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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="qc_lang1.html#NM8" title="QC_LANG1:NM.8">QC-formula</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c1">p</font> is  <a href="qc_lang1.html#V6" title="QC_LANG1:attr.6">closed</a>  &amp; <font color="Maroon" title="c2">q</font> is  <a href="qc_lang1.html#V6" title="QC_LANG1:attr.6">closed</a>  ) iff <font color="Maroon" title="c1">p</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <font color="Maroon" title="c2">q</font> is  <a href="qc_lang1.html#V6" title="QC_LANG1:attr.6">closed</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E1:32"/>
( <font color="Maroon" title="c1">p</font> is  <a href="qc_lang1.html#V6" title="QC_LANG1:attr.6">closed</a>  &amp; <font color="Maroon" title="c2">q</font> is  <a href="qc_lang1.html#V6" title="QC_LANG1:attr.6">closed</a>  implies <font color="Maroon" title="c1">p</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <font color="Maroon" title="c2">q</font> is  <a href="qc_lang1.html#V6" title="QC_LANG1:attr.6">closed</a>  )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">p</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <font color="Maroon" title="c2">q</font> is  <a href="qc_lang1.html#V6" title="QC_LANG1:attr.6">closed</a>  implies ( <font color="Maroon" title="c1">p</font> is  <a href="qc_lang1.html#V6" title="QC_LANG1:attr.6">closed</a>  &amp; <font color="Maroon" title="c2">q</font> is  <a href="qc_lang1.html#V6" title="QC_LANG1:attr.6">closed</a>  ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:32_1"/>
(  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <font color="Maroon" title="c2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="qc_lang1.html#D30" title="QC_LANG1:def.30">QC_LANG1:def 30</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">p</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <font color="Maroon" title="c2">q</font> is  <a href="qc_lang1.html#V6" title="QC_LANG1:attr.6">closed</a> </span><br/>

<a NAME="E2:32_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
;<br/>
<span class="kw">hence </span><a NAME="E3:32_1"/>
 <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="qc_lang3.html#T14" target="_self" title="QC_LANG3:th.14">Th14</a></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="qc_lang1.html#D30" title="QC_LANG1:def.30">QC_LANG1:def 30</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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

<span class="kw">assume </span><a NAME="E2:32"/><span class="lab"><font color="Green" title="E18">A1</font></span>: 
 <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="qc_lang1.html#D30" title="QC_LANG1:def.30">QC_LANG1:def 30</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">p</font> is  <a href="qc_lang1.html#V6" title="QC_LANG1:attr.6">closed</a>  &amp; <font color="Maroon" title="c2">q</font> is  <a href="qc_lang1.html#V6" title="QC_LANG1:attr.6">closed</a>  )</span><br/>

<a NAME="E3:32"/>
 <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <font color="Maroon" title="c2">q</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="qc_lang3.html#T14" target="_self" title="QC_LANG3:th.14">Th14</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:32"/>
(  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <font color="Maroon" title="c2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:32"><span class="lab"><font color="Green" title="E18">A1</font></span></a>, <a class="ref" href="xboole_1.html#T15" title="XBOOLE_1:th.15">XBOOLE_1:15</a></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="qc_lang1.html#D30" title="QC_LANG1:def.30">QC_LANG1:def 30</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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