<?xml version="1.0"?>
<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="E1:31"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K1" title="QC_LANG4:func.1">list_of_immediate_constituents</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">G</font> where <font color="Olive" title="b1">G</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>  : <font color="Olive" title="b1">G</font> <a href="qc_lang2.html#R1" title="QC_LANG2:pred.1">is_immediate_constituent_of</a> <font color="Maroon" title="c1">F</font> </span>}</span> 
  <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="31_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="E1:31_1"/><i><font color="Green" title="E28">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#K1" title="QC_LANG4:func.1">list_of_immediate_constituents</a> <font color="Maroon" title="c1">F</font></span>)</span>
 ;<br/>

<a NAME="E2:31_1"/><b>then </b><i><font color="Green" title="E29">A2</font></i>: 
 ex <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K1" title="QC_LANG4:func.1">list_of_immediate_constituents</a> <font color="Maroon" title="c1">F</font></span>)</span> &amp; <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K1" title="QC_LANG4:func.1">list_of_immediate_constituents</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">x</font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E3:31_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K1" title="QC_LANG4:func.1">list_of_immediate_constituents</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="qc_lang1.html#K8" title="QC_LANG1:func.8">QC-WFF</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c3">G</font> = <font color="Maroon" title="c2">y</font> as    <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="txt" href="qc_lang4.html#E1:31_1"><i><font color="Green" title="E28">A1</font></i></a></i>;<br/>
<a NAME="E5:31_1"/>
<font color="Maroon" title="c3">G</font> <a href="qc_lang2.html#R1" title="QC_LANG2:pred.1">is_immediate_constituent_of</a> <font color="Maroon" title="c1">F</font>
 
<b>by </b><i><a class="txt" href="qc_lang4.html#E2:31_1"><i><font color="Green" title="E29">A2</font></i></a>, <a class="ref" href="qc_lang4.html#T30" target="_self" title="QC_LANG4:th.30">Th30</a></i>;<br/>
<b>hence </b><a NAME="E6:31_1"/>
<font color="Maroon" title="c2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">G1</font> where <font color="Olive" title="b1">G1</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>  : <font color="Olive" title="b1">G1</font> <a href="qc_lang2.html#R1" title="QC_LANG2:pred.1">is_immediate_constituent_of</a> <font color="Maroon" title="c1">F</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" title="b1">G</font> where <font color="Olive" title="b1">G</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>  : <font color="Olive" title="b1">G</font> <a href="qc_lang2.html#R1" title="QC_LANG2:pred.1">is_immediate_constituent_of</a> <font color="Maroon" title="c1">F</font> </span>}</span>  <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#K1" title="QC_LANG4:func.1">list_of_immediate_constituents</a> <font color="Maroon" title="c1">F</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" title="c2">x</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="E1:31_2"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">G</font> where <font color="Olive" title="b1">G</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>  : <font color="Olive" title="b1">G</font> <a href="qc_lang2.html#R1" title="QC_LANG2:pred.1">is_immediate_constituent_of</a> <font color="Maroon" title="c1">F</font> </span>}</span> 
 ;<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="E3:31_2"/><i><font color="Green" title="E28">A3</font></i>: 
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">G</font> &amp; <font color="Maroon" title="c3">G</font> <a href="qc_lang2.html#R1" title="QC_LANG2:pred.1">is_immediate_constituent_of</a> <font color="Maroon" title="c1">F</font> )
 ;<br/>
<a NAME="E4:31_2"/>
 ex <font color="Olive" title="b1">n</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>  st <br/>( <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K1" title="QC_LANG4:func.1">list_of_immediate_constituents</a> <font color="Maroon" title="c1">F</font></span>)</span> &amp; <font color="Maroon" title="c3">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K1" title="QC_LANG4:func.1">list_of_immediate_constituents</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</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" title="E29">A4</font></i>: 
<font color="Maroon" title="c1">F</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> 
 
<b>by </b><i><a class="txt" href="qc_lang4.html#E3:31_2"><i><font color="Green" title="E28">A3</font></i></a>, <a class="ref" href="qc_lang2.html#T58" title="QC_LANG2:th.58">QC_LANG2:58</a></i>;<br/>
<a NAME="E2:31_2_1"/><i><font color="Green" title="E30">A5</font></i>: 
not <font color="Maroon" title="c1">F</font> is <a href="qc_lang1.html#V2" title="QC_LANG1:attr.2">atomic</a>
 
<b>by </b><i><a class="txt" href="qc_lang4.html#E3:31_2"><i><font color="Green" title="E28">A3</font></i></a>, <a class="ref" href="qc_lang2.html#T64" title="QC_LANG2:th.64">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" title="c1">F</font> is <a href="qc_lang1.html#V3" title="QC_LANG1:attr.3">negative</a> or <font color="Maroon" title="c1">F</font> is <a href="qc_lang1.html#V4" title="QC_LANG1:attr.4">conjunctive</a> or <font color="Maroon" title="c1">F</font> is <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a> )
 </span><b>by </b><i><a class="txt" href="qc_lang4.html#E1:31_2_1"><i><font color="Green" title="E29">A4</font></i></a>, <a class="txt" href="qc_lang4.html#E2:31_2_1"><i><font color="Green" title="E30">A5</font></i></a>, <a class="ref" href="qc_lang1.html#T33" title="QC_LANG1:th.33">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" title="E31">A6</font></i>: 
<font color="Maroon" title="c1">F</font> is <a href="qc_lang1.html#V3" title="QC_LANG1:attr.3">negative</a>
 ;<br/><div class="add"><a NAME="E2:31_2_1_1_1"/><b>then </b><i><font color="Green" title="E32">A7</font></i>: 
 <a href="qc_lang4.html#K1" title="QC_LANG4:func.1">list_of_immediate_constituents</a> <font color="Maroon" title="c1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><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">F</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="qc_lang4.html#D1" target="_self" title="QC_LANG4:def.1">Def1</a></i>;<br/><a NAME="E3:31_2_1_1_1"/><i><font color="Green" title="E33">A8</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><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">F</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#D8" title="FINSEQ_1:def.8">FINSEQ_1:def 8</a></i>;<br/><b>consider </b><font color="Maroon" title="c4">n</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> <b> such that </b><br/><a NAME="E5:31_2_1_1_1"/><i><font color="Green" title="E34">A9</font></i>: 
<font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><a NAME="E6:31_2_1_1_1"/><i><font color="Green" title="E35">A10</font></i>: 
<font color="Maroon" title="c4">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c4">n</font>
 <b>by </b><i><a class="txt" href="qc_lang4.html#E5:31_2_1_1_1"><i><font color="Green" title="E34">A9</font></i></a>, <a class="ref" href="finseq_1.html#T5" title="FINSEQ_1:th.5">FINSEQ_1:5</a></i>;<br/><a NAME="E7:31_2_1_1_1"/><i><font color="Green" title="E36">A11</font></i>: 
<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><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">F</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K17" title="QC_LANG1:func.17">the_argument_of</a> <font color="Maroon" title="c1">F</font>
 <b>by </b><i><a class="txt" href="qc_lang4.html#E5:31_2_1_1_1"><i><font color="Green" title="E34">A9</font></i></a>, <a class="ref" href="finseq_1.html#D8" title="FINSEQ_1:def.8">FINSEQ_1:def 8</a></i>;<br/><a NAME="E8:31_2_1_1_1"/>
<font color="Maroon" title="c3">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K17" title="QC_LANG1:func.17">the_argument_of</a> <font color="Maroon" title="c1">F</font>
 <b>by </b><i><a class="txt" href="qc_lang4.html#E3:31_2"><i><font color="Green" title="E28">A3</font></i></a>, <a class="txt" href="qc_lang4.html#E1:31_2_1_1_1"><i><font color="Green" title="E31">A6</font></i></a>, <a class="ref" href="qc_lang2.html#T65" title="QC_LANG2:th.65">QC_LANG2:65</a></i>;<br/><b>hence </b><a NAME="E9:31_2_1_1_1"/>
 ex <font color="Olive" title="b1">n</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>  st <br/>( <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K1" title="QC_LANG4:func.1">list_of_immediate_constituents</a> <font color="Maroon" title="c1">F</font></span>)</span> &amp; <font color="Maroon" title="c3">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K1" title="QC_LANG4:func.1">list_of_immediate_constituents</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> )
 <b>by </b><i><a class="txt" href="qc_lang4.html#E2:31_2_1_1_1"><i><font color="Green" title="E32">A7</font></i></a>, <a class="txt" href="qc_lang4.html#E3:31_2_1_1_1"><i><font color="Green" title="E33">A8</font></i></a>, <a class="txt" href="qc_lang4.html#E5:31_2_1_1_1"><i><font color="Green" title="E34">A9</font></i></a>, <a class="txt" href="qc_lang4.html#E6:31_2_1_1_1"><i><font color="Green" title="E35">A10</font></i></a>, <a class="txt" href="qc_lang4.html#E7:31_2_1_1_1"><i><font color="Green" title="E36">A11</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" title="E31">A12</font></i>: 
<font color="Maroon" title="c1">F</font> is <a href="qc_lang1.html#V4" title="QC_LANG1:attr.4">conjunctive</a>
 ;<br/><div class="add"><a NAME="E2:31_2_1_1_2"/><b>then </b><i><font color="Green" title="E32">A13</font></i>: 
 <a href="qc_lang4.html#K1" title="QC_LANG4:func.1">list_of_immediate_constituents</a> <font color="Maroon" title="c1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><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">F</font></span>)</span>,<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">F</font></span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="qc_lang4.html#D1" target="_self" title="QC_LANG4:def.1">Def1</a></i>;<br/><a NAME="E3:31_2_1_1_2"/><i><font color="Green" title="E33">A14</font></i>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><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">F</font></span>)</span>,<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">F</font></span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><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">F</font></span>)</span>,<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">F</font></span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_left_argument_of</a> <font color="Maroon" title="c1">F</font> &amp; <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><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">F</font></span>)</span>,<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">F</font></span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_right_argument_of</a> <font color="Maroon" title="c1">F</font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T61" title="FINSEQ_1:th.61">FINSEQ_1:61</a></i>;<br/><a NAME="E4:31_2_1_1_2"/><b>then </b><i><font color="Green" title="E34">A15</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><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">F</font></span>)</span>,<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">F</font></span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 2
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">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" title="c3">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_left_argument_of</a> <font color="Maroon" title="c1">F</font> or <font color="Maroon" title="c3">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_right_argument_of</a> <font color="Maroon" title="c1">F</font> )
 </span><b>by </b><i><a class="txt" href="qc_lang4.html#E3:31_2"><i><font color="Green" title="E28">A3</font></i></a>, <a class="txt" href="qc_lang4.html#E1:31_2_1_1_2"><i><font color="Green" title="E31">A12</font></i></a>, <a class="ref" href="qc_lang2.html#T66" title="QC_LANG2:th.66">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" title="E35">A16</font></i>: 
<font color="Maroon" title="c3">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_left_argument_of</a> <font color="Maroon" title="c1">F</font>
 ;<br/><div class="add"><a NAME="E2:31_2_1_1_2_1_1_1"/>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default">1,2</span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/><b>hence </b><a NAME="E3:31_2_1_1_2_1_1_1"/>
 ex <font color="Olive" title="b1">n</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>  st <br/>( <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K1" title="QC_LANG4:func.1">list_of_immediate_constituents</a> <font color="Maroon" title="c1">F</font></span>)</span> &amp; <font color="Maroon" title="c3">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K1" title="QC_LANG4:func.1">list_of_immediate_constituents</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> )
 <b>by </b><i><a class="txt" href="qc_lang4.html#E2:31_2_1_1_2"><i><font color="Green" title="E32">A13</font></i></a>, <a class="txt" href="qc_lang4.html#E3:31_2_1_1_2"><i><font color="Green" title="E33">A14</font></i></a>, <a class="txt" href="qc_lang4.html#E4:31_2_1_1_2"><i><font color="Green" title="E34">A15</font></i></a>, <a class="txt" href="qc_lang4.html#E1:31_2_1_1_2_1_1_1"><i><font color="Green" title="E35">A16</font></i></a>, <a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">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" title="E35">A17</font></i>: 
<font color="Maroon" title="c3">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_right_argument_of</a> <font color="Maroon" title="c1">F</font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c4">n</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> <b> such that </b><br/><a NAME="E3:31_2_1_1_2_1_1_2"/><i><font color="Green" title="E36">A18</font></i>: 
<font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ;<br/><a NAME="E4:31_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c4">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K1" title="QC_LANG4:func.1">list_of_immediate_constituents</a> <font color="Maroon" title="c1">F</font></span>)</span>
 <b>by </b><i><a class="txt" href="qc_lang4.html#E2:31_2_1_1_2"><i><font color="Green" title="E32">A13</font></i></a>, <a class="txt" href="qc_lang4.html#E4:31_2_1_1_2"><i><font color="Green" title="E34">A15</font></i></a>, <a class="txt" href="qc_lang4.html#E3:31_2_1_1_2_1_1_2"><i><font color="Green" title="E36">A18</font></i></a>, <a class="ref" href="finseq_1.html#T5" title="FINSEQ_1:th.5">FINSEQ_1:5</a></i>;<br/><b>hence </b><a NAME="E5:31_2_1_1_2_1_1_2"/>
 ex <font color="Olive" title="b1">n</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>  st <br/>( <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K1" title="QC_LANG4:func.1">list_of_immediate_constituents</a> <font color="Maroon" title="c1">F</font></span>)</span> &amp; <font color="Maroon" title="c3">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K1" title="QC_LANG4:func.1">list_of_immediate_constituents</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> )
 <b>by </b><i><a class="txt" href="qc_lang4.html#E2:31_2_1_1_2"><i><font color="Green" title="E32">A13</font></i></a>, <a class="txt" href="qc_lang4.html#E3:31_2_1_1_2"><i><font color="Green" title="E33">A14</font></i></a>, <a class="txt" href="qc_lang4.html#E1:31_2_1_1_2_1_1_2"><i><font color="Green" title="E35">A17</font></i></a>, <a class="txt" href="qc_lang4.html#E3:31_2_1_1_2_1_1_2"><i><font color="Green" title="E36">A18</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" title="b1">n</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>  st <br/>( <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K1" title="QC_LANG4:func.1">list_of_immediate_constituents</a> <font color="Maroon" title="c1">F</font></span>)</span> &amp; <font color="Maroon" title="c3">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K1" title="QC_LANG4:func.1">list_of_immediate_constituents</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</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" title="E31">A19</font></i>: 
<font color="Maroon" title="c1">F</font> is <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a>
 ;<br/><div class="add"><a NAME="E2:31_2_1_1_3"/><b>then </b>
( not <font color="Maroon" title="c1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K11" title="QC_LANG1:func.11">VERUM</a>  &amp; not <font color="Maroon" title="c1">F</font> is <a href="qc_lang1.html#V2" title="QC_LANG1:attr.2">atomic</a> &amp; not <font color="Maroon" title="c1">F</font> is <a href="qc_lang1.html#V3" title="QC_LANG1:attr.3">negative</a> &amp; not <font color="Maroon" title="c1">F</font> is <a href="qc_lang1.html#V4" title="QC_LANG1:attr.4">conjunctive</a> )
 <b>by </b><i><a class="ref" href="qc_lang1.html#T51" title="QC_LANG1:th.51">QC_LANG1:51</a></i>;<br/><a NAME="E3:31_2_1_1_3"/><b>then </b><i><font color="Green" title="E32">A20</font></i>: 
 <a href="qc_lang4.html#K1" title="QC_LANG4:func.1">list_of_immediate_constituents</a> <font color="Maroon" title="c1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><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">F</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="qc_lang4.html#D1" target="_self" title="QC_LANG4:def.1">Def1</a></i>;<br/><a NAME="E4:31_2_1_1_3"/><i><font color="Green" title="E33">A21</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><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">F</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#D8" title="FINSEQ_1:def.8">FINSEQ_1:def 8</a></i>;<br/><b>consider </b><font color="Maroon" title="c4">n</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> <b> such that </b><br/><a NAME="E6:31_2_1_1_3"/><i><font color="Green" title="E34">A22</font></i>: 
<font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><a NAME="E7:31_2_1_1_3"/><i><font color="Green" title="E35">A23</font></i>: 
<font color="Maroon" title="c4">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c4">n</font>
 <b>by </b><i><a class="txt" href="qc_lang4.html#E6:31_2_1_1_3"><i><font color="Green" title="E34">A22</font></i></a>, <a class="ref" href="finseq_1.html#T5" title="FINSEQ_1:th.5">FINSEQ_1:5</a></i>;<br/><a NAME="E8:31_2_1_1_3"/><i><font color="Green" title="E36">A24</font></i>: 
<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><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">F</font></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K21" title="QC_LANG1:func.21">the_scope_of</a> <font color="Maroon" title="c1">F</font>
 <b>by </b><i><a class="txt" href="qc_lang4.html#E6:31_2_1_1_3"><i><font color="Green" title="E34">A22</font></i></a>, <a class="ref" href="finseq_1.html#D8" title="FINSEQ_1:def.8">FINSEQ_1:def 8</a></i>;<br/><a NAME="E9:31_2_1_1_3"/>
<font color="Maroon" title="c3">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K21" title="QC_LANG1:func.21">the_scope_of</a> <font color="Maroon" title="c1">F</font>
 <b>by </b><i><a class="txt" href="qc_lang4.html#E3:31_2"><i><font color="Green" title="E28">A3</font></i></a>, <a class="txt" href="qc_lang4.html#E1:31_2_1_1_3"><i><font color="Green" title="E31">A19</font></i></a>, <a class="ref" href="qc_lang2.html#T67" title="QC_LANG2:th.67">QC_LANG2:67</a></i>;<br/><b>hence </b><a NAME="E10:31_2_1_1_3"/>
 ex <font color="Olive" title="b1">n</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>  st <br/>( <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K1" title="QC_LANG4:func.1">list_of_immediate_constituents</a> <font color="Maroon" title="c1">F</font></span>)</span> &amp; <font color="Maroon" title="c3">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="qc_lang4.html#K1" title="QC_LANG4:func.1">list_of_immediate_constituents</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> )
 <b>by </b><i><a class="txt" href="qc_lang4.html#E3:31_2_1_1_3"><i><font color="Green" title="E32">A20</font></i></a>, <a class="txt" href="qc_lang4.html#E4:31_2_1_1_3"><i><font color="Green" title="E33">A21</font></i></a>, <a class="txt" href="qc_lang4.html#E6:31_2_1_1_3"><i><font color="Green" title="E34">A22</font></i></a>, <a class="txt" href="qc_lang4.html#E7:31_2_1_1_3"><i><font color="Green" title="E35">A23</font></i></a>, <a class="txt" href="qc_lang4.html#E8:31_2_1_1_3"><i><font color="Green" title="E36">A24</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" title="c2">x</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#K1" title="QC_LANG4:func.1">list_of_immediate_constituents</a> <font color="Maroon" title="c1">F</font></span>)</span>
 <b>by </b><i><a class="txt" href="qc_lang4.html#E3:31_2"><i><font color="Green" title="E28">A3</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>


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


</div>
