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<span class="kw">let </span><font color="Maroon" title="c1">A</font> be    <a href="qc_lang1.html#M1" title="QC_LANG1:mode.1">QC-alphabet</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font> being    <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> <font color="Maroon" title="c1">A</font><br/>  for <font color="Olive" title="b3">x</font> being   <a href="qc_lang1.html#NM5" title="QC_LANG1:NM.5">bound_QC-variable</a> of <font color="Maroon" title="c1">A</font>  st  not <font color="Olive" title="b3">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="qc_lang1.html#K24" title="QC_LANG1:func.24">still_not-bound_in</a> <font color="Olive" title="b2">q</font> holds <br/><span class="p1">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">Ex</a> (<font color="Olive" title="b3">x</font>,<span class="p2">(<span class="default"><font color="Olive" title="b1">p</font> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">=&gt;</a> <font color="Olive" title="b2">q</font></span>)</span>)</span>)</span> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">=&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K11" title="CQC_LANG:func.11">All</a> (<font color="Olive" title="b3">x</font>,<font color="Olive" title="b1">p</font>)</span>)</span> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">=&gt;</a> <font color="Olive" title="b2">q</font></span>)</span> is  <a href="cqc_the1.html#V2" title="CQC_THE1:attr.2">valid</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c3">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> <font color="Maroon" title="c1">A</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font> being   <a href="qc_lang1.html#NM5" title="QC_LANG1:NM.5">bound_QC-variable</a> of <font color="Maroon" title="c1">A</font>  st  not <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="qc_lang1.html#K24" title="QC_LANG1:func.24">still_not-bound_in</a> <font color="Maroon" title="c3">q</font> holds <br/><span class="p1">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">Ex</a> (<font color="Olive" title="b1">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">=&gt;</a> <font color="Maroon" title="c3">q</font></span>)</span>)</span>)</span> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">=&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K11" title="CQC_LANG:func.11">All</a> (<font color="Olive" title="b1">x</font>,<font color="Maroon" title="c2">p</font>)</span>)</span> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">=&gt;</a> <font color="Maroon" title="c3">q</font></span>)</span> is  <a href="cqc_the1.html#V2" title="CQC_THE1:attr.2">valid</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c4">x</font> be   <a href="qc_lang1.html#NM5" title="QC_LANG1:NM.5">bound_QC-variable</a> of <font color="Maroon" title="c1">A</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c4">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="qc_lang1.html#K24" title="QC_LANG1:func.24">still_not-bound_in</a> <font color="Maroon" title="c3">q</font> implies <span class="p1">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">Ex</a> (<font color="Maroon" title="c4">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">=&gt;</a> <font color="Maroon" title="c3">q</font></span>)</span>)</span>)</span> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">=&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K11" title="CQC_LANG:func.11">All</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c2">p</font>)</span>)</span> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">=&gt;</a> <font color="Maroon" title="c3">q</font></span>)</span> is  <a href="cqc_the1.html#V2" title="CQC_THE1:attr.2">valid</a>  )</span><br/>







<span class="kw">assume </span><a NAME="E1:83"/><span class="lab"><font color="Green" title="E64">A1</font></span>: 
 not <font color="Maroon" title="c4">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="qc_lang1.html#K24" title="QC_LANG1:func.24">still_not-bound_in</a> <font color="Maroon" title="c3">q</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">Ex</a> (<font color="Maroon" title="c4">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">=&gt;</a> <font color="Maroon" title="c3">q</font></span>)</span>)</span>)</span> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">=&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K11" title="CQC_LANG:func.11">All</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c2">p</font>)</span>)</span> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">=&gt;</a> <font color="Maroon" title="c3">q</font></span>)</span> is  <a href="cqc_the1.html#V2" title="CQC_THE1:attr.2">valid</a> </span><br/>

<a NAME="E2:83"/>
 not <font color="Maroon" title="c4">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="qc_lang1.html#K24" title="QC_LANG1:func.24">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K11" title="CQC_LANG:func.11">All</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c2">p</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cqc_the2.html#T5" target="_self" onmouseover="rs('cqc_the2/T5')" onmouseout="rh()">Th5</a></span>;<br/>
<a NAME="E3:83"/><span class="kw">then </span>
 not <font color="Maroon" title="c4">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="qc_lang1.html#K24" title="QC_LANG1:func.24">still_not-bound_in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K11" title="CQC_LANG:func.11">All</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c2">p</font>)</span>)</span> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">=&gt;</a> <font color="Maroon" title="c3">q</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:83"><span class="lab"><font color="Green" title="E64">A1</font></span></a>, <a class="ref" href="cqc_the2.html#T7" target="_self" onmouseover="rs('cqc_the2/T7')" onmouseout="rh()">Th7</a></span>;<br/>
<a NAME="E4:83"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A2</font></span>: 
<span class="p1">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">Ex</a> (<font color="Maroon" title="c4">x</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cqc_lang.html#K11" title="CQC_LANG:func.11">All</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c2">p</font>)</span>)</span> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">=&gt;</a> <font color="Maroon" title="c3">q</font></span>)</span>)</span>)</span> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">=&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K11" title="CQC_LANG:func.11">All</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c2">p</font>)</span>)</span> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">=&gt;</a> <font color="Maroon" title="c3">q</font></span>)</span> is  <a href="cqc_the1.html#V2" title="CQC_THE1:attr.2">valid</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cqc_the2.html#T20" target="_self" onmouseover="rs('cqc_the2/T20')" onmouseout="rh()">Th20</a></span>;<br/>
<a NAME="E5:83"/>
<span class="p1">(<span class="default"><a href="cqc_lang.html#K11" title="CQC_LANG:func.11">All</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c2">p</font>)</span>)</span> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">=&gt;</a> <font color="Maroon" title="c2">p</font> is  <a href="cqc_the1.html#V2" title="CQC_THE1:attr.2">valid</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cqc_the1.html#T66" onmouseover="rs('cqc_the1/T66')" onmouseout="rh()">CQC_THE1:66</a></span>;<br/>
<a NAME="E6:83"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A3</font></span>: 
 <a href="cqc_lang.html#K11" title="CQC_LANG:func.11">All</a> (<font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">=&gt;</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">=&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cqc_lang.html#K11" title="CQC_LANG:func.11">All</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c2">p</font>)</span>)</span> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">=&gt;</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span>) is  <a href="cqc_the1.html#V2" title="CQC_THE1:attr.2">valid</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cqc_the2.html#T23" target="_self" onmouseover="rs('cqc_the2/T23')" onmouseout="rh()">Th23</a>, <a class="ref" href="lukasi_1.html#T41" onmouseover="rs('lukasi_1/T41')" onmouseout="rh()">LUKASI_1:41</a></span>;<br/>
<a NAME="E7:83"/>
<span class="p1">(<span class="default"><a href="cqc_lang.html#K11" title="CQC_LANG:func.11">All</a> (<font color="Maroon" title="c4">x</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">=&gt;</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">=&gt;</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="cqc_lang.html#K11" title="CQC_LANG:func.11">All</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c2">p</font>)</span>)</span> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">=&gt;</a> <font color="Maroon" title="c3">q</font></span>)</span></span>)</span>)</span>)</span> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">=&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">Ex</a> (<font color="Maroon" title="c4">x</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">=&gt;</a> <font color="Maroon" title="c3">q</font></span>)</span>)</span>)</span> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">=&gt;</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">Ex</a> (<font color="Maroon" title="c4">x</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="cqc_lang.html#K11" title="CQC_LANG:func.11">All</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c2">p</font>)</span>)</span> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">=&gt;</a> <font color="Maroon" title="c3">q</font></span>)</span>)</span>)</span></span>)</span> is  <a href="cqc_the1.html#V2" title="CQC_THE1:attr.2">valid</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cqc_the2.html#T34" target="_self" onmouseover="rs('cqc_the2/T34')" onmouseout="rh()">Th34</a></span>;<br/>
<a NAME="E8:83"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">Ex</a> (<font color="Maroon" title="c4">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">=&gt;</a> <font color="Maroon" title="c3">q</font></span>)</span>)</span>)</span> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">=&gt;</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">Ex</a> (<font color="Maroon" title="c4">x</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cqc_lang.html#K11" title="CQC_LANG:func.11">All</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c2">p</font>)</span>)</span> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">=&gt;</a> <font color="Maroon" title="c3">q</font></span>)</span>)</span>)</span> is  <a href="cqc_the1.html#V2" title="CQC_THE1:attr.2">valid</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:83"><span class="lab"><font color="Green" title="E66">A3</font></span></a>, <a class="ref" href="cqc_the1.html#T65" onmouseover="rs('cqc_the1/T65')" onmouseout="rh()">CQC_THE1:65</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:83"/>
<span class="p1">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">Ex</a> (<font color="Maroon" title="c4">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">=&gt;</a> <font color="Maroon" title="c3">q</font></span>)</span>)</span>)</span> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">=&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K11" title="CQC_LANG:func.11">All</a> (<font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c2">p</font>)</span>)</span> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">=&gt;</a> <font color="Maroon" title="c3">q</font></span>)</span> is  <a href="cqc_the1.html#V2" title="CQC_THE1:attr.2">valid</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:83"><span class="lab"><font color="Green" title="E65">A2</font></span></a>, <a class="ref" href="lukasi_1.html#T42" onmouseover="rs('lukasi_1/T42')" onmouseout="rh()">LUKASI_1:42</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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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