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

<span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">t</font> being   <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</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> holds  <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="trees_2.html#K7" title="TREES_2:func.7">|</a> <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang4.html#K2" title="QC_LANG4:func.2">tree_of_subformulae</a> <span class="p1">(<span class="default"><span class="p2">(<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="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Olive" title="b1">t</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">t</font> be    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</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 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"><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="trees_2.html#K7" title="TREES_2:func.7">|</a> <font color="Maroon" title="c2">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang4.html#K2" title="QC_LANG4:func.2">tree_of_subformulae</a> <span class="p1">(<span class="default"><span class="p2">(<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="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c2">t</font></span>)</span></span><br/>



<span class="kw">set </span><font color="Maroon" title="c3">T1</font> = <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="trees_2.html#K7" title="TREES_2:func.7">|</a> <font color="Maroon" title="c2">t</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">T2</font> =  <a href="qc_lang4.html#K2" title="QC_LANG4:func.2">tree_of_subformulae</a> <span class="p1">(<span class="default"><span class="p2">(<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="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c2">t</font></span>)</span>;<br/>
<span class="kw">thus </span><a NAME="E1:66"/><span class="lab"><font color="Green" title="E52">A1</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<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="trees_2.html#K7" title="TREES_2:func.7">|</a> <font color="Maroon" title="c2">t</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" title="QC_LANG4:func.2">tree_of_subformulae</a> <span class="p2">(<span class="default"><span class="p3">(<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="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c2">t</font></span>)</span></span>)</span>
  <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="trees_4.html#D1" title="TREES_4:def.1">TREES_4:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive">b<sub>1</sub></font> being   <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <span class="p1">(<span class="default"><span class="p2">(<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="trees_2.html#K7" title="TREES_2:func.7">|</a> <font color="Maroon" title="c2">t</font></span>)</span> holds  <span class="p1">(<span class="default"><span class="p2">(<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="trees_2.html#K7" title="TREES_2:func.7">|</a> <font color="Maroon" title="c2">t</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" title="QC_LANG4:func.2">tree_of_subformulae</a> <span class="p2">(<span class="default"><span class="p3">(<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="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c2">t</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">p</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="trees_2.html#D1" title="TREES_2:def.1">TREES_2:def 1</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="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<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="trees_2.html#K7" title="TREES_2:func.7">|</a> <font color="Maroon" title="c2">t</font></span>)</span> or <font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" title="QC_LANG4:func.2">tree_of_subformulae</a> <span class="p2">(<span class="default"><span class="p3">(<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="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c2">t</font></span>)</span></span>)</span> ) &amp; (  not <font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" title="QC_LANG4:func.2">tree_of_subformulae</a> <span class="p2">(<span class="default"><span class="p3">(<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="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c2">t</font></span>)</span></span>)</span> or <font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<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="trees_2.html#K7" title="TREES_2:func.7">|</a> <font color="Maroon" title="c2">t</font></span>)</span> ) )</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">consider </span><font color="Maroon" title="c6">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="E2:66_1_1"/><span class="lab"><font color="Green" title="E53">A2</font></span>: 
<font color="Maroon" title="c6">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c2">t</font>
 ;<br/><a NAME="E3:66_1_1"/><span class="lab"><font color="Green" title="E54">A3</font></span>: 
<font color="Maroon" title="c2">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">F</font> <a href="qc_lang4.html#K3" title="QC_LANG4:func.3">-entry_points_in_subformula_tree_of</a> <font color="Maroon" title="c6">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:66_1_1"><span class="lab"><font color="Green" title="E53">A2</font></span></a>, <a class="ref" href="qc_lang4.html#D3" target="_self" title="QC_LANG4:def.3">Def3</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c7">t9</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E5:66_1_1"/><span class="lab"><font color="Green" title="E55">A4</font></span>: 
<font color="Maroon" title="c7">t9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">t</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c5">p</font>
 ;<br/><span class="kw">assume </span><a NAME="E6:66_1_1"/>
<font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<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="trees_2.html#K7" title="TREES_2:func.7">|</a> <font color="Maroon" title="c2">t</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="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" title="QC_LANG4:func.2">tree_of_subformulae</a> <span class="p2">(<span class="default"><span class="p3">(<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="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c2">t</font></span>)</span></span>)</span></span><br/><a NAME="E7:66_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<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></span>)</span> <a href="trees_1.html#K4" title="TREES_1:func.4">|</a> <font color="Maroon" title="c2">t</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="trees_2.html#D11" title="TREES_2:def.11">TREES_2:def 11</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">t9</font> = <font color="Maroon" title="c7">t9</font> as    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:66_1_1"><span class="lab"><font color="Green" title="E55">A4</font></span></a>, <a class="ref" href="trees_1.html#D9" title="TREES_1:def.9">TREES_1:def 9</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c9">H</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="E10:66_1_1"/><span class="lab"><font color="Green" title="E56">A5</font></span>: 
<font color="Maroon" title="c9">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c8">t9</font>
 ;<br/><a NAME="E11:66_1_1"/><span class="lab"><font color="Green" title="E57">A6</font></span>: 
<font color="Maroon" title="c6">G</font> <a href="qc_lang4.html#K3" title="QC_LANG4:func.3">-entry_points_in_subformula_tree_of</a> <font color="Maroon" title="c9">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" title="QC_LANG4:func.2">tree_of_subformulae</a> <span class="p2">(<span class="default"><span class="p3">(<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="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c2">t</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:66_1_1"><span class="lab"><font color="Green" title="E53">A2</font></span></a>, <a class="ref" href="trees_1.html#D14" title="TREES_1:def.14">TREES_1:def 14</a></span>;<br/><a NAME="E12:66_1_1"/>
<font color="Maroon" title="c8">t9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">F</font> <a href="qc_lang4.html#K3" title="QC_LANG4:func.3">-entry_points_in_subformula_tree_of</a> <font color="Maroon" title="c9">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:66_1_1"><span class="lab"><font color="Green" title="E56">A5</font></span></a>, <a class="ref" href="qc_lang4.html#D3" target="_self" title="QC_LANG4:def.3">Def3</a></span>;<br/><a NAME="E13:66_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">G</font> <a href="qc_lang4.html#K3" title="QC_LANG4:func.3">-entry_points_in_subformula_tree_of</a> <font color="Maroon" title="c9">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:66_1_1"><span class="lab"><font color="Green" title="E54">A3</font></span></a>, <a class="txt" href="#E5:66_1_1"><span class="lab"><font color="Green" title="E55">A4</font></span></a>, <a class="ref" href="qc_lang4.html#T58" target="_self" title="QC_LANG4:th.58">Th58</a></span>;<br/><span class="kw">hence </span><a NAME="E14:66_1_1"/>
<font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" title="QC_LANG4:func.2">tree_of_subformulae</a> <span class="p2">(<span class="default"><span class="p3">(<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="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c2">t</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:66_1_1"><span class="lab"><font color="Green" title="E57">A6</font></span></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">hence </span><a NAME="E2:66_1"/>
( <font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<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="trees_2.html#K7" title="TREES_2:func.7">|</a> <font color="Maroon" title="c2">t</font></span>)</span> implies <font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" title="QC_LANG4:func.2">tree_of_subformulae</a> <span class="p2">(<span class="default"><span class="p3">(<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="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c2">t</font></span>)</span></span>)</span> )
 ; <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="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" title="QC_LANG4:func.2">tree_of_subformulae</a> <span class="p2">(<span class="default"><span class="p3">(<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="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c2">t</font></span>)</span></span>)</span> or <font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<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="trees_2.html#K7" title="TREES_2:func.7">|</a> <font color="Maroon" title="c2">t</font></span>)</span> )</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">consider </span><font color="Maroon" title="c6">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="E2:66_1_2"/><span class="lab"><font color="Green" title="E53">A7</font></span>: 
<font color="Maroon" title="c6">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c2">t</font>
 ;<br/><span class="kw">assume </span><a NAME="E3:66_1_2"/>
<font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" title="QC_LANG4:func.2">tree_of_subformulae</a> <span class="p2">(<span class="default"><span class="p3">(<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="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c2">t</font></span>)</span></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="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<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="trees_2.html#K7" title="TREES_2:func.7">|</a> <font color="Maroon" title="c2">t</font></span>)</span></span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">s</font> = <font color="Maroon" title="c5">p</font> as    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</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="c6">G</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:66_1_2"><span class="lab"><font color="Green" title="E53">A7</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c8">H</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="E6:66_1_2"/><span class="lab"><font color="Green" title="E54">A8</font></span>: 
<font color="Maroon" title="c8">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c6">G</font></span>)</span> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c7">s</font>
 ;<br/><a NAME="E7:66_1_2"/><span class="lab"><font color="Green" title="E55">A9</font></span>: 
<font color="Maroon" title="c7">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">G</font> <a href="qc_lang4.html#K3" title="QC_LANG4:func.3">-entry_points_in_subformula_tree_of</a> <font color="Maroon" title="c8">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:66_1_2"><span class="lab"><font color="Green" title="E54">A8</font></span></a>, <a class="ref" href="qc_lang4.html#D3" target="_self" title="QC_LANG4:def.3">Def3</a></span>;<br/><a NAME="E8:66_1_2"/><span class="lab"><font color="Green" title="E56">A10</font></span>: 
<font color="Maroon" title="c1">F</font> <a href="qc_lang4.html#K3" title="QC_LANG4:func.3">-entry_points_in_subformula_tree_of</a> <font color="Maroon" title="c8">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="trees_1.html#D14" title="TREES_1:def.14">TREES_1:def 14</a></span>;<br/><a NAME="E9:66_1_2"/>
<font color="Maroon" title="c2">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">F</font> <a href="qc_lang4.html#K3" title="QC_LANG4:func.3">-entry_points_in_subformula_tree_of</a> <font color="Maroon" title="c6">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:66_1_2"><span class="lab"><font color="Green" title="E53">A7</font></span></a>, <a class="ref" href="qc_lang4.html#D3" target="_self" title="QC_LANG4:def.3">Def3</a></span>;<br/><a NAME="E10:66_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">t</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c7">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">F</font> <a href="qc_lang4.html#K3" title="QC_LANG4:func.3">-entry_points_in_subformula_tree_of</a> <font color="Maroon" title="c8">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:66_1_2"><span class="lab"><font color="Green" title="E55">A9</font></span></a>, <a class="ref" href="qc_lang4.html#T57" target="_self" title="QC_LANG4:th.57">Th57</a></span>;<br/><a NAME="E11:66_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<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></span>)</span> <a href="trees_1.html#K4" title="TREES_1:func.4">|</a> <font color="Maroon" title="c2">t</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:66_1_2"><span class="lab"><font color="Green" title="E56">A10</font></span></a>, <a class="ref" href="trees_1.html#D9" title="TREES_1:def.9">TREES_1:def 9</a></span>;<br/><span class="kw">hence </span><a NAME="E12:66_1_2"/>
<font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<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="trees_2.html#K7" title="TREES_2:func.7">|</a> <font color="Maroon" title="c2">t</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="trees_2.html#D11" title="TREES_2:def.11">TREES_2:def 11</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">hence </span><a NAME="E4:66_1"/>
(  not <font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" title="QC_LANG4:func.2">tree_of_subformulae</a> <span class="p2">(<span class="default"><span class="p3">(<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="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c2">t</font></span>)</span></span>)</span> or <font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<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="trees_2.html#K7" title="TREES_2:func.7">|</a> <font color="Maroon" title="c2">t</font></span>)</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><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">p</font> be   <a href="trees_4.html#NM1" title="TREES_4:NM.1">Node</a> of <span class="p1">(<span class="default"><span class="p2">(<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="trees_2.html#K7" title="TREES_2:func.7">|</a> <font color="Maroon" title="c2">t</font></span>)</span>; <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_lang4.html#K2" title="QC_LANG4:func.2">tree_of_subformulae</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="trees_2.html#K7" title="TREES_2:func.7">|</a> <font color="Maroon" title="c2">t</font></span>)</span> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" title="QC_LANG4:func.2">tree_of_subformulae</a> <span class="p2">(<span class="default"><span class="p3">(<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="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c2">t</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">p</font></span><br/><span class="kw">consider </span><font color="Maroon" title="c6">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="E2:66_2"/><span class="lab"><font color="Green" title="E53">A11</font></span>: 
<font color="Maroon" title="c6">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c2">t</font>
 ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c7">s</font> = <font color="Maroon" title="c5">p</font> as    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</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="c6">G</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:66"><span class="lab"><font color="Green" title="E52">A1</font></span></a>, <a class="txt" href="#E2:66_2"><span class="lab"><font color="Green" title="E53">A11</font></span></a></span>;<br/><a NAME="E4:66_2"/><span class="lab"><font color="Green" title="E54">A12</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<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="trees_2.html#K7" title="TREES_2:func.7">|</a> <font color="Maroon" title="c2">t</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<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></span>)</span> <a href="trees_1.html#K4" title="TREES_1:func.4">|</a> <font color="Maroon" title="c2">t</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="trees_2.html#D11" title="TREES_2:def.11">TREES_2:def 11</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">t9</font> = <font color="Maroon" title="c2">t</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c7">s</font> as    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</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> <span class="kw">by</span> <span class="lab"><a class="ref" href="trees_1.html#D9" title="TREES_1:def.9">TREES_1:def 9</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c9">H</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="E7:66_2"/><span class="lab"><font color="Green" title="E55">A13</font></span>: 
<font color="Maroon" title="c9">H</font> <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_lang4.html#K2" title="QC_LANG4:func.2">tree_of_subformulae</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="trees_2.html#K7" title="TREES_2:func.7">|</a> <font color="Maroon" title="c2">t</font></span>)</span> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c5">p</font>
 ;<br/><a NAME="E8:66_2"/><span class="lab"><font color="Green" title="E56">A14</font></span>: 
<font color="Maroon" title="c2">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">F</font> <a href="qc_lang4.html#K3" title="QC_LANG4:func.3">-entry_points_in_subformula_tree_of</a> <font color="Maroon" title="c6">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:66_2"><span class="lab"><font color="Green" title="E53">A11</font></span></a>, <a class="ref" href="qc_lang4.html#D3" target="_self" title="QC_LANG4:def.3">Def3</a></span>;<br/><a NAME="E9:66_2"/>
<span class="p1">(<span class="default"><span class="p2">(<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="trees_2.html#K7" title="TREES_2:func.7">|</a> <font color="Maroon" title="c2">t</font></span>)</span> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">t</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c5">p</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:66_2"><span class="lab"><font color="Green" title="E54">A12</font></span></a>, <a class="ref" href="trees_2.html#D11" title="TREES_2:def.11">TREES_2:def 11</a></span>;<br/><a NAME="E10:66_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">t9</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">F</font> <a href="qc_lang4.html#K3" title="QC_LANG4:func.3">-entry_points_in_subformula_tree_of</a> <font color="Maroon" title="c9">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:66_2"><span class="lab"><font color="Green" title="E55">A13</font></span></a>, <a class="ref" href="qc_lang4.html#D3" target="_self" title="QC_LANG4:def.3">Def3</a></span>;<br/><a NAME="E11:66_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">G</font> <a href="qc_lang4.html#K3" title="QC_LANG4:func.3">-entry_points_in_subformula_tree_of</a> <font color="Maroon" title="c9">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:66_2"><span class="lab"><font color="Green" title="E56">A14</font></span></a>, <a class="ref" href="qc_lang4.html#T58" target="_self" title="QC_LANG4:th.58">Th58</a></span>;<br/><span class="kw">hence </span><a NAME="E12:66_2"/>
<span class="p1">(<span class="default"><span class="p2">(<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="trees_2.html#K7" title="TREES_2:func.7">|</a> <font color="Maroon" title="c2">t</font></span>)</span> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c5">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" title="QC_LANG4:func.2">tree_of_subformulae</a> <span class="p2">(<span class="default"><span class="p3">(<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="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c2">t</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:66_2"><span class="lab"><font color="Green" title="E53">A11</font></span></a>, <a class="txt" href="#E7:66_2"><span class="lab"><font color="Green" title="E55">A13</font></span></a>, <a class="ref" href="qc_lang4.html#D3" target="_self" title="QC_LANG4:def.3">Def3</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">hence </span><a NAME="E3:66"/>
for <font color="Olive" title="b1">p</font> being  <a href="trees_4.html#NM1" title="TREES_4:NM.1">Node</a> of <span class="p1">(<span class="default"><span class="p2">(<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="trees_2.html#K7" title="TREES_2:func.7">|</a> <font color="Maroon" title="c2">t</font></span>)</span> holds  <span class="p1">(<span class="default"><span class="p2">(<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="trees_2.html#K7" title="TREES_2:func.7">|</a> <font color="Maroon" title="c2">t</font></span>)</span> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K2" title="QC_LANG4:func.2">tree_of_subformulae</a> <span class="p2">(<span class="default"><span class="p3">(<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="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c2">t</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">p</font>
 ; <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>
