<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">H</font>, <font color="Maroon" title="c2">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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="qc_lang2.html#K15" title="QC_LANG2:func.15">Subformulae</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">H</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang2.html#K15" title="QC_LANG2:func.15">Subformulae</a> <font color="Maroon" title="c1">H</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="qc_lang2.html#K15" title="QC_LANG2:func.15">Subformulae</a> <font color="Maroon" title="c2">F</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">H</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <font color="Maroon" title="c2">F</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/>



<span class="kw">thus </span><a NAME="E1:107"/>
 <a href="qc_lang2.html#K15" title="QC_LANG2:func.15">Subformulae</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">H</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <font color="Maroon" title="c2">F</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang2.html#K15" title="QC_LANG2:func.15">Subformulae</a> <font color="Maroon" title="c1">H</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="qc_lang2.html#K15" title="QC_LANG2:func.15">Subformulae</a> <font color="Maroon" title="c2">F</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">H</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <font color="Maroon" title="c2">F</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
  <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang2.html#K15" title="QC_LANG2:func.15">Subformulae</a> <font color="Maroon" title="c1">H</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="qc_lang2.html#K15" title="QC_LANG2:func.15">Subformulae</a> <font color="Maroon" title="c2">F</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">H</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <font color="Maroon" title="c2">F</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="qc_lang2.html#K15" title="QC_LANG2:func.15">Subformulae</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">H</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <font color="Maroon" title="c2">F</font></span>)</span></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="107_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang2.html#K15" title="QC_LANG2:func.15">Subformulae</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">H</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <font color="Maroon" title="c2">F</font></span>)</span> or <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang2.html#K15" title="QC_LANG2:func.15">Subformulae</a> <font color="Maroon" title="c1">H</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="qc_lang2.html#K15" title="QC_LANG2:func.15">Subformulae</a> <font color="Maroon" title="c2">F</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">H</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <font color="Maroon" title="c2">F</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:107_1"/>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang2.html#K15" title="QC_LANG2:func.15">Subformulae</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">H</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <font color="Maroon" title="c2">F</font></span>)</span>
 ; <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="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang2.html#K15" title="QC_LANG2:func.15">Subformulae</a> <font color="Maroon" title="c1">H</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="qc_lang2.html#K15" title="QC_LANG2:func.15">Subformulae</a> <font color="Maroon" title="c2">F</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">H</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <font color="Maroon" title="c2">F</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">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> <span class="kw"> such that </span><br/><a NAME="E3:107_1"/><span class="lab"><font color="Green" title="E55">A1</font></span>: 
<font color="Maroon" title="c4">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font>
 <span class="kw">and </span><br/><a NAME="E4:107_1"/><span class="lab"><font color="Green" title="E56">A2</font></span>: 
<font color="Maroon" title="c4">G</font> <a href="qc_lang2.html#R2" title="QC_LANG2:pred.2">is_subformula_of</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="c2">F</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="qc_lang2.html#D23" target="_self" title="QC_LANG2:def.23">Def23</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="107_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:107_1_1"/>
<font color="Maroon" title="c4">G</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</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="c2">F</font>
 ; <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="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="qc_lang2.html#K15" title="QC_LANG2:func.15">Subformulae</a> <font color="Maroon" title="c1">H</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="qc_lang2.html#K15" title="QC_LANG2:func.15">Subformulae</a> <font color="Maroon" title="c2">F</font></span>)</span></span><br/><a NAME="E2:107_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">G</font> <a href="qc_lang2.html#R3" title="QC_LANG2:pred.3">is_proper_subformula_of</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="c2">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:107_1"><span class="lab"><font color="Green" title="E56">A2</font></span></a>, <a class="ref" href="qc_lang2.html#D22" target="_self" title="QC_LANG2:def.22">Def22</a></span>;<br/><a NAME="E3:107_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c4">G</font> <a href="qc_lang2.html#R2" title="QC_LANG2:pred.2">is_subformula_of</a> <font color="Maroon" title="c1">H</font> or <font color="Maroon" title="c4">G</font> <a href="qc_lang2.html#R2" title="QC_LANG2:pred.2">is_subformula_of</a> <font color="Maroon" title="c2">F</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="qc_lang2.html#T89" target="_self" title="QC_LANG2:th.89">Th89</a></span>;<br/><a NAME="E4:107_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang2.html#K15" title="QC_LANG2:func.15">Subformulae</a> <font color="Maroon" title="c1">H</font> or <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang2.html#K15" title="QC_LANG2:func.15">Subformulae</a> <font color="Maroon" title="c2">F</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:107_1"><span class="lab"><font color="Green" title="E55">A1</font></span></a>, <a class="ref" href="qc_lang2.html#D23" target="_self" title="QC_LANG2:def.23">Def23</a></span>;<br/><span class="kw">hence </span><a NAME="E5:107_1_1"/>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="qc_lang2.html#K15" title="QC_LANG2:func.15">Subformulae</a> <font color="Maroon" title="c1">H</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="qc_lang2.html#K15" title="QC_LANG2:func.15">Subformulae</a> <font color="Maroon" title="c2">F</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</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="E6:107_1"/><span class="kw">then </span>
( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="qc_lang2.html#K15" title="QC_LANG2:func.15">Subformulae</a> <font color="Maroon" title="c1">H</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="qc_lang2.html#K15" title="QC_LANG2:func.15">Subformulae</a> <font color="Maroon" title="c2">F</font></span>)</span> or <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">H</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <font color="Maroon" title="c2">F</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:107_1"><span class="lab"><font color="Green" title="E55">A1</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:107_1"/>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang2.html#K15" title="QC_LANG2:func.15">Subformulae</a> <font color="Maroon" title="c1">H</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="qc_lang2.html#K15" title="QC_LANG2:func.15">Subformulae</a> <font color="Maroon" title="c2">F</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">H</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <font color="Maroon" title="c2">F</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</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>

<span class="kw">let </span><font color="Maroon" title="c3">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang2.html#K15" title="QC_LANG2:func.15">Subformulae</a> <font color="Maroon" title="c1">H</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="qc_lang2.html#K15" title="QC_LANG2:func.15">Subformulae</a> <font color="Maroon" title="c2">F</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">H</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <font color="Maroon" title="c2">F</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang2.html#K15" title="QC_LANG2:func.15">Subformulae</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">H</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <font color="Maroon" title="c2">F</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E2:107"/><span class="lab"><font color="Green" title="E55">A3</font></span>: 
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang2.html#K15" title="QC_LANG2:func.15">Subformulae</a> <font color="Maroon" title="c1">H</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2">(<span class="default"><a href="qc_lang2.html#K15" title="QC_LANG2:func.15">Subformulae</a> <font color="Maroon" title="c2">F</font></span>)</span></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">H</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <font color="Maroon" title="c2">F</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <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="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang2.html#K15" title="QC_LANG2:func.15">Subformulae</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">H</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <font color="Maroon" title="c2">F</font></span>)</span></span><br/>

<div><span class="lab"><font color="Green" title="E56">A4</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="107_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:107_2"/>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">H</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <font color="Maroon" title="c2">F</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <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="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang2.html#K15" title="QC_LANG2:func.15">Subformulae</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">H</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <font color="Maroon" title="c2">F</font></span>)</span></span><br/><a NAME="E2:107_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">a</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="c2">F</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E3:107_2"/>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang2.html#K15" title="QC_LANG2:func.15">Subformulae</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">H</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <font color="Maroon" title="c2">F</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="qc_lang2.html#D23" target="_self" title="QC_LANG2:def.23">Def23</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>
<div><span class="lab"><font color="Green" title="E57">A5</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="107_3"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:107_3"/>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang2.html#K15" title="QC_LANG2:func.15">Subformulae</a> <font color="Maroon" title="c2">F</font>
 ; <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="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang2.html#K15" title="QC_LANG2:func.15">Subformulae</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">H</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <font color="Maroon" title="c2">F</font></span>)</span></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">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> <span class="kw"> such that </span><br/><a NAME="E3:107_3"/><span class="lab"><font color="Green" title="E58">A6</font></span>: 
<font color="Maroon" title="c4">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font>
 <span class="kw">and </span><br/><a NAME="E4:107_3"/><span class="lab"><font color="Green" title="E59">A7</font></span>: 
<font color="Maroon" title="c4">G</font> <a href="qc_lang2.html#R2" title="QC_LANG2:pred.2">is_subformula_of</a> <font color="Maroon" title="c2">F</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="qc_lang2.html#D23" target="_self" title="QC_LANG2:def.23">Def23</a></span>;<br/><a NAME="E5:107_3"/>
<font color="Maroon" title="c2">F</font> <a href="qc_lang2.html#R1" title="QC_LANG2:pred.1">is_immediate_constituent_of</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="c2">F</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="qc_lang2.html#T62" target="_self" title="QC_LANG2:th.62">Th62</a></span>;<br/><a NAME="E6:107_3"/><span class="kw">then </span>
<font color="Maroon" title="c2">F</font> <a href="qc_lang2.html#R3" title="QC_LANG2:pred.3">is_proper_subformula_of</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="c2">F</font>
 <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/><a NAME="E7:107_3"/><span class="kw">then </span>
<font color="Maroon" title="c2">F</font> <a href="qc_lang2.html#R2" title="QC_LANG2:pred.2">is_subformula_of</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="c2">F</font>
 <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="E8:107_3"/><span class="kw">then </span>
<font color="Maroon" title="c4">G</font> <a href="qc_lang2.html#R2" title="QC_LANG2:pred.2">is_subformula_of</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="c2">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:107_3"><span class="lab"><font color="Green" title="E59">A7</font></span></a>, <a class="ref" href="qc_lang2.html#T77" target="_self" title="QC_LANG2:th.77">Th77</a></span>;<br/><span class="kw">hence </span><a NAME="E9:107_3"/>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang2.html#K15" title="QC_LANG2:func.15">Subformulae</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">H</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <font color="Maroon" title="c2">F</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:107_3"><span class="lab"><font color="Green" title="E58">A6</font></span></a>, <a class="ref" href="qc_lang2.html#D23" target="_self" title="QC_LANG2:def.23">Def23</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>
<div><span class="lab"><font color="Green" title="E58">A8</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="107_4"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:107_4"/>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang2.html#K15" title="QC_LANG2:func.15">Subformulae</a> <font color="Maroon" title="c1">H</font>
 ; <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="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang2.html#K15" title="QC_LANG2:func.15">Subformulae</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">H</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <font color="Maroon" title="c2">F</font></span>)</span></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">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> <span class="kw"> such that </span><br/><a NAME="E3:107_4"/><span class="lab"><font color="Green" title="E59">A9</font></span>: 
<font color="Maroon" title="c4">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font>
 <span class="kw">and </span><br/><a NAME="E4:107_4"/><span class="lab"><font color="Green" title="E60">A10</font></span>: 
<font color="Maroon" title="c4">G</font> <a href="qc_lang2.html#R2" title="QC_LANG2:pred.2">is_subformula_of</a> <font color="Maroon" title="c1">H</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="qc_lang2.html#D23" target="_self" title="QC_LANG2:def.23">Def23</a></span>;<br/><a NAME="E5:107_4"/>
<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="c1">H</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <font color="Maroon" title="c2">F</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="qc_lang2.html#T62" target="_self" title="QC_LANG2:th.62">Th62</a></span>;<br/><a NAME="E6:107_4"/><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> <font color="Maroon" title="c1">H</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <font color="Maroon" title="c2">F</font>
 <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/><a NAME="E7:107_4"/><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> <font color="Maroon" title="c1">H</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <font color="Maroon" title="c2">F</font>
 <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="E8:107_4"/><span class="kw">then </span>
<font color="Maroon" title="c4">G</font> <a href="qc_lang2.html#R2" title="QC_LANG2:pred.2">is_subformula_of</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="c2">F</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:107_4"><span class="lab"><font color="Green" title="E60">A10</font></span></a>, <a class="ref" href="qc_lang2.html#T77" target="_self" title="QC_LANG2:th.77">Th77</a></span>;<br/><span class="kw">hence </span><a NAME="E9:107_4"/>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang2.html#K15" title="QC_LANG2:func.15">Subformulae</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">H</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <font color="Maroon" title="c2">F</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:107_4"><span class="lab"><font color="Green" title="E59">A9</font></span></a>, <a class="ref" href="qc_lang2.html#D23" target="_self" title="QC_LANG2:def.23">Def23</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="E6:107"/>
(  not <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="qc_lang2.html#K15" title="QC_LANG2:func.15">Subformulae</a> <font color="Maroon" title="c1">H</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="qc_lang2.html#K15" title="QC_LANG2:func.15">Subformulae</a> <font color="Maroon" title="c2">F</font></span>)</span> or <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang2.html#K15" title="QC_LANG2:func.15">Subformulae</a> <font color="Maroon" title="c1">H</font> or <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang2.html#K15" title="QC_LANG2:func.15">Subformulae</a> <font color="Maroon" title="c2">F</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:107"/>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang2.html#K15" title="QC_LANG2:func.15">Subformulae</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">H</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <font color="Maroon" title="c2">F</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:107"><span class="lab"><font color="Green" title="E55">A3</font></span></a>, <a class="txt" href="#E5:107"><span class="lab"><font color="Green" title="E58">A8</font></span></a>, <a class="txt" href="#E4:107"><span class="lab"><font color="Green" title="E57">A5</font></span></a>, <a class="txt" href="#E3:107"><span class="lab"><font color="Green" title="E56">A4</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</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>
