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<b>let </b><font color="Maroon" title="c1">F</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:()"><i><font color="Red">::  thesis: </font></i></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#M1" title="QC_LANG1:mode.1">QC-variable_list</a> of <font color="Olive" title="b1">k</font>  holds not <font color="Maroon" title="c1">F</font> <a href="qc_lang2.html#R3" title="QC_LANG2:pred.3">is_proper_subformula_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/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></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#M1" title="QC_LANG1:mode.1">QC-variable_list</a> of <font color="Maroon" title="c2">k</font>  holds not <font color="Maroon" title="c1">F</font> <a href="qc_lang2.html#R3" title="QC_LANG2:pred.3">is_proper_subformula_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/><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">V</font> being   <a href="qc_lang1.html#M1" title="QC_LANG1:mode.1">QC-variable_list</a> of <font color="Maroon" title="c2">k</font>  holds not <font color="Maroon" title="c1">F</font> <a href="qc_lang2.html#R3" title="QC_LANG2:pred.3">is_proper_subformula_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/><b>let </b><font color="Maroon" title="c4">V</font> be    <a href="qc_lang1.html#M1" title="QC_LANG1:mode.1">QC-variable_list</a> of <font color="Maroon" title="c2">k</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not <font color="Maroon" title="c1">F</font> <a href="qc_lang2.html#R3" title="QC_LANG2:pred.3">is_proper_subformula_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/>







<b>assume </b><a NAME="E1:62"/>
<font color="Maroon" title="c1">F</font> <a href="qc_lang2.html#R3" title="QC_LANG2:pred.3">is_proper_subformula_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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:62"/><b>then </b>
 ex <font color="Olive" title="b1">G</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>  st <font color="Olive" title="b1">G</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>
 
<b>by </b><i><a class="ref" href="qc_lang2.html#T75" target="_self" title="QC_LANG2:th.75">Th75</a></i>;<br/>
<b>hence </b><a NAME="E3:62"/>
contradiction
 <b>by </b><i><a class="ref" href="qc_lang2.html#T59" target="_self" title="QC_LANG2:th.59">Th59</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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