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

<b>thus </b><a NAME="E1:83"/>
 <a href="qc_lang2.html#K15" title="QC_LANG2:func.15">Subformulae</a> <a href="qc_lang2.html#K1" title="QC_LANG2:func.1">FALSUM</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">VERUM</a> ,<a href="qc_lang2.html#K1" title="QC_LANG2:func.1">FALSUM</a> </span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
  <i><font color="Red">:: according to </font></i><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">VERUM</a> ,<a href="qc_lang2.html#K1" title="QC_LANG2:func.1">FALSUM</a> </span><a href="tarski.html#K2" title="TARSKI:func.2">}</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> <a href="qc_lang2.html#K1" title="QC_LANG2:func.1">FALSUM</a> </span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="83_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c1">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <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">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> <a href="qc_lang2.html#K1" title="QC_LANG2:func.1">FALSUM</a>  or <font color="Maroon" title="c1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">VERUM</a> ,<a href="qc_lang2.html#K1" title="QC_LANG2:func.1">FALSUM</a> </span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )</span><br/>

<b>assume </b><a NAME="E1:83_1"/>
<font color="Maroon" title="c1">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> <a href="qc_lang2.html#K1" title="QC_LANG2:func.1">FALSUM</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">VERUM</a> ,<a href="qc_lang2.html#K1" title="QC_LANG2:func.1">FALSUM</a> </span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span></span><br/>

<a NAME="E2:83_1"/><b>then </b>
 ex <font color="Olive" title="b1">F</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 <br/>( <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font> &amp; <font color="Olive" title="b1">F</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>  )
 
<b>by </b><i><a class="ref" href="qc_lang2.html#D23" target="_self" title="QC_LANG2:def.23">Def23</a></i>;<br/>
<a NAME="E3:83_1"/><b>then </b>
( <font color="Maroon" title="c1">a</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">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K11" title="QC_LANG1:func.11">VERUM</a>  )
 
<b>by </b><i><a class="ref" href="qc_lang2.html#T101" target="_self" title="QC_LANG2:th.101">Th101</a></i>;<br/>
<b>hence </b><a NAME="E4:83_1"/>
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">VERUM</a> ,<a href="qc_lang2.html#K1" title="QC_LANG2:func.1">FALSUM</a> </span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>let </b><font color="Maroon" title="c1">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <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">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">VERUM</a> ,<a href="qc_lang2.html#K1" title="QC_LANG2:func.1">FALSUM</a> </span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> or <font color="Maroon" title="c1">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> <a href="qc_lang2.html#K1" title="QC_LANG2:func.1">FALSUM</a>  )</span><br/>

<b>assume </b><a NAME="E2:83"/><i><font color="Green" title="E53">A1</font></i>: 
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><a href="qc_lang1.html#K11" title="QC_LANG1:func.11">VERUM</a> ,<a href="qc_lang2.html#K1" title="QC_LANG2:func.1">FALSUM</a> </span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">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> <a href="qc_lang2.html#K1" title="QC_LANG2:func.1">FALSUM</a> </span><br/>

<a NAME="E3:83"/><b>then </b><i><font color="Green" title="E54">A2</font></i>: 
( <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K11" title="QC_LANG1:func.11">VERUM</a>  or <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang2.html#K1" title="QC_LANG2:func.1">FALSUM</a>  )
 
<b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c2">a</font> = <font color="Maroon" title="c1">a</font> as    <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>  <b>by </b><i><a class="txt" href="#E2:83"><i><font color="Green" title="E53">A1</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<a NAME="E5:83"/>
<font color="Maroon" title="c2">a</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> 
 
<b>by </b><i><a class="txt" href="#E3:83"><i><font color="Green" title="E54">A2</font></i></a>, <a class="ref" href="qc_lang2.html#T101" target="_self" title="QC_LANG2:th.101">Th101</a></i>;<br/>
<b>hence </b><a NAME="E6:83"/>
<font color="Maroon" title="c1">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> <a href="qc_lang2.html#K1" title="QC_LANG2:func.1">FALSUM</a> 
 <b>by </b><i><a class="ref" href="qc_lang2.html#D23" target="_self" title="QC_LANG2:def.23">Def23</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
