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<span class="kw">set </span><font color="Maroon" title="c4">f</font> =  the    <a href="ltlaxio4.html#M2" title="LTLAXIO4:mode.2">Enumeration</a> of <font color="Maroon" title="c3">F</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="ltlaxio1.html#NK1" title="LTLAXIO1:NK.1">LTLB_WFF</a>  = <span class="p1">(<span class="default"> the    <a href="ltlaxio4.html#M2" title="LTLAXIO4:mode.2">Enumeration</a> of <font color="Maroon" title="c3">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">F</font></span>)</span> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a> ;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">g</font> being    <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"> such that </span><br/><a NAME="E2:19_1_1"/><span class="lab"><font color="Green" title="E11">A1</font></span>: 
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">g</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#M2" title="LTLAXIO4:mode.2">Enumeration</a> of <font color="Maroon" title="c3">F</font> &amp; (  for <font color="Olive" title="b1">j</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>  st <font color="Olive" title="b1">j</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="c5">g</font> holds <br/><font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b1">j</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="E3:19_1_1"/>
<font color="Maroon" title="c3">F</font> <a href="ltlaxio4.html#K3" title="LTLAXIO4:func.3">^</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c5">g</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="19_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#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="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font> <a href="ltlaxio4.html#K3" title="LTLAXIO4:func.3">^</a>  or <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c5">g</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:19_1_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font> <a href="ltlaxio4.html#K3" title="LTLAXIO4:func.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"> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c5">g</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">P</font> being   <a href="ltlaxio3.html#NM1" title="LTLAXIO3:NM.1">PNPair</a><span class="kw"> such that </span><br/><a NAME="E3:19_1_1_1"/><span class="lab"><font color="Green" title="E12">A2</font></span>: 
<font color="Maroon" title="c7">P</font> <a href="ltlaxio3.html#K12" title="LTLAXIO3:func.12">^</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font>
 <span class="kw">and </span><br/><a NAME="E4:19_1_1_1"/><span class="lab"><font color="Green" title="E13">A3</font></span>: 
<font color="Maroon" title="c7">P</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">F</font>
 ;<br/>
<a NAME="E5:19_1_1_1"/><span class="lab"><font color="Green" title="E14">A4</font></span>: 
<font color="Maroon" title="c7">P</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#M2" title="LTLAXIO4:mode.2">Enumeration</a> of <font color="Maroon" title="c3">F</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>, <a class="txt" href="#E4:19_1_1_1"><span class="lab"><font color="Green" title="E13">A3</font></span></a></span>;<br/>
<a NAME="E6:19_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E15">A5</font></span>: 
<font color="Maroon" title="c7">P</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a>  the    <a href="ltlaxio4.html#M2" title="LTLAXIO4:mode.2">Enumeration</a> of <font color="Maroon" title="c3">F</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#M2" title="LTLAXIO4:mode.2">Enumeration</a> of <font color="Maroon" title="c3">F</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/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E16">A6</font></span>:  the    <a href="ltlaxio4.html#M2" title="LTLAXIO4:mode.2">Enumeration</a> of <font color="Maroon" title="c3">F</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">P</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a>  the    <a href="ltlaxio4.html#M2" title="LTLAXIO4:mode.2">Enumeration</a> of <font color="Maroon" title="c3">F</font></span>)</span> = 
 the    <a href="ltlaxio4.html#M2" title="LTLAXIO4:mode.2">Enumeration</a> of <font color="Maroon" title="c3">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">P</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a>  the    <a href="ltlaxio4.html#M2" title="LTLAXIO4:mode.2">Enumeration</a> of <font color="Maroon" title="c3">F</font></span>)</span>
<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/>.= 
<font color="Maroon" title="c7">P</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="#E5:19_1_1_1"><span class="lab"><font color="Green" title="E14">A4</font></span></a></span>
;<br/>
<a NAME="E8:19_1_1_1"/><span class="lab"><font color="Green" title="E17">A7</font></span>: 
<font color="Maroon" title="c7">P</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a>  the    <a href="ltlaxio4.html#M2" title="LTLAXIO4:mode.2">Enumeration</a> of <font color="Maroon" title="c3">F</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="c5">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:19_1_1_1"><span class="lab"><font color="Green" title="E15">A5</font></span></a>, <a class="txt" href="#E2:19_1_1"><span class="lab"><font color="Green" title="E11">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/>
<a NAME="E9:19_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">P</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">..</a>  the    <a href="ltlaxio4.html#M2" title="LTLAXIO4:mode.2">Enumeration</a> of <font color="Maroon" title="c3">F</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="c5">g</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/>
<span class="kw">hence </span><a NAME="E10:19_1_1_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c5">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:19_1_1"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="txt" href="#E8:19_1_1_1"><span class="lab"><font color="Green" title="E17">A7</font></span></a>, <a class="txt" href="#E7:19_1_1_1"><span class="lab"><font color="Green" title="E16">A6</font></span></a>, <a class="txt" href="#E3:19_1_1_1"><span class="lab"><font color="Green" title="E12">A2</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="E4:19_1_1"/>
<font color="Maroon" title="c3">F</font> <a href="ltlaxio4.html#K3" title="LTLAXIO4:func.3">^</a>  is  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</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/>


</div>
