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<b>let </b><font color="Maroon">D</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">D</font> <a href="finseq_2.html#K3">*</a> ] <b>means </b> <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="wsierp_1.html#K7">Sum</a> <span class="p1">(<span class="default"><a href="polynom1.html#K5" target="_self">Card</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<a NAME="E1:55"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="finseq_1.html#K6">&lt;*&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">D</font> <a href="finseq_2.html#K3">*</a> </span>)</span>]
 
<b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a>, <a class="ref" href="rvsum_1.html#T102">RVSUM_1:102</a></i>;<br/>
<div><i><font color="Green">E42</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">F</font> be    <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">D</font> <a href="finseq_2.html#K3">*</a> ;<br/><b>let </b><font color="Maroon">p</font> be    <a href="finseq_2.html#M2">Element</a> of <font color="Maroon">D</font> <a href="finseq_2.html#K3">*</a> ;<br/><b>assume </b><a NAME="E1:55_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">F</font>]
 ;<br/><a NAME="E2:55_1"/><i><font color="Green">E44</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 ;<br/> <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">p</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <font color="Maroon">F</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">p</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="dtconstr.html#T21">DTCONSTR:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K7">Sum</a> <span class="p2">(<span class="default"><a href="polynom1.html#K5" target="_self">Card</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="dtconstr.html#K15">FlattenSeq</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">p</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="wsierp_1.html#K7">Sum</a> <span class="p2">(<span class="default"><a href="polynom1.html#K5" target="_self">Card</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span>
<b>by </b><i><a class="ref" href="dtconstr.html#T13">DTCONSTR:13</a></i>
<br/>.= 
 <a href="wsierp_1.html#K7">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K5" target="_self">Card</a> <font color="Maroon">F</font></span>)</span> <a href="wsierp_1.html#K4">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>
<b>by </b><i>, <a class="ref" href="rvsum_1.html#T104">RVSUM_1:104</a></i>
<br/>.= 
 <a href="wsierp_1.html#K7">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom1.html#K5" target="_self">Card</a> <font color="Maroon">F</font></span>)</span> <a href="wsierp_1.html#K4">^</a> <span class="p2">(<span class="default"><a href="polynom1.html#K5" target="_self">Card</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">p</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="wsierp_1.html#K7">Sum</a> <span class="p1">(<span class="default"><a href="polynom1.html#K5" target="_self">Card</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">p</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span></span>)</span>
<b>by </b><i/>
;<br/><b>hence </b><a NAME="E4:55_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">p</font></span><a href="binarith.html#K11">*&gt;</a></span>]
 ;<br/></div><b>end;</b></div>
<b>thus </b><a NAME="E3:55"/>
for <font color="Olive">F</font> being   <a href="finseq_1.html#M2">FinSequence</a> of <font color="Maroon">D</font> <a href="finseq_2.html#K3">*</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">F</font>]
 <b>from </b><i><a class="ref" href="finseq_2.html#S2">FINSEQ_2:sch 2</a>(, <a class="txt" href="polynom1.html#E1"><i><font color="Green">Lemma32</font></i></a>);</i><br/>


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