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<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">H</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K8" title="QC_LANG1:func.8">QC-WFF</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/>  for <font color="Olive" title="b2">P</font> being   <a href="qc_lang1.html#NM6" title="QC_LANG1:NM.6">QC-pred_symbol</a> of <font color="Olive" title="b1">k</font><br/>  for <font color="Olive" title="b3">V</font> being   <a href="qc_lang1.html#NM7" title="QC_LANG1:NM.7">QC-variable_list</a> of <font color="Olive" title="b1">k</font>  holds  not <font color="Maroon" title="c1">H</font> <a href="qc_lang2.html#R1" title="QC_LANG2:pred.1">is_immediate_constituent_of</a> <font color="Olive" title="b2">P</font> <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">!</a> <font color="Olive" title="b3">V</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" 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> being   <a href="qc_lang1.html#NM6" title="QC_LANG1:NM.6">QC-pred_symbol</a> of <font color="Maroon" title="c2">k</font><br/>  for <font color="Olive" title="b2">V</font> being   <a href="qc_lang1.html#NM7" title="QC_LANG1:NM.7">QC-variable_list</a> of <font color="Maroon" title="c2">k</font>  holds  not <font color="Maroon" title="c1">H</font> <a href="qc_lang2.html#R1" title="QC_LANG2:pred.1">is_immediate_constituent_of</a> <font color="Olive" title="b1">P</font> <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">!</a> <font color="Olive" title="b2">V</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">P</font> be   <a href="qc_lang1.html#NM6" title="QC_LANG1:NM.6">QC-pred_symbol</a> of <font color="Maroon" title="c2">k</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">V</font> being   <a href="qc_lang1.html#NM7" title="QC_LANG1:NM.7">QC-variable_list</a> of <font color="Maroon" title="c2">k</font>  holds  not <font color="Maroon" title="c1">H</font> <a href="qc_lang2.html#R1" title="QC_LANG2:pred.1">is_immediate_constituent_of</a> <font color="Maroon" title="c3">P</font> <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">!</a> <font color="Olive" title="b1">V</font></span><br/><span class="kw">let </span><font color="Maroon" title="c4">V</font> be   <a href="qc_lang1.html#NM7" title="QC_LANG1:NM.7">QC-variable_list</a> of <font color="Maroon" title="c2">k</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c1">H</font> <a href="qc_lang2.html#R1" title="QC_LANG2:pred.1">is_immediate_constituent_of</a> <font color="Maroon" title="c3">P</font> <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">!</a> <font color="Maroon" title="c4">V</font></span><br/>







<span class="kw">assume </span><a NAME="E1:40"/><span class="lab"><font color="Green" title="E20">A1</font></span>: 
<font color="Maroon" title="c1">H</font> <a href="qc_lang2.html#R1" title="QC_LANG2:pred.1">is_immediate_constituent_of</a> <font color="Maroon" title="c3">P</font> <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">!</a> <font color="Maroon" title="c4">V</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:40"/><span class="lab"><font color="Green" title="E21">A2</font></span>: 
<font color="Maroon" title="c3">P</font> <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">!</a> <font color="Maroon" title="c4">V</font> is  <a href="qc_lang1.html#V2" title="QC_LANG1:attr.2">atomic</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="qc_lang1.html#D16" title="QC_LANG1:def.16">QC_LANG1:def 16</a></span>;<br/>
<a NAME="E3:40"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A3</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">!</a> <font color="Maroon" title="c4">V</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 3
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="qc_lang1.html#T17" title="QC_LANG1:th.17">QC_LANG1:17</a></span>;<br/>
<a NAME="E4:40"/><span class="lab"><font color="Green" title="E23">A4</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">!</a> <font color="Maroon" title="c4">V</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="ref" href="qc_lang1.html#T17" title="QC_LANG1:th.17">QC_LANG1:17</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E24">A5</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1"><span class="kw">now </span></a><div class="add"><span class="kw">given </span><font color="Maroon" title="c5">H1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K8" title="QC_LANG1:func.8">QC-WFF</a> <span class="kw"> such that </span><a NAME="E2:40_1"/><span class="lab"><font color="Green" title="E25">A6</font></span>: 
( <font color="Maroon" title="c3">P</font> <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">!</a> <font color="Maroon" title="c4">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">H</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <font color="Maroon" title="c5">H1</font> or <font color="Maroon" title="c3">P</font> <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">!</a> <font color="Maroon" title="c4">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">H1</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <font color="Maroon" title="c1">H</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E3:40_1"/>
( <font color="Maroon" title="c1">H</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <font color="Maroon" title="c5">H1</font> is  <a href="qc_lang1.html#V4" title="QC_LANG1:attr.4">conjunctive</a>  &amp; <font color="Maroon" title="c5">H1</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <font color="Maroon" title="c1">H</font> is  <a href="qc_lang1.html#V4" title="QC_LANG1:attr.4">conjunctive</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="qc_lang1.html#D18" title="QC_LANG1:def.18">QC_LANG1:def 18</a></span>;<br/><span class="kw">hence </span><a NAME="E4:40_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:40"><span class="lab"><font color="Green" title="E23">A4</font></span></a>, <a class="txt" href="#E2:40_1"><span class="lab"><font color="Green" title="E25">A6</font></span></a>, <a class="ref" href="qc_lang1.html#T16" title="QC_LANG1:th.16">QC_LANG1:16</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>
<a NAME="E6:40"/>
 <a href="qc_lang1.html#K12" title="QC_LANG1:func.12">'not'</a> <font color="Maroon" title="c1">H</font> is  <a href="qc_lang1.html#V3" title="QC_LANG1:attr.3">negative</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="qc_lang1.html#D17" title="QC_LANG1:def.17">QC_LANG1:def 17</a></span>;<br/>
<a NAME="E7:40"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A7</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p3">(<span class="default"><a href="qc_lang1.html#K12" title="QC_LANG1:func.12">'not'</a> <font color="Maroon" title="c1">H</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="qc_lang1.html#T16" title="QC_LANG1:th.16">QC_LANG1:16</a></span>;<br/>
<a NAME="E8:40"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">!</a> <font color="Maroon" title="c4">V</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:40"><span class="lab"><font color="Green" title="E21">A2</font></span></a>, <a class="ref" href="qc_lang1.html#T17" title="QC_LANG1:th.17">QC_LANG1:17</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">z</font> being   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a><span class="kw"> such that </span><br/><a NAME="E10:40"/><span class="lab"><font color="Green" title="E26">A8</font></span>: 
<font color="Maroon" title="c3">P</font> <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">!</a> <font color="Maroon" title="c4">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> (<font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c1">H</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40"><span class="lab"><font color="Green" title="E20">A1</font></span></a>, <a class="txt" href="#E7:40"><span class="lab"><font color="Green" title="E25">A7</font></span></a>, <a class="txt" href="#E5:40"><span class="lab"><font color="Green" title="E24">A5</font></span></a>, <a class="ref" href="qc_lang2.html#D19" target="_self" title="QC_LANG2:def.19">Def20</a></span>;<br/>
<a NAME="E11:40"/>
 <a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> (<font color="Maroon" title="c5">z</font>,<font color="Maroon" title="c1">H</font>) is  <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="qc_lang1.html#D19" title="QC_LANG1:def.19">QC_LANG1:def 19</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:40"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:40"><span class="lab"><font color="Green" title="E22">A3</font></span></a>, <a class="txt" href="#E10:40"><span class="lab"><font color="Green" title="E26">A8</font></span></a>, <a class="ref" href="qc_lang1.html#T16" title="QC_LANG1:th.16">QC_LANG1:16</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/>


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