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<div class="add">

<b>let </b><font color="Maroon" title="c1">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> ;<br/>

<b>thus </b><a NAME="E1:39"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" title="QC_LANG4:func.2">tree_of_subformulae</a> <font color="Maroon" title="c1">F</font></span>)</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> <font color="Maroon" title="c1">F</font>
  <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="39_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">y</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="E1:39_1"/><i><font color="Green" title="E35">A1</font></i>: 
<font color="Maroon" title="c2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" title="QC_LANG4:func.2">tree_of_subformulae</a> <font color="Maroon" title="c1">F</font></span>)</span>
 ;<br/>

<a NAME="E2:39_1"/><b>then </b>
<font color="Maroon" title="c2">y</font> is    <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="ref" href="qc_lang4.html#T26" target="_self" title="QC_LANG4:th.26">Th26</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c3">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> <b> such that </b><br/><a NAME="E4:39_1"/><i><font color="Green" title="E36">A2</font></i>: 
<font color="Maroon" title="c3">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font>
 ;<br/>
<a NAME="E5:39_1"/>
<font color="Maroon" title="c3">G</font> <a href="qc_lang2.html#R2" title="QC_LANG2:pred.2">is_subformula_of</a> <font color="Maroon" title="c1">F</font>
 
<b>by </b><i><a class="txt" href="qc_lang4.html#E1:39_1"><i><font color="Green" title="E35">A1</font></i></a>, <a class="txt" href="qc_lang4.html#E4:39_1"><i><font color="Green" title="E36">A2</font></i></a>, <a class="ref" href="qc_lang4.html#T39" target="_self" title="QC_LANG4:th.39">Th39</a></i>;<br/>
<b>hence </b><a NAME="E6:39_1"/>
<font color="Maroon" title="c2">y</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">F</font>
 <b>by </b><i><a class="txt" href="qc_lang4.html#E4:39_1"><i><font color="Green" title="E36">A2</font></i></a>, <a class="ref" href="qc_lang2.html#D23" title="QC_LANG2:def.23">QC_LANG2:def 23</a></i>;<br/>


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

<b>thus </b><a NAME="E2:39"/>
 <a href="qc_lang2.html#K15" title="QC_LANG2:func.15">Subformulae</a> <font color="Maroon" title="c1">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" title="QC_LANG4:func.2">tree_of_subformulae</a> <font color="Maroon" title="c1">F</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">y</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="E1:39_2"/>
<font color="Maroon" title="c2">y</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">F</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c3">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> <b> such that </b><br/><a NAME="E3:39_2"/><i><font color="Green" title="E35">A3</font></i>: 
( <font color="Maroon" title="c3">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">y</font> &amp; <font color="Maroon" title="c3">G</font> <a href="qc_lang2.html#R2" title="QC_LANG2:pred.2">is_subformula_of</a> <font color="Maroon" title="c1">F</font> )
 <b>by </b><i><a class="ref" href="qc_lang2.html#D23" title="QC_LANG2:def.23">QC_LANG2:def 23</a></i>;<br/>
<b>thus </b><a NAME="E4:39_2"/>
<font color="Maroon" title="c2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" title="QC_LANG4:func.2">tree_of_subformulae</a> <font color="Maroon" title="c1">F</font></span>)</span>
 <b>by </b><i><a class="txt" href="qc_lang4.html#E3:39_2"><i><font color="Green" title="E35">A3</font></i></a>, <a class="ref" href="qc_lang4.html#T39" target="_self" title="QC_LANG4:th.39">Th39</a></i>;<br/>


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


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