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<span class="kw">let </span><font color="Maroon" title="c1">p1</font>, <font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c3">q1</font>, <font color="Maroon" title="c4">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">p1</font> <a href="cqc_the3.html#R6" title="CQC_THE3:pred.6">is_an_universal_closure_of</a> <font color="Maroon" title="c2">p</font> &amp; <font color="Maroon" title="c3">q1</font> <a href="cqc_the3.html#R6" title="CQC_THE3:pred.6">is_an_universal_closure_of</a> <font color="Maroon" title="c4">q</font> implies ( <font color="Maroon" title="c2">p</font> <a href="cqc_the3.html#R1" title="CQC_THE3:pred.1">|-</a> <font color="Maroon" title="c4">q</font> iff  <a href="cqc_lang.html#K6" title="CQC_LANG:func.6">'not'</a> <font color="Maroon" title="c3">q1</font> <a href="cqc_the3.html#R1" title="CQC_THE3:pred.1">|-</a>  <a href="cqc_lang.html#K6" title="CQC_LANG:func.6">'not'</a> <font color="Maroon" title="c1">p1</font> ) )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:53"/><span class="lab"><font color="Green" title="E44">A1</font></span>: 
<font color="Maroon" title="c1">p1</font> <a href="cqc_the3.html#R6" title="CQC_THE3:pred.6">is_an_universal_closure_of</a> <font color="Maroon" title="c2">p</font>
 <span class="kw">and </span><br/><a NAME="E2:53"/><span class="lab"><font color="Green" title="E45">A2</font></span>: 
<font color="Maroon" title="c3">q1</font> <a href="cqc_the3.html#R6" title="CQC_THE3:pred.6">is_an_universal_closure_of</a> <font color="Maroon" title="c4">q</font>
 ; <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">p</font> <a href="cqc_the3.html#R1" title="CQC_THE3:pred.1">|-</a> <font color="Maroon" title="c4">q</font> iff  <a href="cqc_lang.html#K6" title="CQC_LANG:func.6">'not'</a> <font color="Maroon" title="c3">q1</font> <a href="cqc_the3.html#R1" title="CQC_THE3:pred.1">|-</a>  <a href="cqc_lang.html#K6" title="CQC_LANG:func.6">'not'</a> <font color="Maroon" title="c1">p1</font> )</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><span class="kw">now </span></a><div class="add"><a NAME="E1:53_1"/>
<font color="Maroon" title="c4">q</font> <a href="cqc_the3.html#R5" title="CQC_THE3:pred.5">|-|</a> <font color="Maroon" title="c3">q1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:53"><span class="lab"><font color="Green" title="E45">A2</font></span></a>, <a class="ref" href="cqc_the3.html#T38" target="_self" title="CQC_THE3:th.38">Th38</a></span>;<br/><a NAME="E2:53_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A3</font></span>: 
<font color="Maroon" title="c4">q</font> <a href="cqc_the3.html#R1" title="CQC_THE3:pred.1">|-</a> <font color="Maroon" title="c3">q1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cqc_the3.html#D5" target="_self" title="CQC_THE3:def.5">Def5</a></span>;<br/><a NAME="E3:53_1"/>
<font color="Maroon" title="c1">p1</font> <a href="cqc_the3.html#R5" title="CQC_THE3:pred.5">|-|</a> <font color="Maroon" title="c2">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="ref" href="cqc_the3.html#T38" target="_self" title="CQC_THE3:th.38">Th38</a></span>;<br/><a NAME="E4:53_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A4</font></span>: 
<font color="Maroon" title="c1">p1</font> <a href="cqc_the3.html#R1" title="CQC_THE3:pred.1">|-</a> <font color="Maroon" title="c2">p</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cqc_the3.html#D5" target="_self" title="CQC_THE3:def.5">Def5</a></span>;<br/><span class="kw">assume </span><a NAME="E5:53_1"/>
<font color="Maroon" title="c2">p</font> <a href="cqc_the3.html#R1" title="CQC_THE3:pred.1">|-</a> <font color="Maroon" title="c4">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="cqc_lang.html#K6" title="CQC_LANG:func.6">'not'</a> <font color="Maroon" title="c3">q1</font> <a href="cqc_the3.html#R1" title="CQC_THE3:pred.1">|-</a>  <a href="cqc_lang.html#K6" title="CQC_LANG:func.6">'not'</a> <font color="Maroon" title="c1">p1</font></span><br/><a NAME="E6:53_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">p1</font> <a href="cqc_the3.html#R1" title="CQC_THE3:pred.1">|-</a> <font color="Maroon" title="c4">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:53_1"><span class="lab"><font color="Green" title="E47">A4</font></span></a>, <a class="ref" href="cqc_the3.html#T6" target="_self" title="CQC_THE3:th.6">Th6</a></span>;<br/><a NAME="E7:53_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A5</font></span>: 
<font color="Maroon" title="c1">p1</font> <a href="cqc_the3.html#R1" title="CQC_THE3:pred.1">|-</a> <font color="Maroon" title="c3">q1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:53_1"><span class="lab"><font color="Green" title="E46">A3</font></span></a>, <a class="ref" href="cqc_the3.html#T6" target="_self" title="CQC_THE3:th.6">Th6</a></span>;<br/><a NAME="E8:53_1"/>
<font color="Maroon" title="c1">p1</font> is  <a href="qc_lang1.html#V6" title="QC_LANG1:attr.6">closed</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="ref" href="cqc_the3.html#D6" target="_self" title="CQC_THE3:def.6">Def6</a></span>;<br/><span class="kw">hence </span><a NAME="E9:53_1"/>
 <a href="cqc_lang.html#K6" title="CQC_LANG:func.6">'not'</a> <font color="Maroon" title="c3">q1</font> <a href="cqc_the3.html#R1" title="CQC_THE3:pred.1">|-</a>  <a href="cqc_lang.html#K6" title="CQC_LANG:func.6">'not'</a> <font color="Maroon" title="c1">p1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:53_1"><span class="lab"><font color="Green" title="E48">A5</font></span></a>, <a class="ref" href="cqc_the3.html#T43" target="_self" title="CQC_THE3:th.43">Th43</a></span>; <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">hence </span><a NAME="E4:53"/>
( <font color="Maroon" title="c2">p</font> <a href="cqc_the3.html#R1" title="CQC_THE3:pred.1">|-</a> <font color="Maroon" title="c4">q</font> implies  <a href="cqc_lang.html#K6" title="CQC_LANG:func.6">'not'</a> <font color="Maroon" title="c3">q1</font> <a href="cqc_the3.html#R1" title="CQC_THE3:pred.1">|-</a>  <a href="cqc_lang.html#K6" title="CQC_LANG:func.6">'not'</a> <font color="Maroon" title="c1">p1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="cqc_lang.html#K6" title="CQC_LANG:func.6">'not'</a> <font color="Maroon" title="c3">q1</font> <a href="cqc_the3.html#R1" title="CQC_THE3:pred.1">|-</a>  <a href="cqc_lang.html#K6" title="CQC_LANG:func.6">'not'</a> <font color="Maroon" title="c1">p1</font> implies <font color="Maroon" title="c2">p</font> <a href="cqc_the3.html#R1" title="CQC_THE3:pred.1">|-</a> <font color="Maroon" title="c4">q</font> )</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_2"><span class="kw">now </span></a><div class="add"><a NAME="E1:53_2"/>
<font color="Maroon" title="c3">q1</font> <a href="cqc_the3.html#R5" title="CQC_THE3:pred.5">|-|</a> <font color="Maroon" title="c4">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:53"><span class="lab"><font color="Green" title="E45">A2</font></span></a>, <a class="ref" href="cqc_the3.html#T38" target="_self" title="CQC_THE3:th.38">Th38</a></span>;<br/><a NAME="E2:53_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A6</font></span>: 
<font color="Maroon" title="c3">q1</font> <a href="cqc_the3.html#R1" title="CQC_THE3:pred.1">|-</a> <font color="Maroon" title="c4">q</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cqc_the3.html#D5" target="_self" title="CQC_THE3:def.5">Def5</a></span>;<br/><a NAME="E3:53_2"/>
<font color="Maroon" title="c1">p1</font> <a href="cqc_the3.html#R5" title="CQC_THE3:pred.5">|-|</a> <font color="Maroon" title="c2">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="ref" href="cqc_the3.html#T38" target="_self" title="CQC_THE3:th.38">Th38</a></span>;<br/><a NAME="E4:53_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A7</font></span>: 
<font color="Maroon" title="c2">p</font> <a href="cqc_the3.html#R1" title="CQC_THE3:pred.1">|-</a> <font color="Maroon" title="c1">p1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cqc_the3.html#D5" target="_self" title="CQC_THE3:def.5">Def5</a></span>;<br/><span class="kw">assume </span><a NAME="E5:53_2"/><span class="lab"><font color="Green" title="E48">A8</font></span>: 
 <a href="cqc_lang.html#K6" title="CQC_LANG:func.6">'not'</a> <font color="Maroon" title="c3">q1</font> <a href="cqc_the3.html#R1" title="CQC_THE3:pred.1">|-</a>  <a href="cqc_lang.html#K6" title="CQC_LANG:func.6">'not'</a> <font color="Maroon" title="c1">p1</font>
 ; <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">p</font> <a href="cqc_the3.html#R1" title="CQC_THE3:pred.1">|-</a> <font color="Maroon" title="c4">q</font></span><br/><a NAME="E6:53_2"/>
<font color="Maroon" title="c3">q1</font> is  <a href="qc_lang1.html#V6" title="QC_LANG1:attr.6">closed</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:53"><span class="lab"><font color="Green" title="E45">A2</font></span></a>, <a class="ref" href="cqc_the3.html#D6" target="_self" title="CQC_THE3:def.6">Def6</a></span>;<br/><a NAME="E7:53_2"/><span class="kw">then </span>
<font color="Maroon" title="c1">p1</font> <a href="cqc_the3.html#R1" title="CQC_THE3:pred.1">|-</a> <font color="Maroon" title="c3">q1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:53_2"><span class="lab"><font color="Green" title="E48">A8</font></span></a>, <a class="ref" href="cqc_the3.html#T45" target="_self" title="CQC_THE3:th.45">Th45</a></span>;<br/><a NAME="E8:53_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">p</font> <a href="cqc_the3.html#R1" title="CQC_THE3:pred.1">|-</a> <font color="Maroon" title="c3">q1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:53_2"><span class="lab"><font color="Green" title="E47">A7</font></span></a>, <a class="ref" href="cqc_the3.html#T6" target="_self" title="CQC_THE3:th.6">Th6</a></span>;<br/><span class="kw">hence </span><a NAME="E9:53_2"/>
<font color="Maroon" title="c2">p</font> <a href="cqc_the3.html#R1" title="CQC_THE3:pred.1">|-</a> <font color="Maroon" title="c4">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:53_2"><span class="lab"><font color="Green" title="E46">A6</font></span></a>, <a class="ref" href="cqc_the3.html#T6" target="_self" title="CQC_THE3:th.6">Th6</a></span>; <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">hence </span><a NAME="E6:53"/>
(  <a href="cqc_lang.html#K6" title="CQC_LANG:func.6">'not'</a> <font color="Maroon" title="c3">q1</font> <a href="cqc_the3.html#R1" title="CQC_THE3:pred.1">|-</a>  <a href="cqc_lang.html#K6" title="CQC_LANG:func.6">'not'</a> <font color="Maroon" title="c1">p1</font> implies <font color="Maroon" title="c2">p</font> <a href="cqc_the3.html#R1" title="CQC_THE3:pred.1">|-</a> <font color="Maroon" title="c4">q</font> )
 ; <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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