<?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>thus </b><a NAME="E1:31"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K1" target="_self">list_of_immediate_constituents</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a>  : <font color="Olive">b<sub>1</sub></font> <a href="qc_lang2.html#R1">is_immediate_constituent_of</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:31_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K1" target="_self">list_of_immediate_constituents</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:31_1"/><b>then </b><i><font color="Green">E29</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K1" target="_self">list_of_immediate_constituents</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K1" target="_self">list_of_immediate_constituents</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E3:31_1"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K1" target="_self">list_of_immediate_constituents</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="qc_lang1.html#K8">QC-WFF</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font> as    <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a>  <b>by </b><i><a class="txt" href="qc_lang4.html#E1:31_1"><i><font color="Green">E28</font></i></a></i>;<br/>
<a NAME="E5:31_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="qc_lang2.html#R1">is_immediate_constituent_of</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="qc_lang4.html#E2:31_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="qc_lang4.html#T30" target="_self">Th30</a></i>;<br/>
<b>hence </b><a NAME="E6:31_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a>  : <font color="Olive">b<sub>1</sub></font> <a href="qc_lang2.html#R1">is_immediate_constituent_of</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span> 
 ;<br/>


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

<b>thus </b><a NAME="E2:31"/>
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a>  : <font color="Olive">b<sub>1</sub></font> <a href="qc_lang2.html#R1">is_immediate_constituent_of</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K1" target="_self">list_of_immediate_constituents</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:31_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a>  : <font color="Olive">b<sub>1</sub></font> <a href="qc_lang2.html#R1">is_immediate_constituent_of</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>3</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a> <b> such that </b><br/><a NAME="E3:31_2"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="qc_lang2.html#R1">is_immediate_constituent_of</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/>
<a NAME="E4:31_2"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K1" target="_self">list_of_immediate_constituents</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K1" target="_self">list_of_immediate_constituents</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:31_2_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="qc_lang1.html#K11">VERUM</a> 
 
<b>by </b><i><a class="txt" href="qc_lang4.html#E3:31_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="qc_lang2.html#T58">QC_LANG2:58</a></i>;<br/>
<a NAME="E2:31_2_1"/><i><font color="Green">E30</font></i>: 
not <font color="Maroon">c<sub>1</sub></font> is <a href="qc_lang1.html#V2">atomic</a>
 
<b>by </b><i><a class="txt" href="qc_lang4.html#E3:31_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="qc_lang2.html#T64">QC_LANG2:64</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:31_2_1_1"/>
( <font color="Maroon">c<sub>1</sub></font> is <a href="qc_lang1.html#V3">negative</a> or <font color="Maroon">c<sub>1</sub></font> is <a href="qc_lang1.html#V4">conjunctive</a> or <font color="Maroon">c<sub>1</sub></font> is <a href="qc_lang1.html#V5">universal</a> )
 </span><b>by </b><i><a class="txt" href="qc_lang4.html#E1:31_2_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="qc_lang4.html#E2:31_2_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="qc_lang1.html#T33">QC_LANG1:33</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_1_1_1"><b>suppose </b></a><a NAME="E1:31_2_1_1_1"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="qc_lang1.html#V3">negative</a>
 ;<br/><div class="add"><a NAME="E2:31_2_1_1_1"/><b>then </b><i><font color="Green">E32</font></i>: 
 <a href="qc_lang4.html#K1" target="_self">list_of_immediate_constituents</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K17">the_argument_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="qc_lang4.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E3:31_2_1_1_1"/><i><font color="Green">E33</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K17">the_argument_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#D8">FINSEQ_1:def 8</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E5:31_2_1_1_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><a NAME="E6:31_2_1_1_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="qc_lang4.html#E5:31_2_1_1_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="finseq_1.html#T5">FINSEQ_1:5</a></i>;<br/><a NAME="E7:31_2_1_1_1"/><i><font color="Green">E36</font></i>: 
<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K17">the_argument_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K17">the_argument_of</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="qc_lang4.html#E5:31_2_1_1_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="finseq_1.html#D8">FINSEQ_1:def 8</a></i>;<br/><a NAME="E8:31_2_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K17">the_argument_of</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="qc_lang4.html#E3:31_2"><i><font color="Green">E28</font></i></a>, <a class="txt" href="qc_lang4.html#E1:31_2_1_1_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="qc_lang2.html#T65">QC_LANG2:65</a></i>;<br/><b>hence </b><a NAME="E9:31_2_1_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K1" target="_self">list_of_immediate_constituents</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K1" target="_self">list_of_immediate_constituents</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="qc_lang4.html#E2:31_2_1_1_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="qc_lang4.html#E3:31_2_1_1_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="qc_lang4.html#E5:31_2_1_1_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="qc_lang4.html#E6:31_2_1_1_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="qc_lang4.html#E7:31_2_1_1_1"><i><font color="Green">E36</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_1_1_2"><b>suppose </b></a><a NAME="E1:31_2_1_1_2"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="qc_lang1.html#V4">conjunctive</a>
 ;<br/><div class="add"><a NAME="E2:31_2_1_1_2"/><b>then </b><i><font color="Green">E32</font></i>: 
 <a href="qc_lang4.html#K1" target="_self">list_of_immediate_constituents</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K18">the_left_argument_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="qc_lang1.html#K19">the_right_argument_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="qc_lang4.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E3:31_2_1_1_2"/><i><font color="Green">E33</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K18">the_left_argument_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="qc_lang1.html#K19">the_right_argument_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="hidden.html#R1">=</a> 2 &amp; <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K18">the_left_argument_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="qc_lang1.html#K19">the_right_argument_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K18">the_left_argument_of</a> <font color="Maroon">c<sub>1</sub></font> &amp; <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K18">the_left_argument_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="qc_lang1.html#K19">the_right_argument_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 2 <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K19">the_right_argument_of</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/><a NAME="E4:31_2_1_1_2"/><b>then </b><i><font color="Green">E34</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K18">the_left_argument_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="qc_lang1.html#K19">the_right_argument_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> 2
 <b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:31_2_1_1_2_1_1"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K18">the_left_argument_of</a> <font color="Maroon">c<sub>1</sub></font> or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K19">the_right_argument_of</a> <font color="Maroon">c<sub>1</sub></font> )
 </span><b>by </b><i><a class="txt" href="qc_lang4.html#E3:31_2"><i><font color="Green">E28</font></i></a>, <a class="txt" href="qc_lang4.html#E1:31_2_1_1_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="qc_lang2.html#T66">QC_LANG2:66</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:31_2_1_1_2_1_1_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K18">the_left_argument_of</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><a NAME="E2:31_2_1_1_2_1_1_1"/>
1 <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default">1,2</span><a href="tarski.html#K2">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E3:31_2_1_1_2_1_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K1" target="_self">list_of_immediate_constituents</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K1" target="_self">list_of_immediate_constituents</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="qc_lang4.html#E2:31_2_1_1_2"><i><font color="Green">E32</font></i></a>, <a class="txt" href="qc_lang4.html#E3:31_2_1_1_2"><i><font color="Green">E33</font></i></a>, <a class="txt" href="qc_lang4.html#E4:31_2_1_1_2"><i><font color="Green">E34</font></i></a>, <a class="txt" href="qc_lang4.html#E1:31_2_1_1_2_1_1_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:31_2_1_1_2_1_1_2"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K19">the_right_argument_of</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:31_2_1_1_2_1_1_2"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 2
 ;<br/><a NAME="E4:31_2_1_1_2_1_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K1" target="_self">list_of_immediate_constituents</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="qc_lang4.html#E2:31_2_1_1_2"><i><font color="Green">E32</font></i></a>, <a class="txt" href="qc_lang4.html#E4:31_2_1_1_2"><i><font color="Green">E34</font></i></a>, <a class="txt" href="qc_lang4.html#E3:31_2_1_1_2_1_1_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="finseq_1.html#T5">FINSEQ_1:5</a></i>;<br/><b>hence </b><a NAME="E5:31_2_1_1_2_1_1_2"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K1" target="_self">list_of_immediate_constituents</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K1" target="_self">list_of_immediate_constituents</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="qc_lang4.html#E2:31_2_1_1_2"><i><font color="Green">E32</font></i></a>, <a class="txt" href="qc_lang4.html#E3:31_2_1_1_2"><i><font color="Green">E33</font></i></a>, <a class="txt" href="qc_lang4.html#E1:31_2_1_1_2_1_1_2"><i><font color="Green">E35</font></i></a>, <a class="txt" href="qc_lang4.html#E3:31_2_1_1_2_1_1_2"><i><font color="Green">E36</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:31_2_1_1_2"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K1" target="_self">list_of_immediate_constituents</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K1" target="_self">list_of_immediate_constituents</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="31_2_1_1_3"><b>suppose </b></a><a NAME="E1:31_2_1_1_3"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="qc_lang1.html#V5">universal</a>
 ;<br/><div class="add"><a NAME="E2:31_2_1_1_3"/><b>then </b>
( not <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K11">VERUM</a>  &amp; not <font color="Maroon">c<sub>1</sub></font> is <a href="qc_lang1.html#V2">atomic</a> &amp; not <font color="Maroon">c<sub>1</sub></font> is <a href="qc_lang1.html#V3">negative</a> &amp; not <font color="Maroon">c<sub>1</sub></font> is <a href="qc_lang1.html#V4">conjunctive</a> )
 <b>by </b><i><a class="ref" href="qc_lang1.html#T51">QC_LANG1:51</a></i>;<br/><a NAME="E3:31_2_1_1_3"/><b>then </b><i><font color="Green">E32</font></i>: 
 <a href="qc_lang4.html#K1" target="_self">list_of_immediate_constituents</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K21">the_scope_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="qc_lang4.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E4:31_2_1_1_3"/><i><font color="Green">E33</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K21">the_scope_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#D8">FINSEQ_1:def 8</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E6:31_2_1_1_3"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><a NAME="E7:31_2_1_1_3"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="qc_lang4.html#E6:31_2_1_1_3"><i><font color="Green">E34</font></i></a>, <a class="ref" href="finseq_1.html#T5">FINSEQ_1:5</a></i>;<br/><a NAME="E8:31_2_1_1_3"/><i><font color="Green">E36</font></i>: 
<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K21">the_scope_of</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K21">the_scope_of</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="qc_lang4.html#E6:31_2_1_1_3"><i><font color="Green">E34</font></i></a>, <a class="ref" href="finseq_1.html#D8">FINSEQ_1:def 8</a></i>;<br/><a NAME="E9:31_2_1_1_3"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K21">the_scope_of</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="qc_lang4.html#E3:31_2"><i><font color="Green">E28</font></i></a>, <a class="txt" href="qc_lang4.html#E1:31_2_1_1_3"><i><font color="Green">E31</font></i></a>, <a class="ref" href="qc_lang2.html#T67">QC_LANG2:67</a></i>;<br/><b>hence </b><a NAME="E10:31_2_1_1_3"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K1" target="_self">list_of_immediate_constituents</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K1" target="_self">list_of_immediate_constituents</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="qc_lang4.html#E3:31_2_1_1_3"><i><font color="Green">E32</font></i></a>, <a class="txt" href="qc_lang4.html#E4:31_2_1_1_3"><i><font color="Green">E33</font></i></a>, <a class="txt" href="qc_lang4.html#E6:31_2_1_1_3"><i><font color="Green">E34</font></i></a>, <a class="txt" href="qc_lang4.html#E7:31_2_1_1_3"><i><font color="Green">E35</font></i></a>, <a class="txt" href="qc_lang4.html#E8:31_2_1_1_3"><i><font color="Green">E36</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E5:31_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K1" target="_self">list_of_immediate_constituents</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="qc_lang4.html#E3:31_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


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


</div>
