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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="qc_lang1.html#NM6">QC-pred_symbol</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="qc_lang1.html#M1">QC-variable_list</a> of <font color="Maroon">c<sub>2</sub></font>;<br/>







<b>thus </b><a NAME="E1:77"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="qc_lang2.html#R2" target="_self">is_subformula_of</a> <font color="Maroon">c<sub>3</sub></font> <a href="qc_lang1.html#K9">!</a> <font color="Maroon">c<sub>4</sub></font> implies <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="qc_lang1.html#K9">!</a> <font color="Maroon">c<sub>4</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:77_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="qc_lang2.html#R2" target="_self">is_subformula_of</a> <font color="Maroon">c<sub>3</sub></font> <a href="qc_lang1.html#K9">!</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<b>assume </b><a NAME="E2:77_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> <a href="qc_lang1.html#K9">!</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E3:77_1"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="qc_lang2.html#R3" target="_self">is_proper_subformula_of</a> <font color="Maroon">c<sub>3</sub></font> <a href="qc_lang1.html#K9">!</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="qc_lang2.html#E1:77_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="qc_lang2.html#D22" target="_self">Def22</a></i>;<br/>
<a NAME="E4:77_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a>  st <font color="Olive">b<sub>1</sub></font> <a href="qc_lang2.html#R1" target="_self">is_immediate_constituent_of</a> <font color="Maroon">c<sub>3</sub></font> <a href="qc_lang1.html#K9">!</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="qc_lang2.html#T75" target="_self">Th75</a></i>;<br/>
<b>hence </b><a NAME="E5:77_1"/>
contradiction
 <b>by </b><i><a class="ref" href="qc_lang2.html#T59" target="_self">Th59</a></i>;<br/>


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

<b>thus </b><a NAME="E2:77"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="qc_lang1.html#K9">!</a> <font color="Maroon">c<sub>4</sub></font> implies <font color="Maroon">c<sub>1</sub></font> <a href="qc_lang2.html#R2" target="_self">is_subformula_of</a> <font color="Maroon">c<sub>3</sub></font> <a href="qc_lang1.html#K9">!</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/>


</div>
