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<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">L</font> be   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">H</font> being  <a href="modelc_2.html#NM1" title="MODELC_2:NM.1">LTL-formula</a> holds   <a href="modelc_3.html#K17" title="MODELC_3:func.17">len</a> <font color="Maroon" title="c1">L</font>,<span class="p1">(<span class="default"><a href="modelc_3.html#K7" title="MODELC_3:func.7">{}</a> <font color="Olive" title="b1">H</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">H</font> be   <a href="modelc_2.html#NM1" title="MODELC_2:NM.1">LTL-formula</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="modelc_3.html#K17" title="MODELC_3:func.17">len</a> <font color="Maroon" title="c1">L</font>,<span class="p1">(<span class="default"><a href="modelc_3.html#K7" title="MODELC_3:func.7">{}</a> <font color="Maroon" title="c2">H</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/>



<span class="kw">set </span><font color="Maroon" title="c3">s</font> =  <a href="modelc_3.html#K16" title="MODELC_3:func.16">Partial_seq</a> <font color="Maroon" title="c1">L</font>,<span class="p1">(<span class="default"><a href="modelc_3.html#K7" title="MODELC_3:func.7">{}</a> <font color="Maroon" title="c2">H</font></span>)</span>;<br/>
<a NAME="E1:53"/><span class="lab"><font color="Green" title="E44">A1</font></span>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <span class="p1">(<span class="default"><a href="modelc_3.html#K16" title="MODELC_3:func.16">Partial_seq</a> <font color="Maroon" title="c1">L</font>,<span class="p2">(<span class="default"><a href="modelc_3.html#K7" title="MODELC_3:func.7">{}</a> <font color="Maroon" title="c2">H</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_3.html#D24" target="_self" title="MODELC_3:def.24">Def24</a></span>;<br/>
<a NAME="E2:53"/>
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><a href="modelc_3.html#K16" title="MODELC_3:func.16">Partial_seq</a> <font color="Maroon" title="c1">L</font>,<span class="p3">(<span class="default"><a href="modelc_3.html#K7" title="MODELC_3:func.7">{}</a> <font color="Maroon" title="c2">H</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><a href="modelc_3.html#K16" title="MODELC_3:func.16">Partial_seq</a> <font color="Maroon" title="c1">L</font>,<span class="p3">(<span class="default"><a href="modelc_3.html#K7" title="MODELC_3:func.7">{}</a> <font color="Maroon" title="c2">H</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c5">n</font> = <font color="Maroon" title="c4">n</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/>
<a NAME="E2:53_1"/><span class="lab"><font color="Green" title="E45">A2</font></span>: 
<span class="p1">(<span class="default"><a href="modelc_3.html#K16" title="MODELC_3:func.16">Partial_seq</a> <font color="Maroon" title="c1">L</font>,<span class="p2">(<span class="default"><a href="modelc_3.html#K7" title="MODELC_3:func.7">{}</a> <font color="Maroon" title="c2">H</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="modelc_3.html#D24" target="_self" title="MODELC_3:def.24">Def24</a></span>;<br/>
<span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><a href="modelc_3.html#K16" title="MODELC_3:func.16">Partial_seq</a> <font color="Maroon" title="c1">L</font>,<span class="p3">(<span class="default"><a href="modelc_3.html#K7" title="MODELC_3:func.7">{}</a> <font color="Maroon" title="c2">H</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">n</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="modelc_3.html#K16" title="MODELC_3:func.16">Partial_seq</a> <font color="Maroon" title="c1">L</font>,<span class="p5">(<span class="default"><a href="modelc_3.html#K7" title="MODELC_3:func.7">{}</a> <font color="Maroon" title="c2">H</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="modelc_3.html#K16" title="MODELC_3:func.16">Partial_seq</a> <font color="Maroon" title="c1">L</font>,<span class="p5">(<span class="default"><a href="modelc_3.html#K7" title="MODELC_3:func.7">{}</a> <font color="Maroon" title="c2">H</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53"><span class="lab"><font color="Green" title="E44">A1</font></span></a>, <a class="ref" href="series_2.html#T42" title="SERIES_2:th.42">SERIES_2:42</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> 2
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:53_1"><span class="lab"><font color="Green" title="E45">A2</font></span></a>, <a class="ref" href="modelc_3.html#D24" target="_self" title="MODELC_3:def.24">Def24</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E4:53_1"/>
<span class="p1">(<span class="default"><a href="series_1.html#K1" title="SERIES_1:func.1">Partial_Sums</a> <span class="p2">(<span class="default"><a href="modelc_3.html#K16" title="MODELC_3:func.16">Partial_seq</a> <font color="Maroon" title="c1">L</font>,<span class="p3">(<span class="default"><a href="modelc_3.html#K7" title="MODELC_3:func.7">{}</a> <font color="Maroon" title="c2">H</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c4">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:53"/>
 <a href="modelc_3.html#K17" title="MODELC_3:func.17">len</a> <font color="Maroon" title="c1">L</font>,<span class="p1">(<span class="default"><a href="modelc_3.html#K7" title="MODELC_3:func.7">{}</a> <font color="Maroon" title="c2">H</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
