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

<span class="kw">let </span><font color="Maroon" title="c1">k</font> be    <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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">l</font> being   <a href="cqc_lang.html#NM2" title="CQC_LANG:NM.2">CQC-variable_list</a> of <font color="Maroon" title="c1">k</font><br/>  for <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="Maroon" title="c1">k</font> holds   <a href="cqc_sim1.html#K14" title="CQC_SIM1:func.14">SepQuadruples</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">P</font> <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">!</a> <font color="Olive" title="b1">l</font></span>)</span> <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"><span class="p2"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b2">P</font> <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">!</a> <font color="Olive" title="b1">l</font></span>)</span>,<span class="p3">(<span class="default"><a href="cqc_sim1.html#K11" title="CQC_SIM1:func.11">index</a> <span class="p4">(<span class="default"><font color="Olive" title="b2">P</font> <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">!</a> <font color="Olive" title="b1">l</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="setwiseo.html#K1" title="SETWISEO:func.1">{}.</a> <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a></span>)</span>,<span class="p3">(<span class="default"><a href="cqc_sim1.html#K12" title="CQC_SIM1:func.12">id</a> <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a></span>)</span></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">l</font> be   <a href="cqc_lang.html#NM2" title="CQC_LANG:NM.2">CQC-variable_list</a> of <font color="Maroon" title="c1">k</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">P</font> being   <a href="qc_lang1.html#NM6" title="QC_LANG1:NM.6">QC-pred_symbol</a> of <font color="Maroon" title="c1">k</font> holds   <a href="cqc_sim1.html#K14" title="CQC_SIM1:func.14">SepQuadruples</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">P</font> <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">!</a> <font color="Maroon" title="c2">l</font></span>)</span> <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"><span class="p2"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">P</font> <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">!</a> <font color="Maroon" title="c2">l</font></span>)</span>,<span class="p3">(<span class="default"><a href="cqc_sim1.html#K11" title="CQC_SIM1:func.11">index</a> <span class="p4">(<span class="default"><font color="Olive" title="b1">P</font> <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">!</a> <font color="Maroon" title="c2">l</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="setwiseo.html#K1" title="SETWISEO:func.1">{}.</a> <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a></span>)</span>,<span class="p3">(<span class="default"><a href="cqc_sim1.html#K12" title="CQC_SIM1:func.12">id</a> <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a></span>)</span></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">P</font> be   <a href="qc_lang1.html#NM6" title="QC_LANG1:NM.6">QC-pred_symbol</a> of <font color="Maroon" title="c1">k</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="cqc_sim1.html#K14" title="CQC_SIM1:func.14">SepQuadruples</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">!</a> <font color="Maroon" title="c2">l</font></span>)</span> <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"><span class="p2"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">!</a> <font color="Maroon" title="c2">l</font></span>)</span>,<span class="p3">(<span class="default"><a href="cqc_sim1.html#K11" title="CQC_SIM1:func.11">index</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">!</a> <font color="Maroon" title="c2">l</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="setwiseo.html#K1" title="SETWISEO:func.1">{}.</a> <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a></span>)</span>,<span class="p3">(<span class="default"><a href="cqc_sim1.html#K12" title="CQC_SIM1:func.12">id</a> <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a></span>)</span></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/>





<a NAME="E1:60"/><span class="lab"><font color="Green" title="E34">A1</font></span>: 
<font color="Maroon" title="c3">P</font> <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">!</a> <font color="Maroon" title="c2">l</font> is  <a href="qc_lang1.html#V2" title="QC_LANG1:attr.2">atomic</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="qc_lang1.html#D16" title="QC_LANG1:def.16">QC_LANG1:def 16</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cqc_sim1.html#K14" title="CQC_SIM1:func.14">SepQuadruples</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">!</a> <font color="Maroon" title="c2">l</font></span>)</span> implies <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">!</a> <font color="Maroon" title="c2">l</font></span>)</span>,<span class="p2">(<span class="default"><a href="cqc_sim1.html#K11" title="CQC_SIM1:func.11">index</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">!</a> <font color="Maroon" title="c2">l</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="setwiseo.html#K1" title="SETWISEO:func.1">{}.</a> <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a></span>)</span>,<span class="p2">(<span class="default"><a href="cqc_sim1.html#K12" title="CQC_SIM1:func.12">id</a> <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a></span>)</span></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> ) &amp; ( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">!</a> <font color="Maroon" title="c2">l</font></span>)</span>,<span class="p2">(<span class="default"><a href="cqc_sim1.html#K11" title="CQC_SIM1:func.11">index</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">!</a> <font color="Maroon" title="c2">l</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="setwiseo.html#K1" title="SETWISEO:func.1">{}.</a> <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a></span>)</span>,<span class="p2">(<span class="default"><a href="cqc_sim1.html#K12" title="CQC_SIM1:func.12">id</a> <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a></span>)</span></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> implies <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cqc_sim1.html#K14" title="CQC_SIM1:func.14">SepQuadruples</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">!</a> <font color="Maroon" title="c2">l</font></span>)</span> ) )</span><br/><span class="kw">thus </span><a NAME="E1:60_1"/>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cqc_sim1.html#K14" title="CQC_SIM1:func.14">SepQuadruples</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">!</a> <font color="Maroon" title="c2">l</font></span>)</span> implies <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">!</a> <font color="Maroon" title="c2">l</font></span>)</span>,<span class="p2">(<span class="default"><a href="cqc_sim1.html#K11" title="CQC_SIM1:func.11">index</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">!</a> <font color="Maroon" title="c2">l</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="setwiseo.html#K1" title="SETWISEO:func.1">{}.</a> <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a></span>)</span>,<span class="p2">(<span class="default"><a href="cqc_sim1.html#K12" title="CQC_SIM1:func.12">id</a> <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a></span>)</span></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">!</a> <font color="Maroon" title="c2">l</font></span>)</span>,<span class="p2">(<span class="default"><a href="cqc_sim1.html#K11" title="CQC_SIM1:func.11">index</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">!</a> <font color="Maroon" title="c2">l</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="setwiseo.html#K1" title="SETWISEO:func.1">{}.</a> <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a></span>)</span>,<span class="p2">(<span class="default"><a href="cqc_sim1.html#K12" title="CQC_SIM1:func.12">id</a> <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a></span>)</span></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> implies <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cqc_sim1.html#K14" title="CQC_SIM1:func.14">SepQuadruples</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">!</a> <font color="Maroon" title="c2">l</font></span>)</span> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:60_1_1"/><span class="lab"><font color="Green" title="E35">A2</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cqc_sim1.html#K14" title="CQC_SIM1:func.14">SepQuadruples</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">!</a> <font color="Maroon" title="c2">l</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">!</a> <font color="Maroon" title="c2">l</font></span>)</span>,<span class="p2">(<span class="default"><a href="cqc_sim1.html#K11" title="CQC_SIM1:func.11">index</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">!</a> <font color="Maroon" title="c2">l</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="setwiseo.html#K1" title="SETWISEO:func.1">{}.</a> <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a></span>)</span>,<span class="p2">(<span class="default"><a href="cqc_sim1.html#K12" title="CQC_SIM1:func.12">id</a> <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a></span>)</span></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">q</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">CQC-WFF</a> , <font color="Maroon" title="c6">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> , <font color="Maroon" title="c7">K</font> being   <a href="finsub_1.html#NM1" title="FINSUB_1:NM.1">Finite_Subset</a> of <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>, <font color="Maroon" title="c8">f</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>,<a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>)<span class="kw"> such that </span><br/><a NAME="E3:60_1_1"/><span class="lab"><font color="Green" title="E36">A3</font></span>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><font color="Maroon" title="c5">q</font>,<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">K</font>,<font color="Maroon" title="c8">f</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="domain_1.html#T10" title="DOMAIN_1:th.10">DOMAIN_1:10</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E37">A4</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">given </span><font color="Maroon" title="c9">x</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> , <font color="Maroon" title="c10">i</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> , <font color="Maroon" title="c11">h</font> being    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>,<a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>)<span class="kw"> such that </span><a NAME="E2:60_1_1_1"/>
<font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">k</font>
 <span class="kw">and </span><br/><a NAME="E3:60_1_1_1"/>
<font color="Maroon" title="c11">h</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c9">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p2">(<span class="default"><a href="qc_lang3.html#K2" title="QC_LANG3:func.2">x.</a> <font color="Maroon" title="c10">i</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f</font>
 <span class="kw">and </span><br/><a NAME="E4:60_1_1_1"/><span class="lab"><font color="Green" title="E38">A5</font></span>: 
( <span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K11" title="CQC_LANG:func.11">All</a> (<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c5">q</font>)</span>)</span>,<font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c7">K</font>,<font color="Maroon" title="c11">h</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cqc_sim1.html#K14" title="CQC_SIM1:func.14">SepQuadruples</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">!</a> <font color="Maroon" title="c2">l</font></span>)</span> or <span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K11" title="CQC_LANG:func.11">All</a> (<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c5">q</font>)</span>)</span>,<font color="Maroon" title="c10">i</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">K</font> <a href="setwiseo.html#K6" title="SETWISEO:func.6">\</a> <span class="p3"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><font color="Maroon" title="c9">x</font></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span></span>)</span>,<font color="Maroon" title="c11">h</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cqc_sim1.html#K14" title="CQC_SIM1:func.14">SepQuadruples</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">!</a> <font color="Maroon" title="c2">l</font></span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E5:60_1_1_1"/>
 <a href="cqc_lang.html#K11" title="CQC_LANG:func.11">All</a> (<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c5">q</font>) <a href="qc_lang2.html#R2" title="QC_LANG2:pred.2">is_subformula_of</a> <font color="Maroon" title="c3">P</font> <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">!</a> <font color="Maroon" title="c2">l</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:60_1_1_1"><span class="lab"><font color="Green" title="E38">A5</font></span></a>, <a class="ref" href="cqc_sim1.html#T35" target="_self" title="CQC_SIM1:th.35">Th36</a></span>;<br/><a NAME="E6:60_1_1_1"/><span class="kw">then </span>
 <a href="cqc_lang.html#K11" title="CQC_LANG:func.11">All</a> (<font color="Maroon" title="c9">x</font>,<font color="Maroon" title="c5">q</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">P</font> <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">!</a> <font color="Maroon" title="c2">l</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="qc_lang2.html#T80" title="QC_LANG2:th.80">QC_LANG2:80</a></span>;<br/><a NAME="E7:60_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">P</font> <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">!</a> <font color="Maroon" title="c2">l</font> is  <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="qc_lang1.html#D19" title="QC_LANG1:def.19">QC_LANG1:def 19</a></span>;<br/><span class="kw">hence </span><a NAME="E8:60_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="ref" href="qc_lang1.html#T18" title="QC_LANG1:th.18">QC_LANG1:18</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E38">A6</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">given </span><font color="Maroon" title="c9">r</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">CQC-WFF</a> , <font color="Maroon" title="c10">i</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> <span class="kw"> such that </span><a NAME="E2:60_1_1_2"/>
<font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="cqc_sim1.html#K6" title="CQC_SIM1:func.6">QuantNbr</a> <font color="Maroon" title="c9">r</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:60_1_1_2"/><span class="lab"><font color="Green" title="E39">A7</font></span>: 
<span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">r</font> <a href="cqc_lang.html#K7" title="CQC_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c5">q</font></span>)</span>,<font color="Maroon" title="c10">i</font>,<font color="Maroon" title="c7">K</font>,<font color="Maroon" title="c8">f</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cqc_sim1.html#K14" title="CQC_SIM1:func.14">SepQuadruples</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">!</a> <font color="Maroon" title="c2">l</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E4:60_1_1_2"/>
<font color="Maroon" title="c9">r</font> <a href="cqc_lang.html#K7" title="CQC_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c5">q</font> <a href="qc_lang2.html#R2" title="QC_LANG2:pred.2">is_subformula_of</a> <font color="Maroon" title="c3">P</font> <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">!</a> <font color="Maroon" title="c2">l</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:60_1_1_2"><span class="lab"><font color="Green" title="E39">A7</font></span></a>, <a class="ref" href="cqc_sim1.html#T35" target="_self" title="CQC_SIM1:th.35">Th36</a></span>;<br/><a NAME="E5:60_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">r</font> <a href="cqc_lang.html#K7" title="CQC_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c5">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">P</font> <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">!</a> <font color="Maroon" title="c2">l</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="qc_lang2.html#T80" title="QC_LANG2:th.80">QC_LANG2:80</a></span>;<br/><a NAME="E6:60_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">P</font> <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">!</a> <font color="Maroon" title="c2">l</font> is  <a href="qc_lang1.html#V4" title="QC_LANG1:attr.4">conjunctive</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="qc_lang1.html#D18" title="QC_LANG1:def.18">QC_LANG1:def 18</a></span>;<br/><span class="kw">hence </span><a NAME="E7:60_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="ref" href="qc_lang1.html#T18" title="QC_LANG1:th.18">QC_LANG1:18</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E39">A8</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1_1_3"><span class="kw">now </span></a><div class="add"><span class="kw">given </span><font color="Maroon" title="c9">r</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">CQC-WFF</a> <span class="kw"> such that </span><a NAME="E2:60_1_1_3"/><span class="lab"><font color="Green" title="E40">A9</font></span>: 
<span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">q</font> <a href="cqc_lang.html#K7" title="CQC_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c9">r</font></span>)</span>,<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">K</font>,<font color="Maroon" title="c8">f</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cqc_sim1.html#K14" title="CQC_SIM1:func.14">SepQuadruples</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">!</a> <font color="Maroon" title="c2">l</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E3:60_1_1_3"/>
<font color="Maroon" title="c5">q</font> <a href="cqc_lang.html#K7" title="CQC_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c9">r</font> <a href="qc_lang2.html#R2" title="QC_LANG2:pred.2">is_subformula_of</a> <font color="Maroon" title="c3">P</font> <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">!</a> <font color="Maroon" title="c2">l</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:60_1_1_3"><span class="lab"><font color="Green" title="E40">A9</font></span></a>, <a class="ref" href="cqc_sim1.html#T35" target="_self" title="CQC_SIM1:th.35">Th36</a></span>;<br/><a NAME="E4:60_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c5">q</font> <a href="cqc_lang.html#K7" title="CQC_LANG:func.7">'&amp;'</a> <font color="Maroon" title="c9">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">P</font> <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">!</a> <font color="Maroon" title="c2">l</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="qc_lang2.html#T80" title="QC_LANG2:th.80">QC_LANG2:80</a></span>;<br/><a NAME="E5:60_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c3">P</font> <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">!</a> <font color="Maroon" title="c2">l</font> is  <a href="qc_lang1.html#V4" title="QC_LANG1:attr.4">conjunctive</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="qc_lang1.html#D18" title="QC_LANG1:def.18">QC_LANG1:def 18</a></span>;<br/><span class="kw">hence </span><a NAME="E6:60_1_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="ref" href="qc_lang1.html#T18" title="QC_LANG1:th.18">QC_LANG1:18</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="60_1_1_4"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:60_1_1_4"/>
<span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K6" title="CQC_LANG:func.6">'not'</a> <font color="Maroon" title="c5">q</font></span>)</span>,<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">K</font>,<font color="Maroon" title="c8">f</font></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cqc_sim1.html#K14" title="CQC_SIM1:func.14">SepQuadruples</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">!</a> <font color="Maroon" title="c2">l</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:60_1_1_4"/><span class="kw">then </span>
 <a href="cqc_lang.html#K6" title="CQC_LANG:func.6">'not'</a> <font color="Maroon" title="c5">q</font> <a href="qc_lang2.html#R2" title="QC_LANG2:pred.2">is_subformula_of</a> <font color="Maroon" title="c3">P</font> <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">!</a> <font color="Maroon" title="c2">l</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cqc_sim1.html#T35" target="_self" title="CQC_SIM1:th.35">Th36</a></span>;<br/><a NAME="E3:60_1_1_4"/><span class="kw">then </span>
 <a href="cqc_lang.html#K6" title="CQC_LANG:func.6">'not'</a> <font color="Maroon" title="c5">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">P</font> <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">!</a> <font color="Maroon" title="c2">l</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="qc_lang2.html#T80" title="QC_LANG2:th.80">QC_LANG2:80</a></span>;<br/><a NAME="E4:60_1_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c3">P</font> <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">!</a> <font color="Maroon" title="c2">l</font> is  <a href="qc_lang1.html#V3" title="QC_LANG1:attr.3">negative</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="qc_lang1.html#D17" title="QC_LANG1:def.17">QC_LANG1:def 17</a></span>;<br/><span class="kw">hence </span><a NAME="E5:60_1_1_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="ref" href="qc_lang1.html#T18" title="QC_LANG1:th.18">QC_LANG1:18</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E8:60_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">!</a> <font color="Maroon" title="c2">l</font></span>)</span>,<span class="p2">(<span class="default"><a href="cqc_sim1.html#K11" title="CQC_SIM1:func.11">index</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">!</a> <font color="Maroon" title="c2">l</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="setwiseo.html#K1" title="SETWISEO:func.1">{}.</a> <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a></span>)</span>,<span class="p2">(<span class="default"><a href="cqc_sim1.html#K12" title="CQC_SIM1:func.12">id</a> <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a></span>)</span></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:60_1_1"><span class="lab"><font color="Green" title="E35">A2</font></span></a>, <a class="txt" href="#E3:60_1_1"><span class="lab"><font color="Green" title="E36">A3</font></span></a>, <a class="txt" href="#E6:60_1_1"><span class="lab"><font color="Green" title="E39">A8</font></span></a>, <a class="txt" href="#E5:60_1_1"><span class="lab"><font color="Green" title="E38">A6</font></span></a>, <a class="txt" href="#E4:60_1_1"><span class="lab"><font color="Green" title="E37">A4</font></span></a>, <a class="ref" href="cqc_sim1.html#T34" target="_self" title="CQC_SIM1:th.34">Th35</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">thus </span><a NAME="E2:60_1"/>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">!</a> <font color="Maroon" title="c2">l</font></span>)</span>,<span class="p2">(<span class="default"><a href="cqc_sim1.html#K11" title="CQC_SIM1:func.11">index</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">!</a> <font color="Maroon" title="c2">l</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="setwiseo.html#K1" title="SETWISEO:func.1">{}.</a> <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a></span>)</span>,<span class="p2">(<span class="default"><a href="cqc_sim1.html#K12" title="CQC_SIM1:func.12">id</a> <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a></span>)</span></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span> implies <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cqc_sim1.html#K14" title="CQC_SIM1:func.14">SepQuadruples</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">!</a> <font color="Maroon" title="c2">l</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cqc_sim1.html#T30" target="_self" title="CQC_SIM1:th.30">Th31</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:60"/>
 <a href="cqc_sim1.html#K14" title="CQC_SIM1:func.14">SepQuadruples</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">!</a> <font color="Maroon" title="c2">l</font></span>)</span> <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"><span class="p2"><a href="mcart_1.html#K4" title="MCART_1:func.4">[</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">!</a> <font color="Maroon" title="c2">l</font></span>)</span>,<span class="p3">(<span class="default"><a href="cqc_sim1.html#K11" title="CQC_SIM1:func.11">index</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">!</a> <font color="Maroon" title="c2">l</font></span>)</span></span>)</span>,<span class="p3">(<span class="default"><a href="setwiseo.html#K1" title="SETWISEO:func.1">{}.</a> <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a></span>)</span>,<span class="p3">(<span class="default"><a href="cqc_sim1.html#K12" title="CQC_SIM1:func.12">id</a> <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a></span>)</span></span><a href="mcart_1.html#K4" title="MCART_1:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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