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

<span class="kw">let </span><font color="Maroon" title="c3">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">  for <font color="Olive" title="b1">G</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="finseq_2.html#M1" title="FINSEQ_2:mode.1">FinSequenceSet</a> of <font color="Maroon" title="c3">X</font>  ex <font color="Olive" title="b2">A</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>( <font color="Olive" title="b2">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">G</font> &amp; (  for <font color="Olive" title="b3">B</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>  st <font color="Olive" title="b3">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">G</font> holds <br/> <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b3">B</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="Olive" title="b2">A</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">G</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>   <a href="finseq_2.html#M1" title="FINSEQ_2:mode.1">FinSequenceSet</a> of <font color="Maroon" title="c3">X</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">A</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>( <font color="Olive" title="b1">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">G</font> &amp; (  for <font color="Olive" title="b2">B</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>  st <font color="Olive" title="b2">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">G</font> holds <br/> <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">B</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="Olive" title="b1">A</font> ) )</span><br/>



<span class="kw">set </span><font color="Maroon" title="c5">g</font> =  the    <a href="ltlaxio4.html#M1" title="LTLAXIO4:mode.1">Enumeration</a> of <font color="Maroon" title="c4">G</font>;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>3</sub></font>(  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>) <span class="kw">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="ordinal1.html#K4" title="ORDINAL1:func.4">omega</a>  =  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"> the    <a href="ltlaxio4.html#M1" title="LTLAXIO4:mode.1">Enumeration</a> of <font color="Maroon" title="c4">G</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> $1</span>)</span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">f</font> being    <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"> such that </span><br/><a NAME="E2:4"/><span class="lab"><font color="Green" title="E2">A1</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a>  the    <a href="ltlaxio4.html#M1" title="LTLAXIO4:mode.1">Enumeration</a> of <font color="Maroon" title="c4">G</font> &amp; (  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="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="c6">f</font> holds <br/><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</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="finseq_2.html#S1" onmouseover="rs('finseq_2/S1')" onmouseout="rh()">FINSEQ_2:sch 1</a>();<br/></span>
<a NAME="E3:4"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="numbers.html#T19" onmouseover="rs('numbers/T19')" onmouseout="rh()">NUMBERS:19</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">f1</font> = <font color="Maroon" title="c6">f</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  <span class="kw">by</span> <span class="lab"><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">set </span><font color="Maroon" title="c8">m</font> =  <a href="rfinseq2.html#K1" title="RFINSEQ2:func.1">max_p</a> <font color="Maroon" title="c7">f1</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">A</font> =  the    <a href="ltlaxio4.html#M1" title="LTLAXIO4:mode.1">Enumeration</a> of <font color="Maroon" title="c4">G</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K1" title="RFINSEQ2:func.1">max_p</a> <font color="Maroon" title="c7">f1</font></span>)</span>;<br/>
<a NAME="E5:4"/><span class="lab"><font color="Green" title="E3">A2</font></span>: 
 the    <a href="ltlaxio4.html#M1" title="LTLAXIO4:mode.1">Enumeration</a> of <font color="Maroon" title="c4">G</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="ref" href="rlaffin3.html#D1" onmouseover="rs('rlaffin3/D1')" onmouseout="rh()">RLAFFIN3:def 1</a>, <a class="ref" href="relat_1.html#T38" onmouseover="rs('relat_1/T38')" onmouseout="rh()">RELAT_1:38</a></span>;<br/>
<a NAME="E6:4"/>
 for <font color="Olive" title="b1">B</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>  st <font color="Olive" title="b1">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">G</font> holds <br/> <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">B</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"> the    <a href="ltlaxio4.html#M1" title="LTLAXIO4:mode.1">Enumeration</a> of <font color="Maroon" title="c4">G</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="rfinseq2.html#K1" title="RFINSEQ2:func.1">max_p</a> <font color="Maroon" title="c7">f1</font></span>)</span></span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="4_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">B</font> be   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</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="c10">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">G</font> implies  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">B</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"> the    <a href="ltlaxio4.html#M1" title="LTLAXIO4:mode.1">Enumeration</a> of <font color="Maroon" title="c4">G</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="rfinseq2.html#K1" title="RFINSEQ2:func.1">max_p</a> <font color="Maroon" title="c7">f1</font></span>)</span></span>)</span> )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c11">m1</font> = <font color="Maroon" title="c10">B</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a>  the    <a href="ltlaxio4.html#M1" title="LTLAXIO4:mode.1">Enumeration</a> of <font color="Maroon" title="c4">G</font>;<br/>
<a NAME="E1:4_1"/>
 <a href="rfinseq2.html#K1" title="RFINSEQ2:func.1">max_p</a> <font color="Maroon" title="c7">f1</font> <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="c6">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:4"><span class="lab"><font color="Green" title="E3">A2</font></span></a>, <a class="txt" href="#E2:4"><span class="lab"><font color="Green" title="E2">A1</font></span></a>, <a class="ref" href="rfinseq2.html#D1" onmouseover="rs('rfinseq2/D1')" onmouseout="rh()">RFINSEQ2:def 1</a></span>;<br/>
<a NAME="E2:4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E4">A3</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"> the    <a href="ltlaxio4.html#M1" title="LTLAXIO4:mode.1">Enumeration</a> of <font color="Maroon" title="c4">G</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="rfinseq2.html#K1" title="RFINSEQ2:func.1">max_p</a> <font color="Maroon" title="c7">f1</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="rfinseq2.html#K1" title="RFINSEQ2:func.1">max_p</a> <font color="Maroon" title="c7">f1</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:4"><span class="lab"><font color="Green" title="E2">A1</font></span></a></span>;<br/>
<span class="kw">assume </span><a NAME="E3:4_1"/>
<font color="Maroon" title="c10">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">G</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">  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">B</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"> the    <a href="ltlaxio4.html#M1" title="LTLAXIO4:mode.1">Enumeration</a> of <font color="Maroon" title="c4">G</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="rfinseq2.html#K1" title="RFINSEQ2:func.1">max_p</a> <font color="Maroon" title="c7">f1</font></span>)</span></span>)</span></span><br/>

<a NAME="E4:4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E5">A4</font></span>: 
<font color="Maroon" title="c10">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>  the    <a href="ltlaxio4.html#M1" title="LTLAXIO4:mode.1">Enumeration</a> of <font color="Maroon" title="c4">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlaffin3.html#D1" onmouseover="rs('rlaffin3/D1')" onmouseout="rh()">RLAFFIN3:def 1</a></span>;<br/>
<a NAME="E5:4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E6">A5</font></span>: 
<font color="Maroon" title="c10">B</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a>  the    <a href="ltlaxio4.html#M1" title="LTLAXIO4:mode.1">Enumeration</a> of <font color="Maroon" title="c4">G</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a>  the    <a href="ltlaxio4.html#M1" title="LTLAXIO4:mode.1">Enumeration</a> of <font color="Maroon" title="c4">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T20" onmouseover="rs('finseq_4/T20')" onmouseout="rh()">FINSEQ_4:20</a></span>;<br/>
<a NAME="E6:4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E7">A6</font></span>: 
<font color="Maroon" title="c10">B</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a>  the    <a href="ltlaxio4.html#M1" title="LTLAXIO4:mode.1">Enumeration</a> of <font color="Maroon" title="c4">G</font> <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="c6">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:4"><span class="lab"><font color="Green" title="E2">A1</font></span></a>, <a class="ref" href="finseq_3.html#T29" onmouseover="rs('finseq_3/T29')" onmouseout="rh()">FINSEQ_3:29</a></span>;<br/>
 the    <a href="ltlaxio4.html#M1" title="LTLAXIO4:mode.1">Enumeration</a> of <font color="Maroon" title="c4">G</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">B</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a>  the    <a href="ltlaxio4.html#M1" title="LTLAXIO4:mode.1">Enumeration</a> of <font color="Maroon" title="c4">G</font></span>)</span> = 
 the    <a href="ltlaxio4.html#M1" title="LTLAXIO4:mode.1">Enumeration</a> of <font color="Maroon" title="c4">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">B</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a>  the    <a href="ltlaxio4.html#M1" title="LTLAXIO4:mode.1">Enumeration</a> of <font color="Maroon" title="c4">G</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:4_1"><span class="lab"><font color="Green" title="E6">A5</font></span></a>, <a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>
<br/>.= 
<font color="Maroon" title="c10">B</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T19" onmouseover="rs('finseq_4/T19')" onmouseout="rh()">FINSEQ_4:19</a>, <a class="txt" href="#E4:4_1"><span class="lab"><font color="Green" title="E5">A4</font></span></a></span>
;<br/>
<a NAME="E8:4_1"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">B</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">B</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a>  the    <a href="ltlaxio4.html#M1" title="LTLAXIO4:mode.1">Enumeration</a> of <font color="Maroon" title="c4">G</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:4_1"><span class="lab"><font color="Green" title="E7">A6</font></span></a>, <a class="txt" href="#E2:4"><span class="lab"><font color="Green" title="E2">A1</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:4_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c10">B</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"> the    <a href="ltlaxio4.html#M1" title="LTLAXIO4:mode.1">Enumeration</a> of <font color="Maroon" title="c4">G</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><a href="rfinseq2.html#K1" title="RFINSEQ2:func.1">max_p</a> <font color="Maroon" title="c7">f1</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:4_1"><span class="lab"><font color="Green" title="E7">A6</font></span></a>, <a class="ref" href="rfinseq2.html#D1" onmouseover="rs('rfinseq2/D1')" onmouseout="rh()">RFINSEQ2:def 1</a>, <a class="txt" href="#E5:4"><span class="lab"><font color="Green" title="E3">A2</font></span></a>, <a class="txt" href="#E2:4"><span class="lab"><font color="Green" title="E2">A1</font></span></a>, <a class="txt" href="#E2:4_1"><span class="lab"><font color="Green" title="E4">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">hence </span><a NAME="E7:4"/>
 ex <font color="Olive" title="b1">A</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <br/>( <font color="Olive" title="b1">A</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">G</font> &amp; (  for <font color="Olive" title="b2">B</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>  st <font color="Olive" title="b2">B</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c4">G</font> holds <br/> <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b2">B</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="Olive" title="b1">A</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/>


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