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

<span class="kw">set </span><font color="Maroon" title="c5">YN</font> = <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <font color="Olive" title="b1">R</font></span>)</span> where <font color="Olive" title="b1">R</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a> <a href="ltlaxio3.html#K1" title="LTLAXIO3:func.1">**</a></span>)</span>,<span class="p2">(<span class="default"><a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a> <a href="ltlaxio3.html#K1" title="LTLAXIO3:func.1">**</a></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> : <font color="Olive" title="b1">R</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c4">T</font> </span> } </span> ;<br/>
<a NAME="E1:70_1_1"/><span class="lab"><font color="Green" title="E37">A1</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c4">T</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 
;<br/>
<a NAME="E2:70_1_1"/><span class="lab"><font color="Green" title="E38">A2</font></span>: 
<span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <font color="Olive" title="b1">R</font></span>)</span> where <font color="Olive" title="b1">R</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a> <a href="ltlaxio3.html#K1" title="LTLAXIO3:func.1">**</a></span>)</span>,<span class="p2">(<span class="default"><a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a> <a href="ltlaxio3.html#K1" title="LTLAXIO3:func.1">**</a></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> : <font color="Olive" title="b1">R</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c4">T</font> </span> } </span>  is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="fraenkel.html#S21" onmouseover="rs('fraenkel/S21')" onmouseout="rh()">FRAENKEL:sch 21</a>(<a class="txt" href="#E1:70_1_1"><span class="lab"><font color="Green" title="E37">A1</font></span></a>);<br/></span>
<a NAME="E3:70_1_1"/>
<span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <font color="Olive" title="b1">R</font></span>)</span> where <font color="Olive" title="b1">R</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a> <a href="ltlaxio3.html#K1" title="LTLAXIO3:func.1">**</a></span>)</span>,<span class="p2">(<span class="default"><a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a> <a href="ltlaxio3.html#K1" title="LTLAXIO3:func.1">**</a></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> : <font color="Olive" title="b1">R</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c4">T</font> </span> } </span>  is  <a href="finset_1.html#V5" title="FINSET_1:attr.5">finite-membered</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="70_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="finset_1.html#D6" onmouseover="rs('finset_1/D6')" onmouseout="rh()">FINSET_1:def 6</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c6">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <font color="Olive" title="b1">R</font></span>)</span> where <font color="Olive" title="b1">R</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a> <a href="ltlaxio3.html#K1" title="LTLAXIO3:func.1">**</a></span>)</span>,<span class="p2">(<span class="default"><a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a> <a href="ltlaxio3.html#K1" title="LTLAXIO3:func.1">**</a></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> : <font color="Olive" title="b1">R</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c4">T</font> </span> } </span>  or <font color="Maroon" title="c6">x</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:70_1_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <font color="Olive" title="b1">R</font></span>)</span> where <font color="Olive" title="b1">R</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a> <a href="ltlaxio3.html#K1" title="LTLAXIO3:func.1">**</a></span>)</span>,<span class="p2">(<span class="default"><a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a> <a href="ltlaxio3.html#K1" title="LTLAXIO3:func.1">**</a></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> : <font color="Olive" title="b1">R</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c4">T</font> </span> } </span> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">x</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">R</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a> <a href="ltlaxio3.html#K1" title="LTLAXIO3:func.1">**</a></span>)</span>,<span class="p2">(<span class="default"><a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a> <a href="ltlaxio3.html#K1" title="LTLAXIO3:func.1">**</a></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span><span class="kw"> such that </span><br/><a NAME="E3:70_1_1_1"/><span class="lab"><font color="Green" title="E39">A3</font></span>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <font color="Maroon" title="c7">R</font> &amp; <font color="Maroon" title="c7">R</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c4">T</font> )
 ;<br/>
<span class="kw">thus </span><a NAME="E4:70_1_1_1"/>
<font color="Maroon" title="c6">x</font> is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:70_1_1_1"><span class="lab"><font color="Green" title="E39">A3</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">YN</font> = <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <font color="Olive" title="b1">R</font></span>)</span> where <font color="Olive" title="b1">R</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a> <a href="ltlaxio3.html#K1" title="LTLAXIO3:func.1">**</a></span>)</span>,<span class="p2">(<span class="default"><a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a> <a href="ltlaxio3.html#K1" title="LTLAXIO3:func.1">**</a></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> : <font color="Olive" title="b1">R</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c4">T</font> </span> } </span>  as   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="finset_1.html#V5" title="FINSET_1:attr.5">finite-membered</a>   <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:70_1_1"><span class="lab"><font color="Green" title="E38">A2</font></span></a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="70_1_1_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>   st <font color="Olive" title="b1">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">YN</font> holds <br/><font color="Olive" title="b1">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a> </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">YN</font> implies <font color="Maroon" title="c7">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a>  )</span><br/><span class="kw">assume </span><a NAME="E1:70_1_1_2"/>
<font color="Maroon" title="c7">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">YN</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a> </span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">R</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a> <a href="ltlaxio3.html#K1" title="LTLAXIO3:func.1">**</a></span>)</span>,<span class="p2">(<span class="default"><a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a> <a href="ltlaxio3.html#K1" title="LTLAXIO3:func.1">**</a></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span><span class="kw"> such that </span><br/><a NAME="E3:70_1_1_2"/><span class="lab"><font color="Green" title="E39">A4</font></span>: 
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <font color="Maroon" title="c8">R</font> &amp; <font color="Maroon" title="c8">R</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c4">T</font> )
 ;<br/><span class="kw">thus </span><a NAME="E4:70_1_1_2"/>
<font color="Maroon" title="c7">x</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:70_1_1_2"><span class="lab"><font color="Green" title="E39">A4</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">Tforms</font> =  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <font color="Maroon" title="c6">YN</font> as   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T76" onmouseover="rs('zfmisc_1/T76')" onmouseout="rh()">ZFMISC_1:76</a></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="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a> ] <span class="kw">means </span>( <font color="Maroon" title="c1">P</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">Tforms</font> &amp; <font color="Maroon" title="c1">P</font> is  <a href="ltlaxio4.html#NV1" title="LTLAXIO4:NV.1">s-until</a>  );<br/>
<span class="kw">set </span><font color="Maroon" title="c8">uns</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">p</font>] </span> } </span> ;<br/>
<a NAME="E7:70_1_1"/><span class="lab"><font color="Green" title="E39">A5</font></span>: 
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">p</font>] </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a> 
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="fraenkel.html#S10" onmouseover="rs('fraenkel/S10')" onmouseout="rh()">FRAENKEL:sch 10</a>();<br/></span>
<a NAME="E8:70_1_1"/>
<span class="p1"> { <span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">p</font>] </span> } </span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">Tforms</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="70_1_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">p</font>] </span> } </span>  or <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Tforms</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:70_1_1_3"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">p</font>] </span> } </span> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Tforms</font></span><br/>

<a NAME="E2:70_1_1_3"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a>  st <br/>( <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">p</font>] )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:70_1_1_3"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Tforms</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">uns</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a>  : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">p</font>] </span> } </span>  as   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:70_1_1"><span class="lab"><font color="Green" title="E39">A5</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">e</font> =  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  as    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c4">T</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="trees_1.html#T22" onmouseover="rs('trees_1/T22')" onmouseout="rh()">TREES_1:22</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">f</font> =  the   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>,<span class="p1">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c4">T</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c12">m</font> =  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c9">uns</font>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c13">g</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><br/><a NAME="E12:70_1_1"/><span class="lab"><font color="Green" title="E40">A6</font></span>: 
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c13">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">uns</font>
 <span class="kw">and </span><br/><a NAME="E13:70_1_1"/><span class="lab"><font color="Green" title="E41">A7</font></span>: 
<font color="Maroon" title="c13">g</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T58" onmouseover="rs('finseq_4/T58')" onmouseout="rh()">FINSEQ_4:58</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">g</font> = <font color="Maroon" title="c13">g</font> as   <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:70_1_1"><span class="lab"><font color="Green" title="E40">A6</font></span></a>, <a class="txt" href="#E13:70_1_1"><span class="lab"><font color="Green" title="E41">A7</font></span></a>, <a class="ref" href="finseq_1.html#D4" onmouseover="rs('finseq_1/D4')" onmouseout="rh()">FINSEQ_1:def 4</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>,   <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c4">T</font>,   <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c4">T</font>] <span class="kw">means </span>( (  not <font color="Maroon" title="c14">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">P</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p3">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c9">uns</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c2">T</font></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> implies <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c2">T</font> ) &amp; ( <font color="Maroon" title="c14">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">P</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p3">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c9">uns</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c2">T</font></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> implies ( <font color="Maroon" title="c2">T</font> <a href="trees_1.html#NR3" title="TREES_1:NR.3">is_a_proper_prefix_of</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="ltlaxio4.html#K2" title="LTLAXIO4:func.2">rarg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">P</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p4">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c9">uns</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> ) ) );<br/>
<a NAME="E15:70_1_1"/><span class="lab"><font color="Green" title="E42">A8</font></span>: 
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c14">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c9">uns</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:70_1_1"><span class="lab"><font color="Green" title="E40">A6</font></span></a>, <a class="ref" href="pre_poly.html#T19" onmouseover="rs('pre_poly/T19')" onmouseout="rh()">PRE_POLY:19</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E43">A9</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="70_1_1_4"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b2">x</font> being   <a href="trees_4.html#NM1" title="TREES_4:NM.1">Node</a> of <font color="Maroon" title="c4">T</font>  ex <font color="Olive" title="b3">y</font> being   <a href="trees_4.html#NM1" title="TREES_4:NM.1">Node</a> of <font color="Maroon" title="c4">T</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">y</font>]</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c15">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font> being   <a href="trees_4.html#NM1" title="TREES_4:NM.1">Node</a> of <font color="Maroon" title="c4">T</font>  ex <font color="Olive" title="b2">y</font> being   <a href="trees_4.html#NM1" title="TREES_4:NM.1">Node</a> of <font color="Maroon" title="c4">T</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b2">y</font>,<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font>]</span><br/><span class="kw">let </span><font color="Maroon" title="c16">x</font> be   <a href="trees_4.html#NM1" title="TREES_4:NM.1">Node</a> of <font color="Maroon" title="c4">T</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="trees_4.html#NM1" title="TREES_4:NM.1">Node</a> of <font color="Maroon" title="c4">T</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">y</font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>]</span><br/><a NAME="E1:70_1_1_4"/><span class="lab"><font color="Green" title="E44">A10</font></span>: 
 ex <font color="Olive" title="b1">t</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <font color="Maroon" title="c16">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D1" onmouseover="rs('xboole_0/D1')" onmouseout="rh()">XBOOLE_0:def 1</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c17">gk</font> = <font color="Maroon" title="c14">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">n</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p3">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c9">uns</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_1_4_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:70_1_1_4_1"/>
(  not <font color="Maroon" title="c14">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">n</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p3">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c9">uns</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c16">x</font></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> or <font color="Maroon" title="c14">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">n</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p3">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c9">uns</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c16">x</font></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_1_4_1_1"><span class="kw">suppose </span></a><a NAME="E1:70_1_1_4_1_1"/>
 not <font color="Maroon" title="c14">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">n</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p3">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c9">uns</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c16">x</font></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="trees_4.html#NM1" title="TREES_4:NM.1">Node</a> of <font color="Maroon" title="c4">T</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">y</font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>]</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:70_1_1_4_1_1"/>
 ex <font color="Olive" title="b1">y</font> being   <a href="trees_4.html#NM1" title="TREES_4:NM.1">Node</a> of <font color="Maroon" title="c4">T</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c15">n</font>,<font color="Maroon" title="c16">x</font>,<font color="Olive" title="b1">y</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70_1_1_4"><span class="lab"><font color="Green" title="E44">A10</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_1_4_1_2"><span class="kw">suppose </span></a><a NAME="E1:70_1_1_4_1_2"/><span class="lab"><font color="Green" title="E45">A11</font></span>: 
<font color="Maroon" title="c14">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">n</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p3">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c9">uns</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c16">x</font></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="trees_4.html#NM1" title="TREES_4:NM.1">Node</a> of <font color="Maroon" title="c4">T</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">y</font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>]</span><br/><div class="add"><a NAME="E2:70_1_1_4_1_2"/><span class="lab"><font color="Green" title="E46">A12</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">n</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p2">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c9">uns</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T25" onmouseover="rs('nat_1/T25')" onmouseout="rh()">NAT_1:25</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_1_4_1_2_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:70_1_1_4_1_2_1"/>
( <font color="Maroon" title="c9">uns</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c9">uns</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_1_4_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:70_1_1_4_1_2_1_1"/>
<font color="Maroon" title="c9">uns</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="trees_4.html#NM1" title="TREES_4:NM.1">Node</a> of <font color="Maroon" title="c4">T</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">y</font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>]</span><br/><div class="add"><a NAME="E2:70_1_1_4_1_2_1_1"/><span class="kw">then </span>
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c14">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:70_1_1"><span class="lab"><font color="Green" title="E40">A6</font></span></a>, <a class="ref" href="relat_1.html#T42" onmouseover="rs('relat_1/T42')" onmouseout="rh()">RELAT_1:42</a></span>;<br/><a NAME="E3:70_1_1_4_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A13</font></span>: 
<font color="Maroon" title="c14">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">n</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p3">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c9">uns</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D2" onmouseover="rs('funct_1/D2')" onmouseout="rh()">FUNCT_1:def 2</a></span>;<br/><a NAME="E4:70_1_1_4_1_2_1_1"/>
 <a href="afinsq_1.html#K1" title="AFINSQ_1:func.1">len</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a> <a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ;<br/><span class="kw">hence </span><a NAME="E5:70_1_1_4_1_2_1_1"/>
 ex <font color="Olive" title="b1">y</font> being   <a href="trees_4.html#NM1" title="TREES_4:NM.1">Node</a> of <font color="Maroon" title="c4">T</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c15">n</font>,<font color="Maroon" title="c16">x</font>,<font color="Olive" title="b1">y</font>]
 <span class="kw">by</span> <span class="lab"><a class="ref" href="hilbert2.html#T10" onmouseover="rs('hilbert2/T10')" onmouseout="rh()">HILBERT2:10</a>, <a class="txt" href="#E3:70_1_1_4_1_2_1_1"><span class="lab"><font color="Green" title="E47">A13</font></span></a>, <a class="txt" href="#E1:70_1_1_4_1_2"><span class="lab"><font color="Green" title="E45">A11</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_1_4_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:70_1_1_4_1_2_1_2"/>
<font color="Maroon" title="c9">uns</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being   <a href="trees_4.html#NM1" title="TREES_4:NM.1">Node</a> of <font color="Maroon" title="c4">T</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">y</font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>]</span><br/><div class="add"><a NAME="E2:70_1_1_4_1_2_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c15">n</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p2">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c9">uns</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c9">uns</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T1" onmouseover="rs('nat_d/T1')" onmouseout="rh()">NAT_D:1</a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E3:70_1_1_4_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A14</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c15">n</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p2">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c9">uns</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c14">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:70_1_1"><span class="lab"><font color="Green" title="E42">A8</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a>, <a class="txt" href="#E2:70_1_1_4_1_2"><span class="lab"><font color="Green" title="E46">A12</font></span></a></span>;<br/><a NAME="E4:70_1_1_4_1_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c14">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">n</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p3">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c9">uns</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">uns</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T3" onmouseover="rs('funct_1/T3')" onmouseout="rh()">FUNCT_1:3</a>, <a class="txt" href="#E12:70_1_1"><span class="lab"><font color="Green" title="E40">A6</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">puq</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a> <span class="kw"> such that </span><br/><a NAME="E6:70_1_1_4_1_2_1_2"/><span class="lab"><font color="Green" title="E48">A15</font></span>: 
<font color="Maroon" title="c14">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">n</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p3">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c9">uns</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">puq</font>
 <span class="kw">and </span><br/><a NAME="E7:70_1_1_4_1_2_1_2"/>
<font color="Maroon" title="c18">puq</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">Tforms</font>
 <span class="kw">and </span><br/><a NAME="E8:70_1_1_4_1_2_1_2"/><span class="lab"><font color="Green" title="E49">A16</font></span>: 
<font color="Maroon" title="c18">puq</font> is  <a href="ltlaxio4.html#NV1" title="LTLAXIO4:NV.1">s-until</a> 
 ;<br/><span class="kw">consider </span><font color="Maroon" title="c19">p</font>, <font color="Maroon" title="c20">q</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a> <span class="kw"> such that </span><br/><a NAME="E10:70_1_1_4_1_2_1_2"/><span class="lab"><font color="Green" title="E50">A17</font></span>: 
<font color="Maroon" title="c14">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">n</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p3">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c9">uns</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">p</font> <a href="ltlaxio1.html#NK3" title="LTLAXIO1:NK.3">'U'</a> <font color="Maroon" title="c20">q</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:70_1_1_4_1_2_1_2"><span class="lab"><font color="Green" title="E48">A15</font></span></a>, <a class="txt" href="#E8:70_1_1_4_1_2_1_2"><span class="lab"><font color="Green" title="E49">A16</font></span></a>, <a class="ref" href="hilbert2.html#D6" onmouseover="rs('hilbert2/D6')" onmouseout="rh()">HILBERT2:def 6</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c21">R</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ltlaxio4.html#K11" title="LTLAXIO4:func.11">rngr</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="ltlaxio4.html#K10" title="LTLAXIO4:func.10">|</a> <font color="Maroon" title="c16">x</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E12:70_1_1_4_1_2_1_2"/><span class="lab"><font color="Green" title="E51">A18</font></span>: 
<font color="Maroon" title="c20">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">R</font> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70_1_1_4_1_2"><span class="lab"><font color="Green" title="E45">A11</font></span></a>, <a class="ref" href="ltlaxio4.html#T32" target="_self" onmouseover="rs('ltlaxio4/T32')" onmouseout="rh()">Th32</a>, <a class="txt" href="#E10:70_1_1_4_1_2_1_2"><span class="lab"><font color="Green" title="E50">A17</font></span></a></span>;<br/><a NAME="E13:70_1_1_4_1_2_1_2"/>
<font color="Maroon" title="c21">R</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="ltlaxio4.html#K11" title="LTLAXIO4:func.11">rngr</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="ltlaxio4.html#K10" title="LTLAXIO4:func.10">|</a> <font color="Maroon" title="c16">x</font></span>)</span>
 ;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c22">t</font> being   <a href="trees_4.html#NM1" title="TREES_4:NM.1">Node</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="ltlaxio4.html#K10" title="LTLAXIO4:func.10">|</a> <font color="Maroon" title="c16">x</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E15:70_1_1_4_1_2_1_2"/><span class="lab"><font color="Green" title="E52">A19</font></span>: 
<font color="Maroon" title="c21">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="ltlaxio4.html#K10" title="LTLAXIO4:func.10">|</a> <font color="Maroon" title="c16">x</font></span>)</span> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c22">t</font>
 <span class="kw">and </span><br/><a NAME="E16:70_1_1_4_1_2_1_2"/><span class="lab"><font color="Green" title="E53">A20</font></span>: 
<font color="Maroon" title="c22">t</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ;<br/><a NAME="E17:70_1_1_4_1_2_1_2"/>
<font color="Maroon" title="c22">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="ltlaxio4.html#K10" title="LTLAXIO4:func.10">|</a> <font color="Maroon" title="c16">x</font></span>)</span>
 ;<br/><a NAME="E18:70_1_1_4_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A21</font></span>: 
<font color="Maroon" title="c22">t</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c4">T</font></span>)</span> <a href="trees_1.html#K4" title="TREES_1:func.4">|</a> <font color="Maroon" title="c16">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="trees_2.html#D10" onmouseover="rs('trees_2/D10')" onmouseout="rh()">TREES_2:def 10</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c23">y</font> = <font color="Maroon" title="c16">x</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Maroon" title="c22">t</font> as   <a href="trees_4.html#NM1" title="TREES_4:NM.1">Node</a> of <font color="Maroon" title="c4">T</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="trees_1.html#D6" onmouseover="rs('trees_1/D6')" onmouseout="rh()">TREES_1:def 6</a></span>;<br/><a NAME="E20:70_1_1_4_1_2_1_2"/><span class="lab"><font color="Green" title="E55">A22</font></span>: 
<font color="Maroon" title="c16">x</font> <a href="trees_1.html#NR3" title="TREES_1:NR.3">is_a_proper_prefix_of</a> <font color="Maroon" title="c23">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="trees_1.html#T10" onmouseover="rs('trees_1/T10')" onmouseout="rh()">TREES_1:10</a>, <a class="txt" href="#E16:70_1_1_4_1_2_1_2"><span class="lab"><font color="Green" title="E53">A20</font></span></a></span>;<br/><a NAME="E21:70_1_1_4_1_2_1_2"/>
<font color="Maroon" title="c14">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c15">n</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p3">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c9">uns</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">puq</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:70_1_1_4_1_2_1_2"><span class="lab"><font color="Green" title="E48">A15</font></span></a>, <a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a>, <a class="txt" href="#E3:70_1_1_4_1_2_1_2"><span class="lab"><font color="Green" title="E47">A14</font></span></a></span>;<br/><a NAME="E22:70_1_1_4_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A23</font></span>: 
 <a href="ltlaxio4.html#K2" title="LTLAXIO4:func.2">rarg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c15">n</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p4">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c9">uns</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">q</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio4.html#D1" target="_self" onmouseover="rs('ltlaxio4/D1')" onmouseout="rh()">Def1</a>, <a class="txt" href="#E6:70_1_1_4_1_2_1_2"><span class="lab"><font color="Green" title="E48">A15</font></span></a>, <a class="txt" href="#E8:70_1_1_4_1_2_1_2"><span class="lab"><font color="Green" title="E49">A16</font></span></a>, <a class="txt" href="#E10:70_1_1_4_1_2_1_2"><span class="lab"><font color="Green" title="E50">A17</font></span></a></span>;<br/><a NAME="E23:70_1_1_4_1_2_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c23">y</font></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">R</font> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:70_1_1_4_1_2_1_2"><span class="lab"><font color="Green" title="E52">A19</font></span></a>, <a class="ref" href="trees_2.html#D10" onmouseover="rs('trees_2/D10')" onmouseout="rh()">TREES_2:def 10</a>, <a class="txt" href="#E18:70_1_1_4_1_2_1_2"><span class="lab"><font color="Green" title="E54">A21</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E24:70_1_1_4_1_2_1_2"/>
 ex <font color="Olive" title="b1">y</font> being   <a href="trees_4.html#NM1" title="TREES_4:NM.1">Node</a> of <font color="Maroon" title="c4">T</font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c15">n</font>,<font color="Maroon" title="c16">x</font>,<font color="Olive" title="b1">y</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:70_1_1_4_1_2_1_2"><span class="lab"><font color="Green" title="E51">A18</font></span></a>, <a class="txt" href="#E22:70_1_1_4_1_2_1_2"><span class="lab"><font color="Green" title="E56">A23</font></span></a>, <a class="txt" href="#E1:70_1_1_4_1_2"><span class="lab"><font color="Green" title="E45">A11</font></span></a>, <a class="txt" href="#E20:70_1_1_4_1_2_1_2"><span class="lab"><font color="Green" title="E55">A22</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c15">p</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">sequence</a> of <span class="p1">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c4">T</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E18:70_1_1"/><span class="lab"><font color="Green" title="E44">A24</font></span>: 
( <font color="Maroon" title="c15">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">e</font> &amp; (  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font>,<font color="Maroon" title="c15">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">n</font>,<font color="Maroon" title="c15">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>] ) )
 <span class="kw">from</span> <span class="lab"><a class="ref" href="recdef_1.html#S2" onmouseover="rs('recdef_1/S2')" onmouseout="rh()">RECDEF_1:sch 2</a>(<a class="txt" href="#E16:70_1_1"><span class="lab"><font color="Green" title="E43">A9</font></span></a>);<br/></span>
<span class="kw">defpred </span><font color="Maroon">S<sub>3</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>] <span class="kw">means </span><font color="Maroon" title="c1">P</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"><font color="Maroon" title="c15">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c1">P</font></span>)</span>;<br/>
<div><span class="lab"><font color="Green" title="E45">A25</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="70_1_1_5"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">k</font>]</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c16">k</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>]</span><br/><span class="kw">defpred </span><font color="Maroon">S<sub>4</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>] <span class="kw">means </span> ex <font color="Olive" title="b1">t</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>( <span class="p1">(<span class="default"><font color="Maroon" title="c15">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">P</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <font color="Olive" title="b1">t</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  );<br/><span class="kw">reconsider </span><font color="Maroon" title="c17">pk</font> = <font color="Maroon" title="c15">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c16">k</font>, <font color="Maroon" title="c18">pk1</font> = <font color="Maroon" title="c15">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> as   <a href="trees_4.html#NM1" title="TREES_4:NM.1">Node</a> of <font color="Maroon" title="c4">T</font> ;<br/><a NAME="E2:70_1_1_5"/><span class="lab"><font color="Green" title="E46">A26</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c16">k</font>,<font color="Maroon" title="c17">pk</font>,<font color="Maroon" title="c18">pk1</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:70_1_1"><span class="lab"><font color="Green" title="E44">A24</font></span></a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_1_5_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:70_1_1_5_1"/>
(  not <font color="Maroon" title="c14">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p3">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c9">uns</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c17">pk</font></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> or <font color="Maroon" title="c14">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p3">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c9">uns</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c17">pk</font></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_1_5_1_1"><span class="kw">suppose </span></a><a NAME="E1:70_1_1_5_1_1"/>
 not <font color="Maroon" title="c14">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p3">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c9">uns</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c17">pk</font></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>]</span><br/><div class="add"><a NAME="E2:70_1_1_5_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <font color="Maroon" title="c18">pk1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">pk</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Olive" title="b1">n</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> &amp; <font color="Maroon" title="c17">pk</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Olive" title="b1">n</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c4">T</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:70_1_1_5"><span class="lab"><font color="Green" title="E46">A26</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:70_1_1_5_1_1"/>
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c16">k</font>]
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_1_5_1_2"><span class="kw">suppose </span></a><a NAME="E1:70_1_1_5_1_2"/>
<font color="Maroon" title="c14">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p3">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c9">uns</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c17">pk</font></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>]</span><br/><div class="add"><a NAME="E2:70_1_1_5_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A27</font></span>: 
<font color="Maroon" title="c17">pk</font> <a href="trees_1.html#NR3" title="TREES_1:NR.3">is_a_proper_prefix_of</a> <font color="Maroon" title="c18">pk1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:70_1_1"><span class="lab"><font color="Green" title="E44">A24</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c19">r</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><br/><a NAME="E4:70_1_1_5_1_2"/><span class="lab"><font color="Green" title="E48">A28</font></span>: 
<font color="Maroon" title="c18">pk1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">pk</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c19">r</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="trees_1.html#T1" onmouseover="rs('trees_1/T1')" onmouseout="rh()">TREES_1:1</a></span>;<br/><a NAME="E5:70_1_1_5_1_2"/>
<font color="Maroon" title="c19">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T34" onmouseover="rs('finseq_1/T34')" onmouseout="rh()">FINSEQ_1:34</a>, <a class="txt" href="#E2:70_1_1_5_1_2"><span class="lab"><font color="Green" title="E47">A27</font></span></a>, <a class="txt" href="#E4:70_1_1_5_1_2"><span class="lab"><font color="Green" title="E48">A28</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:70_1_1_5_1_2"/>
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c16">k</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:70_1_1_5_1_2"><span class="lab"><font color="Green" title="E48">A28</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E46">A29</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="70_1_1_6"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c16">k</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c16">k</font>] implies <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/><span class="kw">assume </span><a NAME="E1:70_1_1_6"/><span class="lab"><font color="Green" title="E47">A30</font></span>: 
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c16">k</font>]
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c17">pk</font> = <font color="Maroon" title="c15">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c16">k</font>, <font color="Maroon" title="c18">pk1</font> = <font color="Maroon" title="c15">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> as   <a href="trees_4.html#NM1" title="TREES_4:NM.1">Node</a> of <font color="Maroon" title="c4">T</font> ;<br/><span class="kw">consider </span><font color="Maroon" title="c19">t</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><br/><a NAME="E4:70_1_1_6"/><span class="lab"><font color="Green" title="E48">A31</font></span>: 
<font color="Maroon" title="c18">pk1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">pk</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c19">t</font>
 <span class="kw">and </span><br/><a NAME="E5:70_1_1_6"/><span class="lab"><font color="Green" title="E49">A32</font></span>: 
<font color="Maroon" title="c19">t</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:70_1_1"><span class="lab"><font color="Green" title="E45">A25</font></span></a></span>;<br/><a NAME="E6:70_1_1_6"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">t</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:70_1_1_6"><span class="lab"><font color="Green" title="E49">A32</font></span></a>, <a class="ref" href="nat_1.html#T25" onmouseover="rs('nat_1/T25')" onmouseout="rh()">NAT_1:25</a></span>;<br/><a NAME="E7:70_1_1_6"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">pk</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">t</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70_1_1_6"><span class="lab"><font color="Green" title="E47">A30</font></span></a>, <a class="ref" href="xreal_1.html#T7" onmouseover="rs('xreal_1/T7')" onmouseout="rh()">XREAL_1:7</a></span>;<br/><span class="kw">hence </span><a NAME="E8:70_1_1_6"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c16">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:70_1_1_6"><span class="lab"><font color="Green" title="E48">A31</font></span></a>, <a class="ref" href="finseq_1.html#T22" onmouseover="rs('finseq_1/T22')" onmouseout="rh()">FINSEQ_1:22</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">defpred </span><font color="Maroon">S<sub>4</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>] <span class="kw">means </span> for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">P</font> holds <br/><font color="Maroon" title="c15">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c15">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c1">P</font>;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>3</sub></font>(   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> ) <span class="kw">-&gt; </span>  <a href="trees_4.html#NM1" title="TREES_4:NM.1">Node</a> of <font color="Maroon" title="c4">T</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c15">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="ltlaxio4.html#K1" title="LTLAXIO4:func.1">|</a> <font color="Maroon" title="c1">P</font>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c16">f</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">sequence</a> of <span class="p1">(<span class="default"><a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c4">T</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E22:70_1_1"/><span class="lab"><font color="Green" title="E47">A33</font></span>: 
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  holds  <font color="Maroon" title="c16">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b1">n</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S4" onmouseover="rs('funct_2/S4')" onmouseout="rh()">FUNCT_2:sch 4</a>();<br/></span>
<div><span class="lab"><font color="Green" title="E48">A34</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="70_1_1_7"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c17">k</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c17">k</font>] implies <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c17">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/><span class="kw">reconsider </span><font color="Maroon" title="c18">pk</font> = <font color="Maroon" title="c15">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c17">k</font>, <font color="Maroon" title="c19">pk1</font> = <font color="Maroon" title="c15">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> as   <a href="trees_4.html#NM1" title="TREES_4:NM.1">Node</a> of <font color="Maroon" title="c4">T</font> ;<br/><span class="kw">assume </span><a NAME="E2:70_1_1_7"/><span class="lab"><font color="Green" title="E49">A35</font></span>: 
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c17">k</font>]
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c17">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/><span class="kw">thus </span><a NAME="E3:70_1_1_7"/>
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c17">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="70_1_1_7_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c20">i</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c20">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 implies <font color="Maroon" title="c15">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c15">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )</span><br/>

<div><span class="lab"><font color="Green" title="E50">A36</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="70_1_1_7_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c20">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c17">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 implies <font color="Maroon" title="c15">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c15">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> )</span></a><div class="add"><span class="kw">assume </span><a NAME="E1:70_1_1_7_1_1"/>
<font color="Maroon" title="c20">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c17">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c15">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c15">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><br/><a NAME="E2:70_1_1_7_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c20">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E3:70_1_1_7_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A37</font></span>: 
<font color="Maroon" title="c15">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c15">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c17">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:70_1_1_7"><span class="lab"><font color="Green" title="E49">A35</font></span></a></span>;<br/><a NAME="E4:70_1_1_7_1_1"/>
 ex <font color="Olive" title="b1">t</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>( <font color="Maroon" title="c18">pk</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Olive" title="b1">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">pk1</font> &amp; <font color="Olive" title="b1">t</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:70_1_1"><span class="lab"><font color="Green" title="E45">A25</font></span></a></span>;<br/><a NAME="E5:70_1_1_7_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c18">pk</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c19">pk1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_6.html#T10" onmouseover="rs('finseq_6/T10')" onmouseout="rh()">FINSEQ_6:10</a></span>;<br/><span class="kw">hence </span><a NAME="E6:70_1_1_7_1_1"/>
<font color="Maroon" title="c15">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c15">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:70_1_1_7_1_1"><span class="lab"><font color="Green" title="E51">A37</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">assume </span><a NAME="E2:70_1_1_7_1"/>
<font color="Maroon" title="c20">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c15">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c15">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><br/>

<span class="kw">hence </span><a NAME="E3:70_1_1_7_1"/>
<font color="Maroon" title="c15">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">i</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c15">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70_1_1_7_1"><span class="lab"><font color="Green" title="E50">A36</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<a NAME="E24:70_1_1"/><span class="lab"><font color="Green" title="E49">A38</font></span>: 
<font color="Maroon">S<sub>3</sub></font>[ <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ]
 
;<br/>
<a NAME="E25:70_1_1"/><span class="lab"><font color="Green" title="E50">A39</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">n</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" onmouseover="rs('nat_1/S2')" onmouseout="rh()">NAT_1:sch 2</a>(<a class="txt" href="#E24:70_1_1"><span class="lab"><font color="Green" title="E49">A38</font></span></a>, <a class="txt" href="#E20:70_1_1"><span class="lab"><font color="Green" title="E46">A29</font></span></a>);<br/></span>
<a NAME="E26:70_1_1"/><span class="lab"><font color="Green" title="E51">A40</font></span>: 
<font color="Maroon">S<sub>4</sub></font>[ <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> ]
 
;<br/>
<a NAME="E27:70_1_1"/><span class="lab"><font color="Green" title="E52">A41</font></span>: 
 for <font color="Olive" title="b1">j</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">j</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" onmouseover="rs('nat_1/S2')" onmouseout="rh()">NAT_1:sch 2</a>(<a class="txt" href="#E26:70_1_1"><span class="lab"><font color="Green" title="E51">A40</font></span></a>, <a class="txt" href="#E23:70_1_1"><span class="lab"><font color="Green" title="E48">A34</font></span></a>);<br/></span>
<div><span class="lab"><font color="Green" title="E53">A42</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="70_1_1_8"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon" title="c16">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">f</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">k</font></span>)</span></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c17">k</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c16">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">f</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c17">k</font></span>)</span></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c18">k1</font> = <font color="Maroon" title="c17">k</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c19">fk</font> = <font color="Maroon" title="c16">f</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c17">k</font>, <font color="Maroon" title="c20">fk1</font> = <font color="Maroon" title="c16">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> as    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c4">T</font> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c21">pk</font> = <font color="Maroon" title="c15">p</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c17">k</font>, <font color="Maroon" title="c22">pk1</font> = <font color="Maroon" title="c15">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> as   <a href="trees_4.html#NM1" title="TREES_4:NM.1">Node</a> of <font color="Maroon" title="c4">T</font> ;<br/><a NAME="E4:70_1_1_8"/>
<font color="Maroon" title="c17">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <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="c22">pk1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:70_1_1"><span class="lab"><font color="Green" title="E50">A39</font></span></a></span>;<br/><a NAME="E5:70_1_1_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A43</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">pk1</font> <a href="ltlaxio4.html#K1" title="LTLAXIO4:func.1">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T59" onmouseover="rs('finseq_1/T59')" onmouseout="rh()">FINSEQ_1:59</a></span>;<br/><a NAME="E6:70_1_1_8"/>
<font color="Maroon" title="c18">k1</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="c21">pk</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:70_1_1"><span class="lab"><font color="Green" title="E50">A39</font></span></a></span>;<br/><a NAME="E7:70_1_1_8"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">pk</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c17">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T59" onmouseover="rs('finseq_1/T59')" onmouseout="rh()">FINSEQ_1:59</a></span>;<br/><a NAME="E8:70_1_1_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A44</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">fk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:70_1_1"><span class="lab"><font color="Green" title="E47">A33</font></span></a></span>;<br/><a NAME="E9:70_1_1_8"/><span class="lab"><font color="Green" title="E56">A45</font></span>: 
<font color="Maroon" title="c18">k1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c18">k1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E10:70_1_1_8"/><span class="kw">then </span>
<font color="Maroon" title="c21">pk</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c22">pk1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:70_1_1"><span class="lab"><font color="Green" title="E52">A41</font></span></a></span>;<br/><a NAME="E11:70_1_1_8"/><span class="kw">then </span>
<font color="Maroon" title="c21">pk</font> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c17">k</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c22">pk1</font> <a href="ltlaxio4.html#K1" title="LTLAXIO4:func.1">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T77" onmouseover="rs('relat_1/T77')" onmouseout="rh()">RELAT_1:77</a>, <a class="txt" href="#E9:70_1_1_8"><span class="lab"><font color="Green" title="E56">A45</font></span></a>, <a class="ref" href="finseq_1.html#T5" onmouseover="rs('finseq_1/T5')" onmouseout="rh()">FINSEQ_1:5</a></span>;<br/><a NAME="E12:70_1_1_8"/><span class="kw">then </span>
<font color="Maroon" title="c16">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">k1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c22">pk1</font> <a href="ltlaxio4.html#K1" title="LTLAXIO4:func.1">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:70_1_1"><span class="lab"><font color="Green" title="E47">A33</font></span></a></span>;<br/><a NAME="E13:70_1_1_8"/><span class="kw">then </span>
<font color="Maroon" title="c19">fk</font> <a href="tarski.html#R1" title="TARSKI:pred.1">is_a_prefix_of</a> <font color="Maroon" title="c20">fk1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:70_1_1"><span class="lab"><font color="Green" title="E47">A33</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c23">r</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><span class="kw"> such that </span><br/><a NAME="E15:70_1_1_8"/><span class="lab"><font color="Green" title="E57">A46</font></span>: 
<font color="Maroon" title="c20">fk1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">fk</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c23">r</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="trees_1.html#T1" onmouseover="rs('trees_1/T1')" onmouseout="rh()">TREES_1:1</a></span>;<br/><a NAME="E16:70_1_1_8"/>
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <font color="Maroon" title="c23">r</font> <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="c20">fk1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T30" onmouseover="rs('finseq_1/T30')" onmouseout="rh()">FINSEQ_1:30</a>, <a class="txt" href="#E15:70_1_1_8"><span class="lab"><font color="Green" title="E57">A46</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c24">r</font> = <font color="Maroon" title="c23">r</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T1" onmouseover="rs('xboole_1/T1')" onmouseout="rh()">XBOOLE_1:1</a>, <a class="ref" href="finseq_1.html#D4" onmouseover="rs('finseq_1/D4')" onmouseout="rh()">FINSEQ_1:def 4</a></span>;<br/><a NAME="E18:70_1_1_8"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c20">fk1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">fk</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c24">r</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T22" onmouseover="rs('finseq_1/T22')" onmouseout="rh()">FINSEQ_1:22</a>, <a class="txt" href="#E15:70_1_1_8"><span class="lab"><font color="Green" title="E57">A46</font></span></a></span>;<br/><a NAME="E19:70_1_1_8"/><span class="kw">then </span>
<font color="Maroon" title="c17">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c24">r</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:70_1_1_8"><span class="lab"><font color="Green" title="E54">A43</font></span></a>, <a class="txt" href="#E22:70_1_1"><span class="lab"><font color="Green" title="E47">A33</font></span></a>, <a class="txt" href="#E8:70_1_1_8"><span class="lab"><font color="Green" title="E55">A44</font></span></a></span>;<br/><a NAME="E20:70_1_1_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">Z</font></span>: 
<font color="Maroon" title="c24">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c24">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#T14" onmouseover="rs('finseq_5/T14')" onmouseout="rh()">FINSEQ_5:14</a></span>;<br/><a NAME="E21:70_1_1_8"/><span class="kw">then </span>
1 <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="c24">r</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T39" onmouseover="rs('finseq_1/T39')" onmouseout="rh()">FINSEQ_1:39</a></span>;<br/><a NAME="E22:70_1_1_8"/><span class="kw">then </span>
1 <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c24">r</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a></span>;<br/><a NAME="E23:70_1_1_8"/><span class="kw">then </span>
<font color="Maroon" title="c24">r</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">r</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>;<br/><span class="kw">hence </span><a NAME="E24:70_1_1_8"/>
<font color="Maroon" title="c16">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">f</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c17">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:70_1_1_8"><span class="lab"><font color="Green" title="E57">A46</font></span></a>, <a class="txt" href="#E20:70_1_1_8"><span class="lab"><font color="Green" title="E58">Z</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<font color="Maroon" title="c16">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> = 
<font color="Maroon">H<sub>3</sub></font>( <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> )
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:70_1_1"><span class="lab"><font color="Green" title="E47">A33</font></span></a></span>
<br/>.= 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:70_1_1"><span class="lab"><font color="Green" title="E44">A24</font></span></a></span>
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c17">f</font> = <font color="Maroon" title="c16">f</font> as    <a href="ltlaxio4.html#M4" title="LTLAXIO4:mode.4">path</a> of <font color="Maroon" title="c4">T</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:70_1_1"><span class="lab"><font color="Green" title="E53">A42</font></span></a>, <a class="ref" href="ltlaxio4.html#D13" target="_self" onmouseover="rs('ltlaxio4/D13')" onmouseout="rh()">Def13</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E54">A47</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="70_1_1_10"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  holds  <font color="Maroon">H<sub>3</sub></font>( <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">n</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c18">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">H<sub>3</sub></font>( <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">n</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">n</font></span><br/><span class="kw">reconsider </span><font color="Maroon" title="c19">pn</font> = <font color="Maroon" title="c15">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">n</font> as   <a href="trees_4.html#NM1" title="TREES_4:NM.1">Node</a> of <font color="Maroon" title="c4">T</font> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c20">pln</font> = <font color="Maroon" title="c15">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">pn</font></span>)</span> as   <a href="trees_4.html#NM1" title="TREES_4:NM.1">Node</a> of <font color="Maroon" title="c4">T</font> ;<br/><a NAME="E3:70_1_1_10"/>
<font color="Maroon" title="c19">pn</font> <a href="relset_1.html#R1" title="RELSET_1:pred.1">c=</a> <font color="Maroon" title="c20">pln</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:70_1_1"><span class="lab"><font color="Green" title="E50">A39</font></span></a>, <a class="txt" href="#E27:70_1_1"><span class="lab"><font color="Green" title="E52">A41</font></span></a></span>;<br/><a NAME="E4:70_1_1_10"/><span class="kw">then </span>
(  <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="c19">pn</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c19">pn</font> &amp;  ex <font color="Olive" title="b1">t</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <font color="Maroon" title="c20">pln</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">pn</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Olive" title="b1">t</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" onmouseover="rs('finseq_1/D3')" onmouseout="rh()">FINSEQ_1:def 3</a>, <a class="ref" href="trees_1.html#T1" onmouseover="rs('trees_1/T1')" onmouseout="rh()">TREES_1:1</a></span>;<br/><span class="kw">then </span><font color="Maroon" title="c20">pln</font> <a href="ltlaxio4.html#K1" title="LTLAXIO4:func.1">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">pn</font></span>)</span> = 
<font color="Maroon" title="c19">pn</font> <a href="ltlaxio4.html#K1" title="LTLAXIO4:func.1">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c19">pn</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_6.html#T11" onmouseover="rs('finseq_6/T11')" onmouseout="rh()">FINSEQ_6:11</a></span>
<br/>.= 
<font color="Maroon" title="c19">pn</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T58" onmouseover="rs('finseq_1/T58')" onmouseout="rh()">FINSEQ_1:58</a></span>
;<br/><span class="kw">hence </span><a NAME="E6:70_1_1_10"/>
<font color="Maroon">H<sub>3</sub></font>( <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">n</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c18">n</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E32:70_1_1"/>
<font color="Maroon" title="c17">f</font> is  <a href="ltlaxio4.html#V2" title="LTLAXIO4:attr.2">complete</a> 
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="70_1_1_11"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c18">A</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="ltlaxio4.html#D14" target="_self" onmouseover="rs('ltlaxio4/D14')" onmouseout="rh()">LTLAXIO4:def 14</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">B</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a> <br/>  for <font color="Olive" title="b2">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon" title="c18">A</font> <a href="ltlaxio1.html#NK3" title="LTLAXIO1:NK.3">'U'</a> <font color="Olive" title="b1">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b2">i</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> holds <br/> ex <font color="Olive" title="b3">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  st <br/>( <font color="Olive" title="b3">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b1">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">j</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> &amp; (  for <font color="Olive" title="b4">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b4">k</font> &amp; <font color="Olive" title="b4">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">j</font> holds <br/><font color="Maroon" title="c18">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">k</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c19">B</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">i</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Maroon" title="c18">A</font> <a href="ltlaxio1.html#NK3" title="LTLAXIO1:NK.3">'U'</a> <font color="Maroon" title="c19">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">i</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> holds <br/> ex <font color="Olive" title="b2">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  st <br/>( <font color="Olive" title="b2">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Olive" title="b1">i</font> &amp; <font color="Maroon" title="c19">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">j</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> &amp; (  for <font color="Olive" title="b3">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b3">k</font> &amp; <font color="Olive" title="b3">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">j</font> holds <br/><font color="Maroon" title="c18">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b3">k</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c20">i</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c18">A</font> <a href="ltlaxio1.html#NK3" title="LTLAXIO1:NK.3">'U'</a> <font color="Maroon" title="c19">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Maroon" title="c20">i</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> implies  ex <font color="Olive" title="b1">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  st <br/>( <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c20">i</font> &amp; <font color="Maroon" title="c19">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">j</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> &amp; (  for <font color="Olive" title="b2">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st <font color="Maroon" title="c20">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">k</font> &amp; <font color="Olive" title="b2">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">j</font> holds <br/><font color="Maroon" title="c18">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">k</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> ) ) )</span><br/>





<span class="kw">set </span><font color="Maroon" title="c21">aub</font> = <font color="Maroon" title="c18">A</font> <a href="ltlaxio1.html#NK3" title="LTLAXIO1:NK.3">'U'</a> <font color="Maroon" title="c19">B</font>;<br/>
<span class="kw">assume </span><a NAME="E1:70_1_1_11"/><span class="lab"><font color="Green" title="E55">A48</font></span>: 
<font color="Maroon" title="c18">A</font> <a href="ltlaxio1.html#NK3" title="LTLAXIO1:NK.3">'U'</a> <font color="Maroon" title="c19">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c20">i</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  st <br/>( <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c20">i</font> &amp; <font color="Maroon" title="c19">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">j</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> &amp; (  for <font color="Olive" title="b2">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   st <font color="Maroon" title="c20">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">k</font> &amp; <font color="Olive" title="b2">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">j</font> holds <br/><font color="Maroon" title="c18">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">k</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> ) )</span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>5</sub></font>[  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>] <span class="kw">means </span>( <font color="Maroon" title="c1">P</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c20">i</font> implies (  not <font color="Maroon" title="c19">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c1">P</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> &amp; <font color="Maroon" title="c18">A</font> <a href="ltlaxio1.html#NK3" title="LTLAXIO1:NK.3">'U'</a> <font color="Maroon" title="c19">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c1">P</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> &amp; <font color="Maroon" title="c18">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c1">P</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> ) );<br/>
<span class="kw">assume </span><a NAME="E2:70_1_1_11"/><span class="lab"><font color="Green" title="E56">A49</font></span>: 
 for <font color="Olive" title="b1">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>   holds <br/>(  not <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c20">i</font> or  not <font color="Maroon" title="c19">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Olive" title="b1">j</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> or  ex <font color="Olive" title="b2">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  st <br/>( <font color="Maroon" title="c20">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">k</font> &amp; <font color="Olive" title="b2">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">j</font> &amp;  not <font color="Maroon" title="c18">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Olive" title="b2">k</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> ) )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<div><span class="lab"><font color="Green" title="E57">A50</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="70_1_1_11_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st (  for <font color="Olive" title="b2">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">k</font> holds <br/><font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b2">n</font>] ) holds <br/><font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b1">k</font>]</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c22">k</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c22">k</font> holds <br/><font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b1">n</font>] ) implies <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c22">k</font>] )</span><br/><span class="kw">assume </span><a NAME="E1:70_1_1_11_1"/><span class="lab"><font color="Green" title="E58">A51</font></span>: 
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c22">k</font> holds <br/><font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b1">n</font>]
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c22">k</font>]</span><br/><span class="kw">thus </span><a NAME="E2:70_1_1_11_1"/>
<font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c22">k</font>]
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="70_1_1_11_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c23">k1</font> = <font color="Maroon" title="c22">k</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" onmouseover="rs('ordinal1/D12')" onmouseout="rh()">ORDINAL1:def 12</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c24">fk</font> = <font color="Maroon" title="c4">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c23">k1</font></span>)</span>;<br/>
<span class="kw">assume </span><a NAME="E2:70_1_1_11_1_1"/><span class="lab"><font color="Green" title="E59">A52</font></span>: 
<font color="Maroon" title="c22">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c20">i</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c19">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c22">k</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> &amp; <font color="Maroon" title="c18">A</font> <a href="ltlaxio1.html#NK3" title="LTLAXIO1:NK.3">'U'</a> <font color="Maroon" title="c19">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c22">k</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> &amp; <font color="Maroon" title="c18">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c22">k</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> )</span><br/>

<a NAME="E3:70_1_1_11_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c23">k1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T25" onmouseover="rs('nat_1/T25')" onmouseout="rh()">NAT_1:25</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c25">kk</font> = <font color="Maroon" title="c23">k1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T21" onmouseover="rs('nat_1/T21')" onmouseout="rh()">NAT_1:21</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c26">fkk</font> = <font color="Maroon" title="c4">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c25">kk</font></span>)</span>;<br/>
<a NAME="E5:70_1_1_11_1_1"/><span class="lab"><font color="Green" title="E60">A53</font></span>: 
<font color="Maroon" title="c25">kk</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">k</font>
 
;<br/>
<a NAME="E6:70_1_1_11_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A54</font></span>: 
<font color="Maroon" title="c25">kk</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c20">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a>, <a class="txt" href="#E2:70_1_1_11_1_1"><span class="lab"><font color="Green" title="E59">A52</font></span></a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_1_11_1_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:70_1_1_11_1_1_1"/>
( <font color="Maroon" title="c25">kk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i</font> or <font color="Maroon" title="c25">kk</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c20">i</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:70_1_1_11_1_1"><span class="lab"><font color="Green" title="E61">A54</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_1_11_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:70_1_1_11_1_1_1_1"/><span class="lab"><font color="Green" title="E62">A55</font></span>: 
<font color="Maroon" title="c25">kk</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">i</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c19">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c22">k</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> &amp; <font color="Maroon" title="c18">A</font> <a href="ltlaxio1.html#NK3" title="LTLAXIO1:NK.3">'U'</a> <font color="Maroon" title="c19">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c22">k</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> &amp; <font color="Maroon" title="c18">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c22">k</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> )</span><br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_1_11_1_1_1_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:70_1_1_11_1_1_1_1_1"/>
(  not <font color="Maroon" title="c19">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c23">k1</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> or  ex <font color="Olive" title="b1">m</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  st <br/>( <font color="Maroon" title="c20">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">m</font> &amp; <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">k1</font> &amp;  not <font color="Maroon" title="c18">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Olive" title="b1">m</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:70_1_1_11"><span class="lab"><font color="Green" title="E56">A49</font></span></a>, <a class="txt" href="#E2:70_1_1_11_1_1"><span class="lab"><font color="Green" title="E59">A52</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_1_11_1_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:70_1_1_11_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E63">A56</font></span>: 
 not <font color="Maroon" title="c19">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c23">k1</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c19">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c22">k</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> &amp; <font color="Maroon" title="c18">A</font> <a href="ltlaxio1.html#NK3" title="LTLAXIO1:NK.3">'U'</a> <font color="Maroon" title="c19">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c22">k</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> &amp; <font color="Maroon" title="c18">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c22">k</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> )</span><br/><div class="add"><a NAME="E2:70_1_1_11_1_1_1_1_1_1"/>
<font color="Maroon" title="c17">f</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c22">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c25">kk</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio4.html#D13" target="_self" onmouseover="rs('ltlaxio4/D13')" onmouseout="rh()">Def13</a>, <a class="txt" href="#E5:70_1_1_11_1_1"><span class="lab"><font color="Green" title="E60">A53</font></span></a></span>;<br/><a NAME="E3:70_1_1_11_1_1_1_1_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <font color="Maroon" title="c17">f</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c22">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c25">kk</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Olive" title="b1">n</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c25">kk</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Olive" title="b1">n</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c4">T</font> )
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c27">fk</font> = <font color="Maroon" title="c4">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c23">k1</font></span>)</span> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ltlaxio4.html#K9" title="LTLAXIO4:func.9">compn</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c25">kk</font></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio4.html#T26" target="_self" onmouseover="rs('ltlaxio4/T26')" onmouseout="rh()">Th26</a></span>;<br/><a NAME="E5:70_1_1_11_1_1_1_1_1_1"/>
( <font color="Maroon" title="c19">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">fk</font> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> or ( <font color="Maroon" title="c18">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">fk</font> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> &amp; <font color="Maroon" title="c18">A</font> <a href="ltlaxio1.html#NK3" title="LTLAXIO1:NK.3">'U'</a> <font color="Maroon" title="c19">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">fk</font> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio4.html#T25" target="_self" onmouseover="rs('ltlaxio4/T25')" onmouseout="rh()">Th25</a>, <a class="txt" href="#E1:70_1_1_11_1_1_1_1"><span class="lab"><font color="Green" title="E62">A55</font></span></a>, <a class="txt" href="#E1:70_1_1_11"><span class="lab"><font color="Green" title="E55">A48</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:70_1_1_11_1_1_1_1_1_1"/>
(  not <font color="Maroon" title="c19">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c22">k</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> &amp; <font color="Maroon" title="c18">A</font> <a href="ltlaxio1.html#NK3" title="LTLAXIO1:NK.3">'U'</a> <font color="Maroon" title="c19">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c22">k</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> &amp; <font color="Maroon" title="c18">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c22">k</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70_1_1_11_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E63">A56</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_1_11_1_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:70_1_1_11_1_1_1_1_1_2"/>
 ex <font color="Olive" title="b1">m</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  st <br/>( <font color="Maroon" title="c20">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">m</font> &amp; <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">k1</font> &amp;  not <font color="Maroon" title="c18">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Olive" title="b1">m</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c19">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c22">k</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> &amp; <font color="Maroon" title="c18">A</font> <a href="ltlaxio1.html#NK3" title="LTLAXIO1:NK.3">'U'</a> <font color="Maroon" title="c19">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c22">k</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> &amp; <font color="Maroon" title="c18">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c22">k</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:70_1_1_11_1_1_1_1_1_2"/>
(  not <font color="Maroon" title="c19">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c22">k</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> &amp; <font color="Maroon" title="c18">A</font> <a href="ltlaxio1.html#NK3" title="LTLAXIO1:NK.3">'U'</a> <font color="Maroon" title="c19">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c22">k</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> &amp; <font color="Maroon" title="c18">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c22">k</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a>, <a class="txt" href="#E1:70_1_1_11_1_1_1_1"><span class="lab"><font color="Green" title="E62">A55</font></span></a>, <a class="txt" href="#E5:70_1_1_11_1_1"><span class="lab"><font color="Green" title="E60">A53</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_1_11_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:70_1_1_11_1_1_1_2"/><span class="lab"><font color="Green" title="E62">A57</font></span>: 
<font color="Maroon" title="c25">kk</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c20">i</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c19">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c22">k</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> &amp; <font color="Maroon" title="c18">A</font> <a href="ltlaxio1.html#NK3" title="LTLAXIO1:NK.3">'U'</a> <font color="Maroon" title="c19">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c22">k</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> &amp; <font color="Maroon" title="c18">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c22">k</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> )</span><br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_1_11_1_1_1_2_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:70_1_1_11_1_1_1_2_1"/>
(  not <font color="Maroon" title="c19">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c23">k1</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> or  ex <font color="Olive" title="b1">m</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  st <br/>( <font color="Maroon" title="c20">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">m</font> &amp; <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">k1</font> &amp;  not <font color="Maroon" title="c18">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Olive" title="b1">m</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:70_1_1_11"><span class="lab"><font color="Green" title="E56">A49</font></span></a>, <a class="txt" href="#E2:70_1_1_11_1_1"><span class="lab"><font color="Green" title="E59">A52</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_1_11_1_1_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:70_1_1_11_1_1_1_2_1_1"/><span class="lab"><font color="Green" title="E63">A58</font></span>: 
 not <font color="Maroon" title="c19">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c23">k1</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c19">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c22">k</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> &amp; <font color="Maroon" title="c18">A</font> <a href="ltlaxio1.html#NK3" title="LTLAXIO1:NK.3">'U'</a> <font color="Maroon" title="c19">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c22">k</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> &amp; <font color="Maroon" title="c18">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c22">k</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> )</span><br/><div class="add"><a NAME="E2:70_1_1_11_1_1_1_2_1_1"/>
<font color="Maroon" title="c17">f</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c22">k</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="trees_2.html#K1" title="TREES_2:func.1">succ</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c25">kk</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio4.html#D13" target="_self" onmouseover="rs('ltlaxio4/D13')" onmouseout="rh()">Def13</a>, <a class="txt" href="#E5:70_1_1_11_1_1"><span class="lab"><font color="Green" title="E60">A53</font></span></a></span>;<br/><a NAME="E3:70_1_1_11_1_1_1_2_1_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>( <font color="Maroon" title="c17">f</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c22">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c25">kk</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Olive" title="b1">n</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c25">kk</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Olive" title="b1">n</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <font color="Maroon" title="c4">T</font> )
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c27">fk</font> = <font color="Maroon" title="c4">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c23">k1</font></span>)</span> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="ltlaxio4.html#K9" title="LTLAXIO4:func.9">compn</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c25">kk</font></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio4.html#T26" target="_self" onmouseover="rs('ltlaxio4/T26')" onmouseout="rh()">Th26</a></span>;<br/><a NAME="E5:70_1_1_11_1_1_1_2_1_1"/>
<font color="Maroon" title="c25">kk</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c22">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T146" onmouseover="rs('xreal_1/T146')" onmouseout="rh()">XREAL_1:146</a></span>;<br/><a NAME="E6:70_1_1_11_1_1_1_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c18">A</font> <a href="ltlaxio1.html#NK3" title="LTLAXIO1:NK.3">'U'</a> <font color="Maroon" title="c19">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c25">kk</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70_1_1_11_1"><span class="lab"><font color="Green" title="E58">A51</font></span></a>, <a class="txt" href="#E1:70_1_1_11_1_1_1_2"><span class="lab"><font color="Green" title="E62">A57</font></span></a></span>;<br/><a NAME="E7:70_1_1_11_1_1_1_2_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c19">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">fk</font> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> or ( <font color="Maroon" title="c18">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">fk</font> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> &amp; <font color="Maroon" title="c18">A</font> <a href="ltlaxio1.html#NK3" title="LTLAXIO1:NK.3">'U'</a> <font color="Maroon" title="c19">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c27">fk</font> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio4.html#T25" target="_self" onmouseover="rs('ltlaxio4/T25')" onmouseout="rh()">Th25</a></span>;<br/><span class="kw">hence </span><a NAME="E8:70_1_1_11_1_1_1_2_1_1"/>
(  not <font color="Maroon" title="c19">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c22">k</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> &amp; <font color="Maroon" title="c18">A</font> <a href="ltlaxio1.html#NK3" title="LTLAXIO1:NK.3">'U'</a> <font color="Maroon" title="c19">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c22">k</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> &amp; <font color="Maroon" title="c18">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c22">k</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70_1_1_11_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E63">A58</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_1_11_1_1_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:70_1_1_11_1_1_1_2_1_2"/>
 ex <font color="Olive" title="b1">m</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  st <br/>( <font color="Maroon" title="c20">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">m</font> &amp; <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c23">k1</font> &amp;  not <font color="Maroon" title="c18">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Olive" title="b1">m</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c19">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c22">k</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> &amp; <font color="Maroon" title="c18">A</font> <a href="ltlaxio1.html#NK3" title="LTLAXIO1:NK.3">'U'</a> <font color="Maroon" title="c19">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c22">k</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> &amp; <font color="Maroon" title="c18">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c22">k</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:70_1_1_11_1_1_1_2_1_2"/>
(  not <font color="Maroon" title="c19">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c22">k</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> &amp; <font color="Maroon" title="c18">A</font> <a href="ltlaxio1.html#NK3" title="LTLAXIO1:NK.3">'U'</a> <font color="Maroon" title="c19">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c22">k</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> &amp; <font color="Maroon" title="c18">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c22">k</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:70_1_1_11_1"><span class="lab"><font color="Green" title="E58">A51</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<a NAME="E4:70_1_1_11"/><span class="lab"><font color="Green" title="E58">A59</font></span>: 
 for <font color="Olive" title="b1">j</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b1">j</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S4" onmouseover="rs('nat_1/S4')" onmouseout="rh()">NAT_1:sch 4</a>(<a class="txt" href="#E3:70_1_1_11"><span class="lab"><font color="Green" title="E57">A50</font></span></a>);<br/></span>
<a NAME="E5:70_1_1_11"/>
<font color="Maroon" title="c4">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c20">i</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c4">T</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T3" onmouseover="rs('funct_1/T3')" onmouseout="rh()">FUNCT_1:3</a></span>;<br/>
<a NAME="E6:70_1_1_11"/><span class="kw">then </span>
(  <a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c20">i</font></span>)</span></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c6">YN</font> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c20">i</font></span>)</span></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ltlaxio3.html#K10" title="LTLAXIO3:func.10">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <font color="Maroon" title="c20">i</font></span>)</span></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a></span>;<br/>
<a NAME="E7:70_1_1_11"/><span class="kw">then </span>
( <font color="Maroon" title="c18">A</font> <a href="ltlaxio1.html#NK3" title="LTLAXIO1:NK.3">'U'</a> <font color="Maroon" title="c19">B</font> is  <a href="ltlaxio4.html#NV1" title="LTLAXIO4:NV.1">s-until</a>  &amp; <font color="Maroon" title="c18">A</font> <a href="ltlaxio1.html#NK3" title="LTLAXIO1:NK.3">'U'</a> <font color="Maroon" title="c19">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c7">Tforms</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="hilbert2.html#D6" onmouseover="rs('hilbert2/D6')" onmouseout="rh()">HILBERT2:def 6</a>, <a class="txt" href="#E1:70_1_1_11"><span class="lab"><font color="Green" title="E55">A48</font></span></a>, <a class="ref" href="tarski.html#D4" onmouseover="rs('tarski/D4')" onmouseout="rh()">TARSKI:def 4</a></span>;<br/>
<a NAME="E8:70_1_1_11"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A60</font></span>: 
<font color="Maroon" title="c18">A</font> <a href="ltlaxio1.html#NK3" title="LTLAXIO1:NK.3">'U'</a> <font color="Maroon" title="c19">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c9">uns</font>
 
;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c22">n</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E10:70_1_1_11"/><span class="lab"><font color="Green" title="E60">A61</font></span>: 
<font color="Maroon" title="c22">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c14">g</font>
 <span class="kw">and </span><br/><a NAME="E11:70_1_1_11"/><span class="lab"><font color="Green" title="E61">A62</font></span>: 
<font color="Maroon" title="c14">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c22">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">A</font> <a href="ltlaxio1.html#NK3" title="LTLAXIO1:NK.3">'U'</a> <font color="Maroon" title="c19">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:70_1_1"><span class="lab"><font color="Green" title="E40">A6</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c23">n</font> = <font color="Maroon" title="c22">n</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:70_1_1_11"><span class="lab"><font color="Green" title="E60">A61</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c24">k</font> = <font color="Maroon" title="c23">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:70_1_1_11"><span class="lab"><font color="Green" title="E60">A61</font></span></a>, <a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a>, <a class="ref" href="nat_1.html#T21" onmouseover="rs('nat_1/T21')" onmouseout="rh()">NAT_1:21</a></span>;<br/>
<a NAME="E14:70_1_1_11"/>
<font color="Maroon" title="c24">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <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="c14">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" onmouseover="rs('finseq_3/T25')" onmouseout="rh()">FINSEQ_3:25</a>, <a class="txt" href="#E10:70_1_1_11"><span class="lab"><font color="Green" title="E60">A61</font></span></a></span>;<br/>
<a NAME="E15:70_1_1_11"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A63</font></span>: 
<font color="Maroon" title="c24">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c14">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c25">mnk</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c9">uns</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">k</font>;<br/>
<a NAME="E16:70_1_1_11"/>
( <font color="Maroon" title="c20">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c20">i</font> &amp;  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c9">uns</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a>, <a class="txt" href="#E8:70_1_1_11"><span class="lab"><font color="Green" title="E59">A60</font></span></a>, <a class="ref" href="nat_1.html#T25" onmouseover="rs('nat_1/T25')" onmouseout="rh()">NAT_1:25</a></span>;<br/>
<a NAME="E17:70_1_1_11"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c20">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c9">uns</font></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c20">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T97" onmouseover="rs('xreal_1/T97')" onmouseout="rh()">XREAL_1:97</a></span>;<br/>
<a NAME="E18:70_1_1_11"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A64</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c9">uns</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c20">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T40" onmouseover="rs('xreal_1/T40')" onmouseout="rh()">XREAL_1:40</a></span>;<br/>
<a NAME="E19:70_1_1_11"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A65</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c9">uns</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c20">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c20">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/>
<span class="lab"><font color="Green" title="E65">A66</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c9">uns</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c20">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">k</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p2">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c9">uns</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon" title="c24">k</font> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p2">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c9">uns</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T21" onmouseover="rs('nat_d/T21')" onmouseout="rh()">NAT_D:21</a></span>
<br/>.= 
<font color="Maroon" title="c24">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T24" onmouseover="rs('nat_d/T24')" onmouseout="rh()">NAT_D:24</a>, <a class="txt" href="#E15:70_1_1_11"><span class="lab"><font color="Green" title="E62">A63</font></span></a>, <a class="txt" href="#E15:70_1_1"><span class="lab"><font color="Green" title="E42">A8</font></span></a></span>
;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c26">t</font> = <font color="Maroon" title="c15">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c9">uns</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c20">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">k</font></span>)</span>, <font color="Maroon" title="c27">t1</font> = <font color="Maroon" title="c15">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c9">uns</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p4">(<span class="default"><font color="Maroon" title="c20">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> as   <a href="trees_4.html#NM1" title="TREES_4:NM.1">Node</a> of <font color="Maroon" title="c4">T</font> ;<br/>
<a NAME="E22:70_1_1_11"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c9">uns</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">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="c26">t</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:70_1_1"><span class="lab"><font color="Green" title="E50">A39</font></span></a></span>;<br/>
<a NAME="E23:70_1_1_11"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A67</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c26">t</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c20">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:70_1_1_11"><span class="lab"><font color="Green" title="E63">A64</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<a NAME="E24:70_1_1_11"/><span class="lab"><font color="Green" title="E67">A68</font></span>: 
<font color="Maroon" title="c18">A</font> <a href="ltlaxio1.html#NK3" title="LTLAXIO1:NK.3">'U'</a> <font color="Maroon" title="c19">B</font> is  <a href="ltlaxio4.html#NV1" title="LTLAXIO4:NV.1">s-until</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="hilbert2.html#D6" onmouseover="rs('hilbert2/D6')" onmouseout="rh()">HILBERT2:def 6</a></span>;<br/>
<a NAME="E25:70_1_1_11"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c9">uns</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c20">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <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="c27">t1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:70_1_1"><span class="lab"><font color="Green" title="E50">A39</font></span></a></span>;<br/>
<a NAME="E26:70_1_1_11"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A69</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c27">t1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c20">i</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:70_1_1_11"><span class="lab"><font color="Green" title="E64">A65</font></span></a>, <a class="ref" href="xxreal_0.html#T2" onmouseover="rs('xxreal_0/T2')" onmouseout="rh()">XXREAL_0:2</a></span>;<br/>
<span class="lab"><font color="Green" title="E69">A70</font></span>: <font color="Maroon" title="c27">t1</font> = 
<font color="Maroon">H<sub>3</sub></font>( <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c27">t1</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:70_1_1"><span class="lab"><font color="Green" title="E54">A47</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c17">f</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c27">t1</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:70_1_1"><span class="lab"><font color="Green" title="E47">A33</font></span></a></span>
;<br/>
<font color="Maroon" title="c26">t</font> = 
<font color="Maroon">H<sub>3</sub></font>( <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c26">t</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:70_1_1"><span class="lab"><font color="Green" title="E54">A47</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c17">f</font> <a href="ltlaxio4.html#K12" title="LTLAXIO4:func.12">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c26">t</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:70_1_1"><span class="lab"><font color="Green" title="E47">A33</font></span></a></span>
;<br/>
<a NAME="E29:70_1_1_11"/><span class="kw">then </span>
<font color="Maroon" title="c14">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c26">t</font></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:70_1_1_11"><span class="lab"><font color="Green" title="E66">A67</font></span></a>, <a class="txt" href="#E4:70_1_1_11"><span class="lab"><font color="Green" title="E58">A59</font></span></a>, <a class="txt" href="#E11:70_1_1_11"><span class="lab"><font color="Green" title="E61">A62</font></span></a></span>;<br/>
<a NAME="E30:70_1_1_11"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A71</font></span>: 
 <a href="ltlaxio4.html#K2" title="LTLAXIO4:func.2">rarg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c9">uns</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c20">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">k</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p4">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c9">uns</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">T</font> <a href="trees_2.html#K3" title="TREES_2:func.3">.</a> <font color="Maroon" title="c27">t1</font></span>)</span> <a href="ltlaxio3.html#K8" title="LTLAXIO3:func.8">`1</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:70_1_1_11"><span class="lab"><font color="Green" title="E65">A66</font></span></a>, <a class="txt" href="#E18:70_1_1"><span class="lab"><font color="Green" title="E44">A24</font></span></a></span>;<br/>
 <a href="ltlaxio4.html#K2" title="LTLAXIO4:func.2">rarg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c9">uns</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <span class="p0">(<span class="default"><font color="Maroon" title="c20">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c24">k</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p4">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Maroon" title="c9">uns</font></span>)</span></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> = 
 <a href="ltlaxio4.html#K2" title="LTLAXIO4:func.2">rarg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">A</font> <a href="ltlaxio1.html#NK3" title="LTLAXIO1:NK.3">'U'</a> <font color="Maroon" title="c19">B</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:70_1_1_11"><span class="lab"><font color="Green" title="E61">A62</font></span></a>, <a class="txt" href="#E10:70_1_1_11"><span class="lab"><font color="Green" title="E60">A61</font></span></a>, <a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a>, <a class="txt" href="#E20:70_1_1_11"><span class="lab"><font color="Green" title="E65">A66</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c19">B</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ltlaxio4.html#D1" target="_self" onmouseover="rs('ltlaxio4/D1')" onmouseout="rh()">Def1</a>, <a class="txt" href="#E24:70_1_1_11"><span class="lab"><font color="Green" title="E67">A68</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E32:70_1_1_11"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:70_1_1_11"><span class="lab"><font color="Green" title="E68">A69</font></span></a>, <a class="txt" href="#E4:70_1_1_11"><span class="lab"><font color="Green" title="E58">A59</font></span></a>, <a class="txt" href="#E30:70_1_1_11"><span class="lab"><font color="Green" title="E70">A71</font></span></a>, <a class="txt" href="#E27:70_1_1_11"><span class="lab"><font color="Green" title="E69">A70</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E33:70_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being    <a href="ltlaxio4.html#M4" title="LTLAXIO4:mode.4">path</a> of <font color="Maroon" title="c4">T</font> st <font color="Olive">b<sub>1</sub></font> is  <a href="ltlaxio4.html#V2" title="LTLAXIO4:attr.2">complete</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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