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<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a> ;<br/>

<a NAME="E1:92"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="qc_lang1.html#V2">atomic</a> implies  <a href="qc_lang2.html#K15" target="_self">Subformulae</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:92_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a>ex <font color="Olive">b<sub>2</sub></font> being  <a href="qc_lang1.html#NM6">QC-pred_symbol</a> of <font color="Olive">b<sub>1</sub></font>ex <font color="Olive">b<sub>3</sub></font> being   <a href="qc_lang1.html#M1">QC-variable_list</a> of <font color="Olive">b<sub>1</sub></font> st <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="qc_lang1.html#K9">!</a> <font color="Olive">b<sub>3</sub></font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="qc_lang1.html#D17">QC_LANG1:def 17</a><br/>

<b>hence </b><a NAME="E2:92_1"/>
 <a href="qc_lang2.html#K15" target="_self">Subformulae</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="qc_lang2.html#T108" target="_self">Th108</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:92"/>
( ( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K11">VERUM</a>  or <font color="Maroon">c<sub>1</sub></font> is <a href="qc_lang1.html#V2">atomic</a> ) implies  <a href="qc_lang2.html#K15" target="_self">Subformulae</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span> )
 <b>by </b><i><a class="ref" href="qc_lang2.html#T107" target="_self">Th107</a></i>;<br/>

<b>assume </b><a NAME="E3:92"/><i><font color="Green">E60</font></i>: 
 <a href="qc_lang2.html#K15" target="_self">Subformulae</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<b>assume </b><a NAME="E4:92"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="qc_lang1.html#K11">VERUM</a>  &amp; not <font color="Maroon">c<sub>1</sub></font> is <a href="qc_lang1.html#V2">atomic</a> )
 ;<br/>

<a NAME="E5:92"/><b>then </b>
( <font color="Maroon">c<sub>1</sub></font> is <a href="qc_lang1.html#V3">negative</a> or <font color="Maroon">c<sub>1</sub></font> is <a href="qc_lang1.html#V4">conjunctive</a> or <font color="Maroon">c<sub>1</sub></font> is <a href="qc_lang1.html#V5">universal</a> )
 
<b>by </b><i><a class="ref" href="qc_lang1.html#T33">QC_LANG1:33</a></i>;<br/>
<a NAME="E6:92"/><b>then </b><i><font color="Green">E61</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K12">'not'</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K17">the_argument_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> or <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K18">the_left_argument_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="qc_lang1.html#K13">'&amp;'</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K19">the_right_argument_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> or <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K14">All</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K20">bound_in</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="qc_lang1.html#K21">the_scope_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="qc_lang2.html#T4" target="_self">Th4</a>, <a class="ref" href="qc_lang2.html#T7" target="_self">Th7</a>, <a class="ref" href="qc_lang1.html#D23">QC_LANG1:def 23</a></i>;<br/>
<div><i><font color="Green">E62</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="92_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:92_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K12">'not'</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K17">the_argument_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/><a NAME="E2:92_2"/><b>then </b><i><font color="Green">E63</font></i>: 
 <a href="qc_lang1.html#K17">the_argument_of</a> <font color="Maroon">c<sub>1</sub></font> <a href="qc_lang2.html#R1" target="_self">is_immediate_constituent_of</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="qc_lang2.html#T60" target="_self">Th60</a></i>;<br/><a NAME="E3:92_2"/><b>then </b>
 <a href="qc_lang1.html#K17">the_argument_of</a> <font color="Maroon">c<sub>1</sub></font> <a href="qc_lang2.html#R3" target="_self">is_proper_subformula_of</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="qc_lang2.html#T73" target="_self">Th73</a></i>;<br/><a NAME="E4:92_2"/><b>then </b>
 <a href="qc_lang1.html#K17">the_argument_of</a> <font color="Maroon">c<sub>1</sub></font> <a href="qc_lang2.html#R2" target="_self">is_subformula_of</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="qc_lang2.html#D22" target="_self">Def22</a></i>;<br/><a NAME="E5:92_2"/><b>then </b><i><font color="Green">E64</font></i>: 
 <a href="qc_lang1.html#K17">the_argument_of</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="qc_lang2.html#K15" target="_self">Subformulae</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="qc_lang2.html#D23" target="_self">Def23</a></i>;<br/><a NAME="E6:92_2"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K17">the_argument_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="qc_lang2.html#E2:92_2"><i><font color="Green">E63</font></i></a>, <a class="ref" href="qc_lang2.html#T71" target="_self">Th71</a></i>;<br/><b>hence </b><a NAME="E7:92_2"/>
contradiction
 <b>by </b><i><a class="txt" href="qc_lang2.html#E3:92"><i><font color="Green">E60</font></i></a>, <a class="txt" href="qc_lang2.html#E5:92_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:92_3"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K18">the_left_argument_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="qc_lang1.html#K13">'&amp;'</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K19">the_right_argument_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/><a NAME="E2:92_3"/><b>then </b><i><font color="Green">E63</font></i>: 
(  <a href="qc_lang1.html#K18">the_left_argument_of</a> <font color="Maroon">c<sub>1</sub></font> <a href="qc_lang2.html#R1" target="_self">is_immediate_constituent_of</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="qc_lang1.html#K19">the_right_argument_of</a> <font color="Maroon">c<sub>1</sub></font> <a href="qc_lang2.html#R1" target="_self">is_immediate_constituent_of</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="ref" href="qc_lang2.html#T62" target="_self">Th62</a></i>;<br/><a NAME="E3:92_3"/><b>then </b>
(  <a href="qc_lang1.html#K18">the_left_argument_of</a> <font color="Maroon">c<sub>1</sub></font> <a href="qc_lang2.html#R3" target="_self">is_proper_subformula_of</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="qc_lang1.html#K19">the_right_argument_of</a> <font color="Maroon">c<sub>1</sub></font> <a href="qc_lang2.html#R3" target="_self">is_proper_subformula_of</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="ref" href="qc_lang2.html#T73" target="_self">Th73</a></i>;<br/><a NAME="E4:92_3"/><b>then </b>
(  <a href="qc_lang1.html#K18">the_left_argument_of</a> <font color="Maroon">c<sub>1</sub></font> <a href="qc_lang2.html#R2" target="_self">is_subformula_of</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="qc_lang1.html#K19">the_right_argument_of</a> <font color="Maroon">c<sub>1</sub></font> <a href="qc_lang2.html#R2" target="_self">is_subformula_of</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="ref" href="qc_lang2.html#D22" target="_self">Def22</a></i>;<br/><a NAME="E5:92_3"/><b>then </b><i><font color="Green">E64</font></i>: 
(  <a href="qc_lang1.html#K18">the_left_argument_of</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="qc_lang2.html#K15" target="_self">Subformulae</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="qc_lang1.html#K19">the_right_argument_of</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="qc_lang2.html#K15" target="_self">Subformulae</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="ref" href="qc_lang2.html#D23" target="_self">Def23</a></i>;<br/><a NAME="E6:92_3"/>
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K18">the_left_argument_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K19">the_right_argument_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="qc_lang2.html#E2:92_3"><i><font color="Green">E63</font></i></a>, <a class="ref" href="qc_lang2.html#T71" target="_self">Th71</a></i>;<br/><b>hence </b><a NAME="E7:92_3"/>
contradiction
 <b>by </b><i><a class="txt" href="qc_lang2.html#E3:92"><i><font color="Green">E60</font></i></a>, <a class="txt" href="qc_lang2.html#E5:92_3"><i><font color="Green">E64</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div>
<a NAME="E9:92"/><b>then </b><i><font color="Green">E63</font></i>: 
 <a href="qc_lang1.html#K21">the_scope_of</a> <font color="Maroon">c<sub>1</sub></font> <a href="qc_lang2.html#R1" target="_self">is_immediate_constituent_of</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="qc_lang2.html#E6:92"><i><font color="Green">E61</font></i></a>, <a class="txt" href="qc_lang2.html#E7:92"><i><font color="Green">E62</font></i></a>, <a class="ref" href="qc_lang2.html#T63" target="_self">Th63</a></i>;<br/>
<a NAME="E10:92"/><b>then </b>
 <a href="qc_lang1.html#K21">the_scope_of</a> <font color="Maroon">c<sub>1</sub></font> <a href="qc_lang2.html#R3" target="_self">is_proper_subformula_of</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="qc_lang2.html#T73" target="_self">Th73</a></i>;<br/>
<a NAME="E11:92"/><b>then </b>
 <a href="qc_lang1.html#K21">the_scope_of</a> <font color="Maroon">c<sub>1</sub></font> <a href="qc_lang2.html#R2" target="_self">is_subformula_of</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="qc_lang2.html#D22" target="_self">Def22</a></i>;<br/>
<a NAME="E12:92"/><b>then </b><i><font color="Green">E64</font></i>: 
 <a href="qc_lang1.html#K21">the_scope_of</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="qc_lang2.html#K15" target="_self">Subformulae</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="qc_lang2.html#D23" target="_self">Def23</a></i>;<br/>
<a NAME="E13:92"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K21">the_scope_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="qc_lang2.html#E9:92"><i><font color="Green">E63</font></i></a>, <a class="ref" href="qc_lang2.html#T71" target="_self">Th71</a></i>;<br/>
<b>hence </b><a NAME="E14:92"/>
contradiction
 <b>by </b><i><a class="txt" href="qc_lang2.html#E3:92"><i><font color="Green">E60</font></i></a>, <a class="txt" href="qc_lang2.html#E12:92"><i><font color="Green">E64</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


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