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

<span class="kw">let </span><font color="Maroon" title="c1">l1</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="qc_lang1.html#K1" title="QC_LANG1:func.1">QC-variables</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c1">l1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>  implies  <a href="qc_lang1.html#K22" title="QC_LANG1:func.22">still_not-bound_in</a> <font color="Maroon" title="c1">l1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c1">l1</font> )</span><br/>

<a NAME="E1:77"/><span class="lab"><font color="Green" title="E57">A1</font></span>: 
 <a href="qc_lang3.html#K1" title="QC_LANG3:func.1">variables_in</a> <font color="Maroon" title="c1">l1</font>,<a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c1">l1</font> <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> <font color="Maroon" title="c1">l1</font> &amp; <font color="Maroon" title="c1">l1</font> <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">assume </span><a NAME="E2:77"/><span class="lab"><font color="Green" title="E58">A2</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c1">l1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> 
 ; <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#K22" title="QC_LANG1:func.22">still_not-bound_in</a> <font color="Maroon" title="c1">l1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c1">l1</font></span><br/>

<a NAME="E3:77"/><span class="lab"><font color="Green" title="E59">A3</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c1">l1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="qc_lang3.html#K1" title="QC_LANG3:func.1">variables_in</a> <font color="Maroon" title="c1">l1</font>,<a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <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="c2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c1">l1</font> or <font color="Maroon" title="c2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang3.html#K1" title="QC_LANG3:func.1">variables_in</a> <font color="Maroon" title="c1">l1</font>,<a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:77_1"/><span class="lab"><font color="Green" title="E60">A4</font></span>: 
<font color="Maroon" title="c2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c1">l1</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="c2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang3.html#K1" title="QC_LANG3:func.1">variables_in</a> <font color="Maroon" title="c1">l1</font>,<a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:77_1"/><span class="lab"><font color="Green" title="E61">A5</font></span>: 
<font color="Maroon" title="c3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c1">l1</font>
 <span class="kw">and </span><br/><a NAME="E4:77_1"/><span class="lab"><font color="Green" title="E62">A6</font></span>: 
<font color="Maroon" title="c1">l1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T11" title="FINSEQ_2:th.11">FINSEQ_2:11</a></span>;<br/>
<a NAME="E5:77_1"/>
<font color="Maroon" title="c3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">l1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:77_1"><span class="lab"><font color="Green" title="E61">A5</font></span></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E6:77_1"/><span class="kw">then </span>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> &amp; <font color="Maroon" title="c3">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> <font color="Maroon" title="c1">l1</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:77_1"/>
<font color="Maroon" title="c2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang3.html#K1" title="QC_LANG3:func.1">variables_in</a> <font color="Maroon" title="c1">l1</font>,<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="txt" href="#E2:77"><span class="lab"><font color="Green" title="E58">A2</font></span></a>, <a class="txt" href="#E1:77"><span class="lab"><font color="Green" title="E57">A1</font></span></a>, <a class="txt" href="#E1:77_1"><span class="lab"><font color="Green" title="E60">A4</font></span></a>, <a class="txt" href="#E3:77_1"><span class="lab"><font color="Green" title="E61">A5</font></span></a>, <a class="txt" href="#E4:77_1"><span class="lab"><font color="Green" title="E62">A6</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="E4:77"/>
 <a href="qc_lang3.html#K1" title="QC_LANG3:func.1">variables_in</a> <font color="Maroon" title="c1">l1</font>,<a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c1">l1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <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="c2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang3.html#K1" title="QC_LANG3:func.1">variables_in</a> <font color="Maroon" title="c1">l1</font>,<a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>  or <font color="Maroon" title="c2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c1">l1</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:77_2"/>
<font color="Maroon" title="c2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang3.html#K1" title="QC_LANG3:func.1">variables_in</a> <font color="Maroon" title="c1">l1</font>,<a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</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="c2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c1">l1</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">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="E3:77_2"/><span class="lab"><font color="Green" title="E60">A7</font></span>: 
<font color="Maroon" title="c2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">l1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">k</font>
 <span class="kw">and </span><br/><a NAME="E4:77_2"/><span class="lab"><font color="Green" title="E61">A8</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> &amp; <font color="Maroon" title="c3">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> <font color="Maroon" title="c1">l1</font> )
 <span class="kw">and </span><br/><a NAME="E5:77_2"/>
<font color="Maroon" title="c1">l1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E1:77"><span class="lab"><font color="Green" title="E57">A1</font></span></a></span>;<br/>
<a NAME="E6:77_2"/>
<font color="Maroon" title="c3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">l1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:77_2"><span class="lab"><font color="Green" title="E61">A8</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E7:77_2"/><span class="kw">then </span>
<font color="Maroon" title="c3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c1">l1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:77_2"/>
<font color="Maroon" title="c2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c1">l1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:77_2"><span class="lab"><font color="Green" title="E60">A7</font></span></a>, <a class="ref" href="funct_1.html#T12" title="FUNCT_1:th.12">FUNCT_1:12</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="E5:77"/><span class="kw">then </span>
 <a href="qc_lang3.html#K1" title="QC_LANG3:func.1">variables_in</a> <font color="Maroon" title="c1">l1</font>,<a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c1">l1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:77"><span class="lab"><font color="Green" title="E59">A3</font></span></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:77"/>
 <a href="qc_lang1.html#K22" title="QC_LANG1:func.22">still_not-bound_in</a> <font color="Maroon" title="c1">l1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c1">l1</font>
 <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>; <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>
