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<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> 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">c<sub>4</sub></font> be   <a href="subset_1.html#NM2">Subset</a> of <a href="cqc_lang.html#K7">CQC-WFF</a> ;<br/>







<b>assume </b><a NAME="E1:49"/><i><font color="Green">E41</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>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:49_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="cqc_the1.html#R3">|-</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E2:49_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="cqc_the3.html#R5" target="_self">|-|</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="cqc_the3.html#E1:49"><i><font color="Green">E41</font></i></a>, <a class="ref" href="cqc_the3.html#T38" target="_self">Th38</a></i>;<br/><a NAME="E3:49_1"/><b>then </b><i><font color="Green">E43</font></i>: 
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="cqc_the3.html#R4" target="_self">|-|</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="cqc_the3.html#T29" target="_self">Th29</a></i>;<br/><a NAME="E4:49_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="cqc_the3.html#R2" target="_self">|-</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="cqc_the3.html#E1:49_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="cqc_the3.html#T10" target="_self">Th10</a></i>;<br/><a NAME="E5:49_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="cqc_the3.html#R2" target="_self">|-</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="cqc_the3.html#E3:49_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="cqc_the3.html#T26" target="_self">Th26</a></i>;<br/><a NAME="E6:49_1"/><b>then </b><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="cqc_the1.html#R3">|-</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="cqc_the3.html#T10" target="_self">Th10</a></i>;<br/><a NAME="E7:49_1"/>
<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#E1:49"><i><font color="Green">E41</font></i></a>, <a class="ref" href="cqc_the3.html#D6" target="_self">Def6</a></i>;<br/><b>hence </b><a NAME="E8:49_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="cqc_the1.html#R3">|-</a> <font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="cqc_the3.html#E6:49_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="cqc_the2.html#T96">CQC_THE2:96</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:49"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="cqc_the1.html#R3">|-</a> <font color="Maroon">c<sub>3</sub></font> implies <font color="Maroon">c<sub>4</sub></font> <a href="cqc_the1.html#R3">|-</a> <font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:49_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="cqc_the1.html#R3">|-</a> <font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E2:49_2"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="cqc_the1.html#R3">|-</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="cqc_the3.html#T40" target="_self">Th40</a></i>;<br/><a NAME="E3:49_2"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="cqc_the3.html#R2" target="_self">|-</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="cqc_the3.html#T10" target="_self">Th10</a></i>;<br/><a NAME="E4:49_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="cqc_the3.html#R5" target="_self">|-|</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="cqc_the3.html#E1:49"><i><font color="Green">E41</font></i></a>, <a class="ref" href="cqc_the3.html#T38" target="_self">Th38</a></i>;<br/><a NAME="E5:49_2"/><b>then </b>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="cqc_the3.html#R4" target="_self">|-|</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="cqc_the3.html#T29" target="_self">Th29</a></i>;<br/><a NAME="E6:49_2"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="cqc_the3.html#R2" target="_self">|-</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="cqc_the3.html#E3:49_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="cqc_the3.html#T26" target="_self">Th26</a></i>;<br/><b>hence </b><a NAME="E7:49_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="cqc_the1.html#R3">|-</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="cqc_the3.html#T10" target="_self">Th10</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:49"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="cqc_the1.html#R3">|-</a> <font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K12">=&gt;</a> <font color="Maroon">c<sub>3</sub></font> implies <font color="Maroon">c<sub>4</sub></font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="domain_1.html#K6">}</a></span> <a href="cqc_the1.html#R3">|-</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>


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