a class="txt" href="cqc_the3.html#E2:54"><i><font color="Green">E45</font></i></a>, <a class="ref" href="cqc_the3.html#D6" target="_self">Def6</a></i>;<br/><a NAME="E3:54_2"/><b>then </b><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="cqc_the3.html#R1" target="_self">|-</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="cqc_the3.html#E1:54_2"><i><font color="Green">E46</font></i></a>, <a class="ref" href="cqc_the3.html#T45" target="_self">Th45</a></i>;<br/><a NAME="E4:54_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="cqc_the3.html#R5" target="_self">|-|</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="cqc_the3.html#E1:54"><i><font color="Green">E44</font></i></a>, <a class="ref" href="cqc_the3.html#T38" target="_self">Th38</a></i>;<br/><a NAME="E5:54_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="cqc_the3.html#R1" target="_self">|-</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="cqc_the3.html#D5" target="_self">Def5</a></i>;<br/><a NAME="E6:54_2"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="cqc_the3.html#R1" target="_self">|-</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="cqc_the3.html#E3:54_2"><i><font color="Green">E47</font></i></a>, <a class="ref" href="cqc_the3.html#T6" target="_self">Th6</a></i>;<br/><a NAME="E7:54_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="cqc_the3.html#R5" target="_self">|-|</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="cqc_the3.html#E2:54"><i><font color="Green">E45</font></i></a>, <a class="ref" href="cqc_the3.html#T38" target="_self">Th38</a></i>;<br/><a NAME="E8:54_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="cqc_the3.html#R1" target="_self">|-</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="cqc_the3.html#D5" target="_self">Def5</a></i>;<br/><b>hence </b><a NAME="E9:54_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="cqc_the3.html#R1" target="_self">|-</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="cqc_the3.html#E6:54_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="cqc_the3.html#T6" target="_self">Th6</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">c<sub>3</sub></font> <a href="cqc_the3.html#R1" target="_self">|-</a>  <a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">c<sub>1</sub></font> implies <font color="Maroon">c<sub>2</sub></font> <a href="cqc_the3.html#R1" target="_self">|-</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/>


</div>
