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

<b>let </b><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> ;<br/><b>let </b><font color="Maroon" title="c2">x</font> be   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a>;<br/>



<b>thus </b><a NAME="E1:88"/>
 <a href="qc_lang2.html#K15" title="QC_LANG2:func.15">Subformulae</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">H</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">H</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</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><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">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><br/>

<b>assume </b><a NAME="E1:88_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"><a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">H</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">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> <b> such that </b><br/><a NAME="E3:88_1"/><i><font color="Green" title="E59">A1</font></i>: 
( <font color="Maroon" title="c4">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> &amp; <font color="Maroon" title="c4">F</font> <a href="qc_lang2.html#R2" title="QC_LANG2:pred.2">is_subformula_of</a>  <a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">H</font> )
 <b>by </b><i><a class="ref" href="qc_lang2.html#D23" target="_self" title="QC_LANG2:def.23">Def23</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:88_1_1"/>
<font color="Maroon" title="c4">F</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">H</font>
 ;<br/><a NAME="E2:88_1_1"/><b>then </b>
<font color="Maroon" title="c4">F</font> <a href="qc_lang2.html#R3" title="QC_LANG2:pred.3">is_proper_subformula_of</a>  <a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">H</font>
 <b>by </b><i><a class="txt" href="qc_lang2.html#E3:88_1"><i><font color="Green" title="E59">A1</font></i></a>, <a class="ref" href="qc_lang2.html#D22" target="_self" title="QC_LANG2:def.22">Def22</a></i>;<br/><a NAME="E3:88_1_1"/><b>then </b>
<font color="Maroon" title="c4">F</font> <a href="qc_lang2.html#R2" title="QC_LANG2:pred.2">is_subformula_of</a> <font color="Maroon" title="c1">H</font>
 <b>by </b><i><a class="ref" href="qc_lang2.html#T91" target="_self" title="QC_LANG2:th.91">Th91</a></i>;<br/><b>hence </b><a NAME="E4:88_1_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> <font color="Maroon" title="c1">H</font>
 <b>by </b><i><a class="txt" href="qc_lang2.html#E3:88_1"><i><font color="Green" title="E59">A1</font></i></a>, <a class="ref" href="qc_lang2.html#D23" target="_self" title="QC_LANG2:def.23">Def23</a></i>;<br/></div><b>end;</b></div>
<a NAME="E5:88_1"/><b>then </b>
( <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> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">H</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<b>by </b><i><a class="txt" href="qc_lang2.html#E3:88_1"><i><font color="Green" title="E59">A1</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E6:88_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"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">H</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>


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

<b>let </b><font color="Maroon" title="c3">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><br/>

<b>assume </b><a NAME="E2:88"/><i><font color="Green" title="E59">A2</font></i>: 
<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"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">H</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/>

<div><i><font color="Green" title="E60">A3</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="88_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:88_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> <font color="Maroon" title="c1">H</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c4">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> <b> such that </b><br/><a NAME="E3:88_2"/><i><font color="Green" title="E61">A4</font></i>: 
( <font color="Maroon" title="c4">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> &amp; <font color="Maroon" title="c4">F</font> <a href="qc_lang2.html#R2" title="QC_LANG2:pred.2">is_subformula_of</a> <font color="Maroon" title="c1">H</font> )
 <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="E4:88_2"/>
<font color="Maroon" title="c1">H</font> <a href="qc_lang2.html#R1" title="QC_LANG2:pred.1">is_immediate_constituent_of</a>  <a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">H</font>
 <b>by </b><i><a class="ref" href="qc_lang2.html#T63" target="_self" title="QC_LANG2:th.63">Th63</a></i>;<br/><a NAME="E5:88_2"/><b>then </b>
<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_lang1.html#K14" title="QC_LANG1:func.14">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">H</font>
 <b>by </b><i><a class="ref" href="qc_lang2.html#T73" target="_self" title="QC_LANG2:th.73">Th73</a></i>;<br/><a NAME="E6:88_2"/><b>then </b>
<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#K14" title="QC_LANG1:func.14">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">H</font>
 <b>by </b><i><a class="ref" href="qc_lang2.html#D22" target="_self" title="QC_LANG2:def.22">Def22</a></i>;<br/><a NAME="E7:88_2"/><b>then </b>
<font color="Maroon" title="c4">F</font> <a href="qc_lang2.html#R2" title="QC_LANG2:pred.2">is_subformula_of</a>  <a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">H</font>
 <b>by </b><i><a class="txt" href="qc_lang2.html#E3:88_2"><i><font color="Green" title="E61">A4</font></i></a>, <a class="ref" href="qc_lang2.html#T77" target="_self" title="QC_LANG2:th.77">Th77</a></i>;<br/><b>hence </b><a NAME="E8:88_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"><a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">H</font></span>)</span>
 <b>by </b><i><a class="txt" href="qc_lang2.html#E3:88_2"><i><font color="Green" title="E61">A4</font></i></a>, <a class="ref" href="qc_lang2.html#D23" target="_self" title="QC_LANG2:def.23">Def23</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="88_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:88_3"/>
<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"><a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">H</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/><a NAME="E2:88_3"/><b>then </b>
( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">H</font> &amp;  <a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> <font color="Maroon" title="c2">x</font>,<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#K14" title="QC_LANG1:func.14">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">H</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:88_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"><a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">H</font></span>)</span>
 <b>by </b><i><a class="ref" href="qc_lang2.html#D23" target="_self" title="QC_LANG2:def.23">Def23</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:88"/>
<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"><a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c1">H</font></span>)</span>
 <b>by </b><i><a class="txt" href="qc_lang2.html#E2:88"><i><font color="Green" title="E59">A2</font></i></a>, <a class="txt" href="qc_lang2.html#E3:88"><i><font color="Green" title="E60">A3</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>


</div>
