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



<b>thus </b><a NAME="E1:87"/>
 <a href="qc_lang2.html#K15" target="_self">Subformulae</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="qc_lang1.html#K13">'&amp;'</a> <font color="Maroon">F</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang2.html#K15" target="_self">Subformulae</a> <font color="Maroon">H</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="qc_lang2.html#K15" target="_self">Subformulae</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="qc_lang1.html#K13">'&amp;'</a> <font color="Maroon">F</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:87_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang2.html#K15" target="_self">Subformulae</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="qc_lang1.html#K13">'&amp;'</a> <font color="Maroon">F</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">G</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a> <b> such that </b><br/><a NAME="E3:87_1"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">G</font> <a href="qc_lang2.html#R2" target="_self">is_subformula_of</a> <font color="Maroon">H</font> <a href="qc_lang1.html#K13">'&amp;'</a> <font color="Maroon">F</font> )
 <b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:87_1_1"/>
<font color="Maroon">G</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">H</font> <a href="qc_lang1.html#K13">'&amp;'</a> <font color="Maroon">F</font>
 ;<br/><a NAME="E2:87_1_1"/><b>then </b>
<font color="Maroon">G</font> <a href="qc_lang2.html#R3" target="_self">is_proper_subformula_of</a> <font color="Maroon">H</font> <a href="qc_lang1.html#K13">'&amp;'</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="qc_lang2.html#T46" target="_self">Th46</a>, <a class="ref" href="qc_lang2.html#T4" target="_self">Th4</a></i>;<br/><a NAME="E3:87_1_1"/><b>then </b>
( <font color="Maroon">G</font> <a href="qc_lang2.html#R2" target="_self">is_subformula_of</a> <font color="Maroon">H</font> or <font color="Maroon">G</font> <a href="qc_lang2.html#R2" target="_self">is_subformula_of</a> <font color="Maroon">F</font> )
 <b>by </b><i><a class="ref" href="qc_lang2.html#T24" target="_self">Th24</a></i>;<br/><a NAME="E4:87_1_1"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang2.html#K15" target="_self">Subformulae</a> <font color="Maroon">H</font> or <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang2.html#K15" target="_self">Subformulae</a> <font color="Maroon">F</font> )
 <b>by </b><i><a class="ref" href="qc_lang2.html#T46" target="_self">Th46</a>, </i>;<br/><b>hence </b><a NAME="E5:87_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="qc_lang2.html#K15" target="_self">Subformulae</a> <font color="Maroon">H</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="qc_lang2.html#K15" target="_self">Subformulae</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div>
<a NAME="E5:87_1"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="qc_lang2.html#K15" target="_self">Subformulae</a> <font color="Maroon">H</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="qc_lang2.html#K15" target="_self">Subformulae</a> <font color="Maroon">F</font></span>)</span> or <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="qc_lang1.html#K13">'&amp;'</a> <font color="Maroon">F</font></span>)</span></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="qc_lang2.html#T46" target="_self">Th46</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E6:87_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang2.html#K15" target="_self">Subformulae</a> <font color="Maroon">H</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="qc_lang2.html#K15" target="_self">Subformulae</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="qc_lang1.html#K13">'&amp;'</a> <font color="Maroon">F</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:87"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang2.html#K15" target="_self">Subformulae</a> <font color="Maroon">H</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p2">(<span class="default"><a href="qc_lang2.html#K15" target="_self">Subformulae</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="qc_lang1.html#K13">'&amp;'</a> <font color="Maroon">F</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E3:87"/><i><font color="Green">E30</font></i>: 
(  not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="qc_lang2.html#K15" target="_self">Subformulae</a> <font color="Maroon">H</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><a href="qc_lang2.html#K15" target="_self">Subformulae</a> <font color="Maroon">F</font></span>)</span> or <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang2.html#K15" target="_self">Subformulae</a> <font color="Maroon">H</font> or <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang2.html#K15" target="_self">Subformulae</a> <font color="Maroon">F</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<div><i><font color="Green">E34</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="87_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:87_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang2.html#K15" target="_self">Subformulae</a> <font color="Maroon">H</font>
 ;<br/><b>then consider </b><font color="Maroon">G</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a> <b> such that </b><br/><a NAME="E3:87_2"/><i><font color="Green">E69</font></i>: 
( <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">G</font> <a href="qc_lang2.html#R2" target="_self">is_subformula_of</a> <font color="Maroon">H</font> )
 <b>by </b><i/>;<br/><a NAME="E4:87_2"/>
<font color="Maroon">H</font> <a href="qc_lang2.html#R1" target="_self">is_immediate_constituent_of</a> <font color="Maroon">H</font> <a href="qc_lang1.html#K13">'&amp;'</a> <font color="Maroon">F</font>
 <b>by </b><i/>;<br/><a NAME="E5:87_2"/><b>then </b>
<font color="Maroon">H</font> <a href="qc_lang2.html#R3" target="_self">is_proper_subformula_of</a> <font color="Maroon">H</font> <a href="qc_lang1.html#K13">'&amp;'</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="txt" href="qc_lang2.html#E4:87"><i><font color="Green">E34</font></i></a></i>;<br/><a NAME="E6:87_2"/><b>then </b>
<font color="Maroon">H</font> <a href="qc_lang2.html#R2" target="_self">is_subformula_of</a> <font color="Maroon">H</font> <a href="qc_lang1.html#K13">'&amp;'</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="qc_lang2.html#T4" target="_self">Th4</a></i>;<br/><a NAME="E7:87_2"/><b>then </b>
<font color="Maroon">G</font> <a href="qc_lang2.html#R2" target="_self">is_subformula_of</a> <font color="Maroon">H</font> <a href="qc_lang1.html#K13">'&amp;'</a> <font color="Maroon">F</font>
 <b>by </b><i>, <a class="ref" href="qc_lang2.html#T19" target="_self">Th19</a></i>;<br/><b>hence </b><a NAME="E8:87_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang2.html#K15" target="_self">Subformulae</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="qc_lang1.html#K13">'&amp;'</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E86</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="87_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:87_3"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang2.html#K15" target="_self">Subformulae</a> <font color="Maroon">F</font>
 ;<br/><b>then consider </b><font color="Maroon">G</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a> <b> such that </b><br/><a NAME="E3:87_3"/><i><font color="Green">E87</font></i>: 
( <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> &amp; <font color="Maroon">G</font> <a href="qc_lang2.html#R2" target="_self">is_subformula_of</a> <font color="Maroon">F</font> )
 <b>by </b><i/>;<br/><a NAME="E4:87_3"/>
<font color="Maroon">F</font> <a href="qc_lang2.html#R1" target="_self">is_immediate_constituent_of</a> <font color="Maroon">H</font> <a href="qc_lang1.html#K13">'&amp;'</a> <font color="Maroon">F</font>
 <b>by </b><i/>;<br/><a NAME="E5:87_3"/><b>then </b>
<font color="Maroon">F</font> <a href="qc_lang2.html#R3" target="_self">is_proper_subformula_of</a> <font color="Maroon">H</font> <a href="qc_lang1.html#K13">'&amp;'</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="txt" href="qc_lang2.html#E4:87"><i><font color="Green">E34</font></i></a></i>;<br/><a NAME="E6:87_3"/><b>then </b>
<font color="Maroon">F</font> <a href="qc_lang2.html#R2" target="_self">is_subformula_of</a> <font color="Maroon">H</font> <a href="qc_lang1.html#K13">'&amp;'</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="qc_lang2.html#T4" target="_self">Th4</a></i>;<br/><a NAME="E7:87_3"/><b>then </b>
<font color="Maroon">G</font> <a href="qc_lang2.html#R2" target="_self">is_subformula_of</a> <font color="Maroon">H</font> <a href="qc_lang1.html#K13">'&amp;'</a> <font color="Maroon">F</font>
 <b>by </b><i>, <a class="ref" href="qc_lang2.html#T19" target="_self">Th19</a></i>;<br/><b>hence </b><a NAME="E8:87_3"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang2.html#K15" target="_self">Subformulae</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="qc_lang1.html#K13">'&amp;'</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_4"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:87_4"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">H</font> <a href="qc_lang1.html#K13">'&amp;'</a> <font color="Maroon">F</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 ;<br/><a NAME="E2:87_4"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> <a href="qc_lang1.html#K13">'&amp;'</a> <font color="Maroon">F</font> &amp; <font color="Maroon">H</font> <a href="qc_lang1.html#K13">'&amp;'</a> <font color="Maroon">F</font> <a href="qc_lang2.html#R2" target="_self">is_subformula_of</a> <font color="Maroon">H</font> <a href="qc_lang1.html#K13">'&amp;'</a> <font color="Maroon">F</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:87_4"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang2.html#K15" target="_self">Subformulae</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="qc_lang1.html#K13">'&amp;'</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:87"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="qc_lang2.html#K15" target="_self">Subformulae</a> <span class="p1">(<span class="default"><font color="Maroon">H</font> <a href="qc_lang1.html#K13">'&amp;'</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i><a class="ref" href="qc_lang2.html#T46" target="_self">Th46</a>, <a class="ref" href="qc_lang2.html#T47" target="_self">Th47</a>, <a class="ref" href="qc_lang2.html#D20" target="_self">Def20</a>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div>
