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

<span class="kw">let </span><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">y</font> be   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">s9</font> being  <a href="qc_lang1.html#NM8" title="QC_LANG1:NM.8">QC-formula</a>  st <font color="Maroon" title="c1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">y</font> &amp; not <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <font color="Olive" title="b1">s9</font> holds <br/> not <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">s9</font> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c2">y</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">s9</font> be   <a href="qc_lang1.html#NM8" title="QC_LANG1:NM.8">QC-formula</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">y</font> &amp; not <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <font color="Maroon" title="c3">s9</font> implies  not <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s9</font> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c2">y</font></span>)</span> )</span><br/>





<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <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> ] <span class="kw">means </span>( <font color="Maroon" title="c1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">y</font> &amp; not <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> $1 implies  not <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <span class="p1">(<span class="default">$1 <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c2">y</font></span>)</span> );<br/>
<div><span class="lab"><font color="Green" title="E29">A1</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">s</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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c4">s</font> is  <a href="qc_lang1.html#V2" title="QC_LANG1:attr.2">atomic</a>  implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">s</font>] ) &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="cqc_lang.html#K6" title="CQC_LANG:func.6">VERUM</a> ] &amp; ( <font color="Maroon" title="c4">s</font> is  <a href="qc_lang1.html#V3" title="QC_LANG1:attr.3">negative</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K17" title="QC_LANG1:func.17">the_argument_of</a> <font color="Maroon" title="c4">s</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">s</font>] ) &amp; ( <font color="Maroon" title="c4">s</font> is  <a href="qc_lang1.html#V4" title="QC_LANG1:attr.4">conjunctive</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_left_argument_of</a> <font color="Maroon" title="c4">s</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_right_argument_of</a> <font color="Maroon" title="c4">s</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">s</font>] ) &amp; ( <font color="Maroon" title="c4">s</font> is  <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K21" title="QC_LANG1:func.21">the_scope_of</a> <font color="Maroon" title="c4">s</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">s</font>] ) )</span><br/><span class="kw">thus </span><a NAME="E1:54_1"/>
( <font color="Maroon" title="c4">s</font> is  <a href="qc_lang1.html#V2" title="QC_LANG1:attr.2">atomic</a>  implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">s</font>] )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[ <a href="cqc_lang.html#K6" title="CQC_LANG:func.6">VERUM</a> ] &amp; ( <font color="Maroon" title="c4">s</font> is  <a href="qc_lang1.html#V3" title="QC_LANG1:attr.3">negative</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K17" title="QC_LANG1:func.17">the_argument_of</a> <font color="Maroon" title="c4">s</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">s</font>] ) &amp; ( <font color="Maroon" title="c4">s</font> is  <a href="qc_lang1.html#V4" title="QC_LANG1:attr.4">conjunctive</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_left_argument_of</a> <font color="Maroon" title="c4">s</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_right_argument_of</a> <font color="Maroon" title="c4">s</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">s</font>] ) &amp; ( <font color="Maroon" title="c4">s</font> is  <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K21" title="QC_LANG1:func.21">the_scope_of</a> <font color="Maroon" title="c4">s</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">s</font>] ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:54_1_1"/><span class="lab"><font color="Green" title="E30">A2</font></span>: 
<font color="Maroon" title="c4">s</font> is  <a href="qc_lang1.html#V2" title="QC_LANG1:attr.2">atomic</a> 
 <span class="kw">and </span><br/><a NAME="E2:54_1_1"/><span class="lab"><font color="Green" title="E31">A3</font></span>: 
<font color="Maroon" title="c1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">y</font>
 <span class="kw">and </span><br/><a NAME="E3:54_1_1"/><span class="lab"><font color="Green" title="E32">A4</font></span>: 
not <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <font color="Maroon" title="c4">s</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> not <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c2">y</font></span>)</span></span><br/>

<span class="kw">thus </span><a NAME="E4:54_1_1"/>
not <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c2">y</font></span>)</span>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">set </span><font color="Maroon" title="c5">l</font> =  <a href="qc_lang1.html#K16" title="QC_LANG1:func.16">the_arguments_of</a> <font color="Maroon" title="c4">s</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">ll</font> =  <a href="cqc_lang.html#K1" title="CQC_LANG:func.1">Subst</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K16" title="QC_LANG1:func.16">the_arguments_of</a> <font color="Maroon" title="c4">s</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang3.html#K3" title="QC_LANG3:func.3">a.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> <font color="Maroon" title="c2">y</font></span>)</span>;<br/>
<span class="lab"><font color="Green" title="E33">A5</font></span>:  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <font color="Maroon" title="c4">s</font> = 
 <a href="qc_lang1.html#K22" title="QC_LANG1:func.22">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K16" title="QC_LANG1:func.16">the_arguments_of</a> <font color="Maroon" title="c4">s</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54_1_1"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="ref" href="qc_lang3.html#T8" title="QC_LANG3:th.8">QC_LANG3:8</a></span>
<br/>.= 
 <a href="qc_lang3.html#K1" title="QC_LANG3:func.1">variables_in</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K16" title="QC_LANG1:func.16">the_arguments_of</a> <font color="Maroon" title="c4">s</font></span>)</span>,<a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="qc_lang3.html#T6" title="QC_LANG3:th.6">QC_LANG3:6</a></span>
<br/>.= 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang1.html#K16" title="QC_LANG1:func.16">the_arguments_of</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> where <font color="Olive" title="b1">k</font> is    <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>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K16" title="QC_LANG1:func.16">the_arguments_of</a> <font color="Maroon" title="c4">s</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="qc_lang1.html#K16" title="QC_LANG1:func.16">the_arguments_of</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>  ) </span>}</span> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="qc_lang3.html#D2" title="QC_LANG3:def.2">QC_LANG3:def 2</a></span>
;<br/>
<a NAME="E2:54_1_1_1"/><span class="lab"><font color="Green" title="E34">A6</font></span>: 
( <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cqc_lang.html#K1" title="CQC_LANG:func.1">Subst</a> <span class="p4">(<span class="default"><a href="qc_lang1.html#K16" title="QC_LANG1:func.16">the_arguments_of</a> <font color="Maroon" title="c4">s</font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="qc_lang3.html#K3" title="QC_LANG3:func.3">a.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> <font color="Maroon" title="c2">y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> where <font color="Olive" title="b1">k</font> is    <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>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K1" title="CQC_LANG:func.1">Subst</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K16" title="QC_LANG1:func.16">the_arguments_of</a> <font color="Maroon" title="c4">s</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang3.html#K3" title="QC_LANG3:func.3">a.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> <font color="Maroon" title="c2">y</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="cqc_lang.html#K1" title="CQC_LANG:func.1">Subst</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K16" title="QC_LANG1:func.16">the_arguments_of</a> <font color="Maroon" title="c4">s</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang3.html#K3" title="QC_LANG3:func.3">a.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> <font color="Maroon" title="c2">y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>  ) </span>}</span>  implies <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang1.html#K16" title="QC_LANG1:func.16">the_arguments_of</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> where <font color="Olive" title="b1">i</font> is    <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>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K16" title="QC_LANG1:func.16">the_arguments_of</a> <font color="Maroon" title="c4">s</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="qc_lang1.html#K16" title="QC_LANG1:func.16">the_arguments_of</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>  ) </span>}</span>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:54_1_1_1_2"/>
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cqc_lang.html#K1" title="CQC_LANG:func.1">Subst</a> <span class="p4">(<span class="default"><a href="qc_lang1.html#K16" title="QC_LANG1:func.16">the_arguments_of</a> <font color="Maroon" title="c4">s</font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="qc_lang3.html#K3" title="QC_LANG3:func.3">a.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> <font color="Maroon" title="c2">y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> where <font color="Olive" title="b1">k</font> is    <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>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K1" title="CQC_LANG:func.1">Subst</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K16" title="QC_LANG1:func.16">the_arguments_of</a> <font color="Maroon" title="c4">s</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang3.html#K3" title="QC_LANG3:func.3">a.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> <font color="Maroon" title="c2">y</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="cqc_lang.html#K1" title="CQC_LANG:func.1">Subst</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K16" title="QC_LANG1:func.16">the_arguments_of</a> <font color="Maroon" title="c4">s</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang3.html#K3" title="QC_LANG3:func.3">a.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> <font color="Maroon" title="c2">y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>  ) </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang1.html#K16" title="QC_LANG1:func.16">the_arguments_of</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> where <font color="Olive" title="b1">i</font> is    <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>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K16" title="QC_LANG1:func.16">the_arguments_of</a> <font color="Maroon" title="c4">s</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="qc_lang1.html#K16" title="QC_LANG1:func.16">the_arguments_of</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>  ) </span>}</span> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">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> <span class="kw"> such that </span><br/><a NAME="E3:54_1_1_1_2"/><span class="lab"><font color="Green" title="E35">A7</font></span>: 
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K1" title="CQC_LANG:func.1">Subst</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K16" title="QC_LANG1:func.16">the_arguments_of</a> <font color="Maroon" title="c4">s</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang3.html#K3" title="QC_LANG3:func.3">a.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> <font color="Maroon" title="c2">y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font>
 <span class="kw">and </span><br/><a NAME="E4:54_1_1_1_2"/><span class="lab"><font color="Green" title="E36">A8</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">k</font>
 <span class="kw">and </span><br/><a NAME="E5:54_1_1_1_2"/><span class="lab"><font color="Green" title="E37">A9</font></span>: 
<font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K1" title="CQC_LANG:func.1">Subst</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K16" title="QC_LANG1:func.16">the_arguments_of</a> <font color="Maroon" title="c4">s</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang3.html#K3" title="QC_LANG3:func.3">a.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> <font color="Maroon" title="c2">y</font></span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E6:54_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="cqc_lang.html#K1" title="CQC_LANG:func.1">Subst</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K16" title="QC_LANG1:func.16">the_arguments_of</a> <font color="Maroon" title="c4">s</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang3.html#K3" title="QC_LANG3:func.3">a.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> <font color="Maroon" title="c2">y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> 
 ;<br/>
<a NAME="E7:54_1_1_1_2"/><span class="lab"><font color="Green" title="E38">A10</font></span>: 
<font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K16" title="QC_LANG1:func.16">the_arguments_of</a> <font color="Maroon" title="c4">s</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:54_1_1_1_2"><span class="lab"><font color="Green" title="E37">A9</font></span></a>, <a class="ref" href="cqc_lang.html#D3" title="CQC_LANG:def.3">CQC_LANG:def 3</a></span>;<br/>
<a NAME="E8:54_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K16" title="QC_LANG1:func.16">the_arguments_of</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:54_1_1"><span class="lab"><font color="Green" title="E31">A3</font></span></a>, <a class="txt" href="#E3:54_1_1_1_2"><span class="lab"><font color="Green" title="E35">A7</font></span></a>, <a class="txt" href="#E4:54_1_1_1_2"><span class="lab"><font color="Green" title="E36">A8</font></span></a>, <a class="ref" href="cqc_lang.html#T11" title="CQC_LANG:th.11">CQC_LANG:11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:54_1_1_1_2"/>
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang1.html#K16" title="QC_LANG1:func.16">the_arguments_of</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font></span>)</span> where <font color="Olive" title="b1">i</font> is    <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>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K16" title="QC_LANG1:func.16">the_arguments_of</a> <font color="Maroon" title="c4">s</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="qc_lang1.html#K16" title="QC_LANG1:func.16">the_arguments_of</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>  ) </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:54_1_1_1_2"><span class="lab"><font color="Green" title="E36">A8</font></span></a>, <a class="txt" href="#E7:54_1_1_1_2"><span class="lab"><font color="Green" title="E38">A10</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:54_1_1_1"/><span class="lab"><font color="Green" title="E35">A11</font></span>: 
<font color="Maroon" title="c4">s</font> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <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_lang1.html#K15" title="QC_LANG1:func.15">the_pred_symbol_of</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">!</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K1" title="CQC_LANG:func.1">Subst</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K16" title="QC_LANG1:func.16">the_arguments_of</a> <font color="Maroon" title="c4">s</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang3.html#K3" title="QC_LANG3:func.3">a.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> <font color="Maroon" title="c2">y</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54_1_1"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="ref" href="cqc_lang.html#T29" title="CQC_LANG:th.29">CQC_LANG:29</a></span>;<br/>
<a NAME="E4:54_1_1_1"/><span class="lab"><font color="Green" title="E36">A12</font></span>: 
( <font color="Maroon" title="c4">s</font> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c2">y</font> is  <a href="qc_lang1.html#V2" title="QC_LANG1:attr.2">atomic</a>  &amp;  <a href="qc_lang1.html#K16" title="QC_LANG1:func.16">the_arguments_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cqc_lang.html#K1" title="CQC_LANG:func.1">Subst</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K16" title="QC_LANG1:func.16">the_arguments_of</a> <font color="Maroon" title="c4">s</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang3.html#K3" title="QC_LANG3:func.3">a.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> <font color="Maroon" title="c2">y</font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_1_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">consider </span><font color="Maroon" title="c7">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="c8">p</font> being   <a href="qc_lang1.html#NM6" title="QC_LANG1:NM.6">QC-pred_symbol</a> of <font color="Maroon" title="c7">k</font>, <font color="Maroon" title="c9">l2</font> being   <a href="qc_lang1.html#NM7" title="QC_LANG1:NM.7">QC-variable_list</a> of <font color="Maroon" title="c7">k</font><span class="kw"> such that </span><br/><a NAME="E2:54_1_1_1_3"/><span class="lab"><font color="Green" title="E37">A13</font></span>: 
<font color="Maroon" title="c4">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p</font> <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">!</a> <font color="Maroon" title="c9">l2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54_1_1"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="ref" href="qc_lang1.html#D17" title="QC_LANG1:def.17">QC_LANG1:def 17</a></span>;<br/>
<a NAME="E3:54_1_1_1_3"/>
<font color="Maroon" title="c9">l2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K16" title="QC_LANG1:func.16">the_arguments_of</a> <font color="Maroon" title="c4">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54_1_1"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="txt" href="#E2:54_1_1_1_3"><span class="lab"><font color="Green" title="E37">A13</font></span></a>, <a class="ref" href="qc_lang1.html#D22" title="QC_LANG1:def.22">QC_LANG1:def 22</a></span>;<br/>
<a NAME="E4:54_1_1_1_3"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K16" title="QC_LANG1:func.16">the_arguments_of</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D18" title="FINSEQ_1:def.18">FINSEQ_1:def 18</a></span>;<br/>
<a NAME="E5:54_1_1_1_3"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K1" title="CQC_LANG:func.1">Subst</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K16" title="QC_LANG1:func.16">the_arguments_of</a> <font color="Maroon" title="c4">s</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang3.html#K3" title="QC_LANG3:func.3">a.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> <font color="Maroon" title="c2">y</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cqc_lang.html#D3" title="CQC_LANG:def.3">CQC_LANG:def 3</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">l3</font> =  <a href="cqc_lang.html#K1" title="CQC_LANG:func.1">Subst</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K16" title="QC_LANG1:func.16">the_arguments_of</a> <font color="Maroon" title="c4">s</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang3.html#K3" title="QC_LANG3:func.3">a.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> <font color="Maroon" title="c2">y</font></span>)</span> as   <a href="qc_lang1.html#NM7" title="QC_LANG1:NM.7">QC-variable_list</a> of <font color="Maroon" title="c7">k</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D18" title="FINSEQ_1:def.18">FINSEQ_1:def 18</a></span>;<br/>
<a NAME="E7:54_1_1_1_3"/><span class="lab"><font color="Green" title="E38">A14</font></span>: 
<font color="Maroon" title="c4">s</font> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p</font> <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">!</a> <font color="Maroon" title="c10">l3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54_1_1"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="txt" href="#E3:54_1_1_1"><span class="lab"><font color="Green" title="E35">A11</font></span></a>, <a class="txt" href="#E2:54_1_1_1_3"><span class="lab"><font color="Green" title="E37">A13</font></span></a>, <a class="ref" href="qc_lang1.html#D21" title="QC_LANG1:def.21">QC_LANG1:def 21</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:54_1_1_1_3"/>
<font color="Maroon" title="c4">s</font> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c2">y</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#D17" title="QC_LANG1:def.17">QC_LANG1:def 17</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="qc_lang1.html#K16" title="QC_LANG1:func.16">the_arguments_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cqc_lang.html#K1" title="CQC_LANG:func.1">Subst</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K16" title="QC_LANG1:func.16">the_arguments_of</a> <font color="Maroon" title="c4">s</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang3.html#K3" title="QC_LANG3:func.3">a.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> <font color="Maroon" title="c2">y</font></span>)</span></span><br/>

<span class="kw">hence </span><a NAME="E9:54_1_1_1_3"/>
 <a href="qc_lang1.html#K16" title="QC_LANG1:func.16">the_arguments_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cqc_lang.html#K1" title="CQC_LANG:func.1">Subst</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K16" title="QC_LANG1:func.16">the_arguments_of</a> <font color="Maroon" title="c4">s</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang3.html#K3" title="QC_LANG3:func.3">a.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> <font color="Maroon" title="c2">y</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:54_1_1_1_3"><span class="lab"><font color="Green" title="E38">A14</font></span></a>, <a class="ref" href="qc_lang1.html#D22" title="QC_LANG1:def.22">QC_LANG1:def 22</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span> <a href="qc_lang1.html#K22" title="QC_LANG1:func.22">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K16" title="QC_LANG1:func.16">the_arguments_of</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c2">y</font></span>)</span></span>)</span> = 
 <a href="qc_lang3.html#K1" title="QC_LANG3:func.1">variables_in</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K1" title="CQC_LANG:func.1">Subst</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K16" title="QC_LANG1:func.16">the_arguments_of</a> <font color="Maroon" title="c4">s</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang3.html#K3" title="QC_LANG3:func.3">a.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> <font color="Maroon" title="c2">y</font></span>)</span></span>)</span>,<a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="qc_lang3.html#T6" title="QC_LANG3:th.6">QC_LANG3:6</a></span>
<br/>.= 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cqc_lang.html#K1" title="CQC_LANG:func.1">Subst</a> <span class="p4">(<span class="default"><a href="qc_lang1.html#K16" title="QC_LANG1:func.16">the_arguments_of</a> <font color="Maroon" title="c4">s</font></span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="qc_lang3.html#K3" title="QC_LANG3:func.3">a.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> <font color="Maroon" title="c2">y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> where <font color="Olive" title="b1">k</font> is    <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>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K1" title="CQC_LANG:func.1">Subst</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K16" title="QC_LANG1:func.16">the_arguments_of</a> <font color="Maroon" title="c4">s</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang3.html#K3" title="QC_LANG3:func.3">a.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> <font color="Maroon" title="c2">y</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="cqc_lang.html#K1" title="CQC_LANG:func.1">Subst</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K16" title="QC_LANG1:func.16">the_arguments_of</a> <font color="Maroon" title="c4">s</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang3.html#K3" title="QC_LANG3:func.3">a.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="cqc_lang.html#K3" title="CQC_LANG:func.3">.--&gt;</a> <font color="Maroon" title="c2">y</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>  ) </span>}</span> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="qc_lang3.html#D2" title="QC_LANG3:def.2">QC_LANG3:def 2</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E6:54_1_1_1"/>
not <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c2">y</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:54_1_1"><span class="lab"><font color="Green" title="E32">A4</font></span></a>, <a class="txt" href="#E1:54_1_1_1"><span class="lab"><font color="Green" title="E33">A5</font></span></a>, <a class="txt" href="#E4:54_1_1_1"><span class="lab"><font color="Green" title="E36">A12</font></span></a>, <a class="txt" href="#E2:54_1_1_1"><span class="lab"><font color="Green" title="E34">A6</font></span></a>, <a class="ref" href="qc_lang3.html#T8" title="QC_LANG3:th.8">QC_LANG3:8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>


</div><span class="kw">end;</span></div><span class="kw">thus </span><a NAME="E2:54_1"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="cqc_lang.html#K6" title="CQC_LANG:func.6">VERUM</a> ]
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cqc_lang.html#T28" title="CQC_LANG:th.28">CQC_LANG:28</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c4">s</font> is  <a href="qc_lang1.html#V3" title="QC_LANG1:attr.3">negative</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K17" title="QC_LANG1:func.17">the_argument_of</a> <font color="Maroon" title="c4">s</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">s</font>] ) &amp; ( <font color="Maroon" title="c4">s</font> is  <a href="qc_lang1.html#V4" title="QC_LANG1:attr.4">conjunctive</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_left_argument_of</a> <font color="Maroon" title="c4">s</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_right_argument_of</a> <font color="Maroon" title="c4">s</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">s</font>] ) &amp; ( <font color="Maroon" title="c4">s</font> is  <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K21" title="QC_LANG1:func.21">the_scope_of</a> <font color="Maroon" title="c4">s</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">s</font>] ) )</span><br/><span class="kw">thus </span><a NAME="E3:54_1"/>
( <font color="Maroon" title="c4">s</font> is  <a href="qc_lang1.html#V3" title="QC_LANG1:attr.3">negative</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K17" title="QC_LANG1:func.17">the_argument_of</a> <font color="Maroon" title="c4">s</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">s</font>] )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c4">s</font> is  <a href="qc_lang1.html#V4" title="QC_LANG1:attr.4">conjunctive</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_left_argument_of</a> <font color="Maroon" title="c4">s</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_right_argument_of</a> <font color="Maroon" title="c4">s</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">s</font>] ) &amp; ( <font color="Maroon" title="c4">s</font> is  <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K21" title="QC_LANG1:func.21">the_scope_of</a> <font color="Maroon" title="c4">s</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">s</font>] ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:54_1_2"/><span class="lab"><font color="Green" title="E30">A15</font></span>: 
<font color="Maroon" title="c4">s</font> is  <a href="qc_lang1.html#V3" title="QC_LANG1:attr.3">negative</a> 
 <span class="kw">and </span><br/><a NAME="E2:54_1_2"/><span class="lab"><font color="Green" title="E31">A16</font></span>: 
( ( <font color="Maroon" title="c1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">y</font> &amp; not <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K17" title="QC_LANG1:func.17">the_argument_of</a> <font color="Maroon" title="c4">s</font></span>)</span> implies  not <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <span class="p1">(<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="c4">s</font></span>)</span> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c2">y</font></span>)</span> ) &amp; <font color="Maroon" title="c1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">y</font> &amp; not <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <font color="Maroon" title="c4">s</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> not <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c2">y</font></span>)</span></span><br/>

<a NAME="E3:54_1_2"/>
not <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K12" title="QC_LANG1:func.12">'not'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang1.html#K17" title="QC_LANG1:func.17">the_argument_of</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c2">y</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54_1_2"><span class="lab"><font color="Green" title="E30">A15</font></span></a>, <a class="txt" href="#E2:54_1_2"><span class="lab"><font color="Green" title="E31">A16</font></span></a>, <a class="ref" href="qc_lang3.html#T10" title="QC_LANG3:th.10">QC_LANG3:10</a>, <a class="ref" href="qc_lang3.html#T11" title="QC_LANG3:th.11">QC_LANG3:11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:54_1_2"/>
not <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c2">y</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54_1_2"><span class="lab"><font color="Green" title="E30">A15</font></span></a>, <a class="ref" href="cqc_lang.html#T31" title="CQC_LANG:th.31">CQC_LANG:31</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">thus </span><a NAME="E4:54_1"/>
( <font color="Maroon" title="c4">s</font> is  <a href="qc_lang1.html#V4" title="QC_LANG1:attr.4">conjunctive</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_left_argument_of</a> <font color="Maroon" title="c4">s</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_right_argument_of</a> <font color="Maroon" title="c4">s</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">s</font>] )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">s</font> is  <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K21" title="QC_LANG1:func.21">the_scope_of</a> <font color="Maroon" title="c4">s</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">s</font>] )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:54_1_3"/><span class="lab"><font color="Green" title="E30">A17</font></span>: 
<font color="Maroon" title="c4">s</font> is  <a href="qc_lang1.html#V4" title="QC_LANG1:attr.4">conjunctive</a> 
 <span class="kw">and </span><br/><a NAME="E2:54_1_3"/><span class="lab"><font color="Green" title="E31">A18</font></span>: 
( ( <font color="Maroon" title="c1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">y</font> &amp; not <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_left_argument_of</a> <font color="Maroon" title="c4">s</font></span>)</span> implies  not <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <span class="p1">(<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="c4">s</font></span>)</span> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c2">y</font></span>)</span> ) &amp; ( <font color="Maroon" title="c1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">y</font> &amp; not <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_right_argument_of</a> <font color="Maroon" title="c4">s</font></span>)</span> implies  not <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <span class="p1">(<span class="default"><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="c4">s</font></span>)</span> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c2">y</font></span>)</span> ) &amp; <font color="Maroon" title="c1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">y</font> &amp; not <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <font color="Maroon" title="c4">s</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> not <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c2">y</font></span>)</span></span><br/>

<a NAME="E3:54_1_3"/>
 <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <font color="Maroon" title="c4">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_left_argument_of</a> <font color="Maroon" title="c4">s</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_right_argument_of</a> <font color="Maroon" title="c4">s</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54_1_3"><span class="lab"><font color="Green" title="E30">A17</font></span></a>, <a class="ref" href="qc_lang3.html#T13" title="QC_LANG3:th.13">QC_LANG3:13</a></span>;<br/>
<a NAME="E4:54_1_3"/><span class="kw">then </span>
not <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_left_argument_of</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c2">y</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_right_argument_of</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c2">y</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:54_1_3"><span class="lab"><font color="Green" title="E31">A18</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E5:54_1_3"/><span class="kw">then </span>
not <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_left_argument_of</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_right_argument_of</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c2">y</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="qc_lang3.html#T14" title="QC_LANG3:th.14">QC_LANG3:14</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:54_1_3"/>
not <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c2">y</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54_1_3"><span class="lab"><font color="Green" title="E30">A17</font></span></a>, <a class="ref" href="cqc_lang.html#T33" title="CQC_LANG:th.33">CQC_LANG:33</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">thus </span><a NAME="E5:54_1"/>
( <font color="Maroon" title="c4">s</font> is  <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a>  &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K21" title="QC_LANG1:func.21">the_scope_of</a> <font color="Maroon" title="c4">s</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">s</font>] )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:54_1_4"/><span class="lab"><font color="Green" title="E30">A19</font></span>: 
<font color="Maroon" title="c4">s</font> is  <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a> 
 <span class="kw">and </span><br/><a NAME="E2:54_1_4"/><span class="lab"><font color="Green" title="E31">A20</font></span>: 
( <font color="Maroon" title="c1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">y</font> &amp; not <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K21" title="QC_LANG1:func.21">the_scope_of</a> <font color="Maroon" title="c4">s</font></span>)</span> implies  not <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <span class="p1">(<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="c4">s</font></span>)</span> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c2">y</font></span>)</span> )
 <span class="kw">and </span><br/><a NAME="E3:54_1_4"/><span class="lab"><font color="Green" title="E32">A21</font></span>: 
<font color="Maroon" title="c1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">y</font>
 <span class="kw">and </span><br/><a NAME="E4:54_1_4"/><span class="lab"><font color="Green" title="E33">A22</font></span>: 
not <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <font color="Maroon" title="c4">s</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> not <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c2">y</font></span>)</span></span><br/>

<a NAME="E5:54_1_4"/><span class="lab"><font color="Green" title="E34">A23</font></span>: 
 <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <font color="Maroon" title="c4">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K21" title="QC_LANG1:func.21">the_scope_of</a> <font color="Maroon" title="c4">s</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">bound_in</a> <font color="Maroon" title="c4">s</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54_1_4"><span class="lab"><font color="Green" title="E30">A19</font></span></a>, <a class="ref" href="qc_lang3.html#T15" title="QC_LANG3:th.15">QC_LANG3:15</a></span>;<br/>
<span class="kw">thus </span><a NAME="E6:54_1_4"/>
not <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c2">y</font></span>)</span>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_4_1"><span class="kw">proof </span></a><div class="add">

<div><span class="lab"><font color="Green" title="E35">A24</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_4_1_1"><span class="kw">now </span></a><div class="add"><a NAME="E1:54_1_4_1_1"/>
 <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> <font color="Maroon" title="c1">x</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang1.html#K21" title="QC_LANG1:func.21">the_scope_of</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c2">y</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang1.html#K21" title="QC_LANG1:func.21">the_scope_of</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c2">y</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="qc_lang3.html#T16" title="QC_LANG3:th.16">QC_LANG3:16</a></span>;<br/><a NAME="E2:54_1_4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A25</font></span>: 
( (  not <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <span class="p1">(<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="c4">s</font></span>)</span> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c2">y</font></span>)</span> or <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) iff  not <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> <font color="Maroon" title="c1">x</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang1.html#K21" title="QC_LANG1:func.21">the_scope_of</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c2">y</font></span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><span class="kw">assume </span><a NAME="E3:54_1_4_1_1"/>
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">bound_in</a> <font color="Maroon" title="c4">s</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> not <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c2">y</font></span>)</span></span><br/><a NAME="E4:54_1_4_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K20" title="QC_LANG1:func.20">bound_in</a> <font color="Maroon" title="c4">s</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E5:54_1_4_1_1"/>
not <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c2">y</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54_1_4"><span class="lab"><font color="Green" title="E30">A19</font></span></a>, <a class="txt" href="#E3:54_1_4"><span class="lab"><font color="Green" title="E32">A21</font></span></a>, <a class="txt" href="#E2:54_1_4_1_1"><span class="lab"><font color="Green" title="E36">A25</font></span></a>, <a class="ref" href="cqc_lang.html#T36" title="CQC_LANG:th.36">CQC_LANG:36</a>, <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="Red">::  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="54_1_4_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:54_1_4_1_2"/>
not <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K21" title="QC_LANG1:func.21">the_scope_of</a> <font color="Maroon" title="c4">s</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> not <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c2">y</font></span>)</span></span><br/><a NAME="E2:54_1_4_1_2"/><span class="kw">then </span>
not <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang1.html#K21" title="QC_LANG1:func.21">the_scope_of</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c2">y</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">bound_in</a> <font color="Maroon" title="c4">s</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:54_1_4"><span class="lab"><font color="Green" title="E31">A20</font></span></a>, <a class="txt" href="#E3:54_1_4"><span class="lab"><font color="Green" title="E32">A21</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E3:54_1_4_1_2"/><span class="kw">then </span>
not <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">bound_in</a> <font color="Maroon" title="c4">s</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang1.html#K21" title="QC_LANG1:func.21">the_scope_of</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c2">y</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="qc_lang3.html#T16" title="QC_LANG3:th.16">QC_LANG3:16</a></span>;<br/><span class="kw">hence </span><a NAME="E4:54_1_4_1_2"/>
not <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c2">y</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:54_1_4"><span class="lab"><font color="Green" title="E30">A19</font></span></a>, <a class="txt" href="#E4:54_1_4"><span class="lab"><font color="Green" title="E33">A22</font></span></a>, <a class="ref" href="cqc_lang.html#T35" title="CQC_LANG:th.35">CQC_LANG:35</a>, <a class="ref" href="cqc_lang.html#T36" title="CQC_LANG:th.36">CQC_LANG:36</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:54_1_4_1"/>
not <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c2">y</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:54_1_4"><span class="lab"><font color="Green" title="E33">A22</font></span></a>, <a class="txt" href="#E5:54_1_4"><span class="lab"><font color="Green" title="E34">A23</font></span></a>, <a class="txt" href="#E1:54_1_4_1"><span class="lab"><font color="Green" title="E35">A24</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>


</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<a NAME="E2:54"/>
for <font color="Olive" title="b1">s</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>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">s</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="qc_lang1.html#S2" title="QC_LANG1:sch.2">QC_LANG1:sch 2</a>(<a class="txt" href="#E1:54"><span class="lab"><font color="Green" title="E29">A1</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E3:54"/>
( <font color="Maroon" title="c1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">y</font> &amp; not <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <font color="Maroon" title="c3">s9</font> implies  not <font color="Maroon" title="c1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s9</font> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c2">y</font></span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
