<?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/>

<a NAME="E1:92"/>
( <font color="Maroon" title="c1">H</font> is <a href="qc_lang1.html#V2" title="QC_LANG1:attr.2">atomic</a> implies  <a href="qc_lang2.html#K15" title="QC_LANG2:func.15">Subformulae</a> <font color="Maroon" title="c1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">H</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:92_1"/>
 ex <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ex <font color="Olive" title="b2">P</font> being  <a href="qc_lang1.html#NM6" title="QC_LANG1:NM.6">QC-pred_symbol</a> of <font color="Olive" title="b1">k</font> ex <font color="Olive" title="b3">V</font> being   <a href="qc_lang1.html#M1" title="QC_LANG1:mode.1">QC-variable_list</a> of <font color="Olive" title="b1">k</font> st <font color="Maroon" title="c1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">P</font> <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">!</a> <font color="Olive" title="b3">V</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="qc_lang1.html#D17" title="QC_LANG1:def.17">QC_LANG1:def 17</a><br/>

<b>hence </b><a NAME="E2:92_1"/>
 <a href="qc_lang2.html#K15" title="QC_LANG2:func.15">Subformulae</a> <font color="Maroon" title="c1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">H</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="qc_lang2.html#T108" target="_self" title="QC_LANG2:th.108">Th108</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:92"/>
( ( <font color="Maroon" title="c1">H</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">H</font> is <a href="qc_lang1.html#V2" title="QC_LANG1:attr.2">atomic</a> ) implies  <a href="qc_lang2.html#K15" title="QC_LANG2:func.15">Subformulae</a> <font color="Maroon" title="c1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">H</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <b>by </b><i><a class="ref" href="qc_lang2.html#T107" target="_self" title="QC_LANG2:th.107">Th107</a></i>;<br/>

<b>assume </b><a NAME="E3:92"/><i><font color="Green" title="E57">A1</font></i>: 
 <a href="qc_lang2.html#K15" title="QC_LANG2:func.15">Subformulae</a> <font color="Maroon" title="c1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">H</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ;<br/>

<b>assume </b><a NAME="E4:92"/>
( <font color="Maroon" title="c1">H</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="qc_lang1.html#K11" title="QC_LANG1:func.11">VERUM</a>  &amp; not <font color="Maroon" title="c1">H</font> is <a href="qc_lang1.html#V2" title="QC_LANG1:attr.2">atomic</a> )
 ;<br/>

<a NAME="E5:92"/><b>then </b>
( <font color="Maroon" title="c1">H</font> is <a href="qc_lang1.html#V3" title="QC_LANG1:attr.3">negative</a> or <font color="Maroon" title="c1">H</font> is <a href="qc_lang1.html#V4" title="QC_LANG1:attr.4">conjunctive</a> or <font color="Maroon" title="c1">H</font> is <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a> )
 
<b>by </b><i><a class="ref" href="qc_lang1.html#T33" title="QC_LANG1:th.33">QC_LANG1:33</a></i>;<br/>
<a NAME="E6:92"/><b>then </b><i><font color="Green" title="E58">A2</font></i>: 
( <font color="Maroon" title="c1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K12" title="QC_LANG1:func.12">'not'</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K17" title="QC_LANG1:func.17">the_argument_of</a> <font color="Maroon" title="c1">H</font></span>)</span> or <font color="Maroon" title="c1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_left_argument_of</a> <font color="Maroon" title="c1">H</font></span>)</span> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_right_argument_of</a> <font color="Maroon" title="c1">H</font></span>)</span> or <font color="Maroon" title="c1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">bound_in</a> <font color="Maroon" title="c1">H</font></span>)</span>,<span class="p1">(<span class="default"><a href="qc_lang1.html#K21" title="QC_LANG1:func.21">the_scope_of</a> <font color="Maroon" title="c1">H</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="qc_lang2.html#T4" target="_self" title="QC_LANG2:th.4">Th4</a>, <a class="ref" href="qc_lang2.html#T7" target="_self" title="QC_LANG2:th.7">Th7</a>, <a class="ref" href="qc_lang1.html#D23" title="QC_LANG1:def.23">QC_LANG1:def 23</a></i>;<br/>
<div><i><font color="Green" title="E59">A3</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="92_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:92_2"/>
<font color="Maroon" title="c1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K12" title="QC_LANG1:func.12">'not'</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K17" title="QC_LANG1:func.17">the_argument_of</a> <font color="Maroon" title="c1">H</font></span>)</span>
 ;<br/><a NAME="E2:92_2"/><b>then </b><i><font color="Green" title="E60">A4</font></i>: 
 <a href="qc_lang1.html#K17" title="QC_LANG1:func.17">the_argument_of</a> <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>
 <b>by </b><i><a class="ref" href="qc_lang2.html#T60" target="_self" title="QC_LANG2:th.60">Th60</a></i>;<br/><a NAME="E3:92_2"/><b>then </b>
 <a href="qc_lang1.html#K17" title="QC_LANG1:func.17">the_argument_of</a> <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>
 <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="E4:92_2"/><b>then </b>
 <a href="qc_lang1.html#K17" title="QC_LANG1:func.17">the_argument_of</a> <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>
 <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="E5:92_2"/><b>then </b><i><font color="Green" title="E61">A5</font></i>: 
 <a href="qc_lang1.html#K17" title="QC_LANG1:func.17">the_argument_of</a> <font color="Maroon" title="c1">H</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="ref" href="qc_lang2.html#D23" target="_self" title="QC_LANG2:def.23">Def23</a></i>;<br/><a NAME="E6:92_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K17" title="QC_LANG1:func.17">the_argument_of</a> <font color="Maroon" title="c1">H</font></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <font color="Maroon" title="c1">H</font></span>)</span>
 <b>by </b><i><a class="txt" href="qc_lang2.html#E2:92_2"><i><font color="Green" title="E60">A4</font></i></a>, <a class="ref" href="qc_lang2.html#T71" target="_self" title="QC_LANG2:th.71">Th71</a></i>;<br/><b>hence </b><a NAME="E7:92_2"/>
contradiction
 <b>by </b><i><a class="txt" href="qc_lang2.html#E3:92"><i><font color="Green" title="E57">A1</font></i></a>, <a class="txt" href="qc_lang2.html#E5:92_2"><i><font color="Green" title="E61">A5</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="92_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:92_3"/>
<font color="Maroon" title="c1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_left_argument_of</a> <font color="Maroon" title="c1">H</font></span>)</span> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_right_argument_of</a> <font color="Maroon" title="c1">H</font></span>)</span>
 ;<br/><a NAME="E2:92_3"/><b>then </b><i><font color="Green" title="E60">A6</font></i>: 
(  <a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_left_argument_of</a> <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> &amp;  <a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_right_argument_of</a> <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> )
 <b>by </b><i><a class="ref" href="qc_lang2.html#T62" target="_self" title="QC_LANG2:th.62">Th62</a></i>;<br/><a NAME="E3:92_3"/><b>then </b>
(  <a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_left_argument_of</a> <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> &amp;  <a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_right_argument_of</a> <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> )
 <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="E4:92_3"/><b>then </b>
(  <a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_left_argument_of</a> <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> &amp;  <a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_right_argument_of</a> <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> )
 <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="E5:92_3"/><b>then </b><i><font color="Green" title="E61">A7</font></i>: 
(  <a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_left_argument_of</a> <font color="Maroon" title="c1">H</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> &amp;  <a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_right_argument_of</a> <font color="Maroon" title="c1">H</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="ref" href="qc_lang2.html#D23" target="_self" title="QC_LANG2:def.23">Def23</a></i>;<br/><a NAME="E6:92_3"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_left_argument_of</a> <font color="Maroon" title="c1">H</font></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <font color="Maroon" title="c1">H</font></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_right_argument_of</a> <font color="Maroon" title="c1">H</font></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <font color="Maroon" title="c1">H</font></span>)</span> )
 <b>by </b><i><a class="txt" href="qc_lang2.html#E2:92_3"><i><font color="Green" title="E60">A6</font></i></a>, <a class="ref" href="qc_lang2.html#T71" target="_self" title="QC_LANG2:th.71">Th71</a></i>;<br/><b>hence </b><a NAME="E7:92_3"/>
contradiction
 <b>by </b><i><a class="txt" href="qc_lang2.html#E3:92"><i><font color="Green" title="E57">A1</font></i></a>, <a class="txt" href="qc_lang2.html#E5:92_3"><i><font color="Green" title="E61">A7</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div>
<a NAME="E9:92"/><b>then </b><i><font color="Green" title="E60">A8</font></i>: 
 <a href="qc_lang1.html#K21" title="QC_LANG1:func.21">the_scope_of</a> <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>
 
<b>by </b><i><a class="txt" href="qc_lang2.html#E6:92"><i><font color="Green" title="E58">A2</font></i></a>, <a class="txt" href="qc_lang2.html#E7:92"><i><font color="Green" title="E59">A3</font></i></a>, <a class="ref" href="qc_lang2.html#T63" target="_self" title="QC_LANG2:th.63">Th63</a></i>;<br/>
<a NAME="E10:92"/><b>then </b>
 <a href="qc_lang1.html#K21" title="QC_LANG1:func.21">the_scope_of</a> <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>
 
<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="E11:92"/><b>then </b>
 <a href="qc_lang1.html#K21" title="QC_LANG1:func.21">the_scope_of</a> <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>
 
<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="E12:92"/><b>then </b><i><font color="Green" title="E61">A9</font></i>: 
 <a href="qc_lang1.html#K21" title="QC_LANG1:func.21">the_scope_of</a> <font color="Maroon" title="c1">H</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="ref" href="qc_lang2.html#D23" target="_self" title="QC_LANG2:def.23">Def23</a></i>;<br/>
<a NAME="E13:92"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K21" title="QC_LANG1:func.21">the_scope_of</a> <font color="Maroon" title="c1">H</font></span>)</span></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10" title="QC_LANG1:func.10">@</a> <font color="Maroon" title="c1">H</font></span>)</span>
 
<b>by </b><i><a class="txt" href="qc_lang2.html#E9:92"><i><font color="Green" title="E60">A8</font></i></a>, <a class="ref" href="qc_lang2.html#T71" target="_self" title="QC_LANG2:th.71">Th71</a></i>;<br/>
<b>hence </b><a NAME="E14:92"/>
contradiction
 <b>by </b><i><a class="txt" href="qc_lang2.html#E3:92"><i><font color="Green" title="E57">A1</font></i></a>, <a class="txt" href="qc_lang2.html#E12:92"><i><font color="Green" title="E61">A9</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>


</div>
