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<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="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">H</font> <a href="qc_lang2.html#R2" title="QC_LANG2:pred.2">is_subformula_of</a>  <a href="qc_lang2.html#K1" title="QC_LANG2:func.1">FALSUM</a>  iff ( <font color="Maroon" title="c1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang2.html#K1" title="QC_LANG2:func.1">FALSUM</a>  or <font color="Maroon" title="c1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K11" title="QC_LANG1:func.11">VERUM</a>  ) )</span><br/>

<span class="kw">thus </span><a NAME="E1:97"/>
(  not <font color="Maroon" title="c1">H</font> <a href="qc_lang2.html#R2" title="QC_LANG2:pred.2">is_subformula_of</a>  <a href="qc_lang2.html#K1" title="QC_LANG2:func.1">FALSUM</a>  or <font color="Maroon" title="c1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang2.html#K1" title="QC_LANG2:func.1">FALSUM</a>  or <font color="Maroon" title="c1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K11" title="QC_LANG1:func.11">VERUM</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">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang2.html#K1" title="QC_LANG2:func.1">FALSUM</a>  or <font color="Maroon" title="c1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K11" title="QC_LANG1:func.11">VERUM</a>  ) implies <font color="Maroon" title="c1">H</font> <a href="qc_lang2.html#R2" title="QC_LANG2:pred.2">is_subformula_of</a>  <a href="qc_lang2.html#K1" title="QC_LANG2:func.1">FALSUM</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="97_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:97_1"/>
( <font color="Maroon" title="c1">H</font> <a href="qc_lang2.html#R2" title="QC_LANG2:pred.2">is_subformula_of</a>  <a href="qc_lang2.html#K1" title="QC_LANG2:func.1">FALSUM</a>  &amp; <font color="Maroon" title="c1">H</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="qc_lang2.html#K1" title="QC_LANG2:func.1">FALSUM</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">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K11" title="QC_LANG1:func.11">VERUM</a> </span><br/>

<a NAME="E2:97_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">H</font> <a href="qc_lang2.html#R3" title="QC_LANG2:pred.3">is_proper_subformula_of</a>  <a href="qc_lang2.html#K1" title="QC_LANG2:func.1">FALSUM</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="qc_lang2.html#D22" target="_self" title="QC_LANG2:def.22">Def22</a></span>;<br/>
<a NAME="E3:97_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">H</font> <a href="qc_lang2.html#R2" title="QC_LANG2:pred.2">is_subformula_of</a>  <a href="qc_lang1.html#K11" title="QC_LANG1:func.11">VERUM</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="qc_lang2.html#T86" target="_self" title="QC_LANG2:th.86">Th86</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:97_1"/>
<font color="Maroon" title="c1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K11" title="QC_LANG1:func.11">VERUM</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="qc_lang2.html#T99" target="_self" title="QC_LANG2:th.99">Th99</a></span>; <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>

<a NAME="E2:97"/>
 <a href="qc_lang1.html#K11" title="QC_LANG1:func.11">VERUM</a>  <a href="qc_lang2.html#R1" title="QC_LANG2:pred.1">is_immediate_constituent_of</a>  <a href="qc_lang2.html#K1" title="QC_LANG2:func.1">FALSUM</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="qc_lang2.html#D20" target="_self" title="QC_LANG2:def.20">Def20</a></span>;<br/>
<a NAME="E3:97"/><span class="kw">then </span>
 <a href="qc_lang1.html#K11" title="QC_LANG1:func.11">VERUM</a>  <a href="qc_lang2.html#R3" title="QC_LANG2:pred.3">is_proper_subformula_of</a>  <a href="qc_lang2.html#K1" title="QC_LANG2:func.1">FALSUM</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="qc_lang2.html#T73" target="_self" title="QC_LANG2:th.73">Th73</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:97"/>
( ( <font color="Maroon" title="c1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang2.html#K1" title="QC_LANG2:func.1">FALSUM</a>  or <font color="Maroon" title="c1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K11" title="QC_LANG1:func.11">VERUM</a>  ) implies <font color="Maroon" title="c1">H</font> <a href="qc_lang2.html#R2" title="QC_LANG2:pred.2">is_subformula_of</a>  <a href="qc_lang2.html#K1" title="QC_LANG2:func.1">FALSUM</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="qc_lang2.html#D22" target="_self" title="QC_LANG2:def.22">Def22</a></span>; <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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