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<b>let </b><font color="Maroon" title="c1">G</font> be   <a href="group_1.html#NM1" title="GROUP_1:NM.1">Group</a>;<br/><b>let </b><font color="Maroon" title="c2">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">G</font>;<br/><b>let </b><font color="Maroon" title="c3">I</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> ;<br/><b>let </b><font color="Maroon" title="c4">w</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K4" title="NUMBERS:func.4">INT</a> ;<br/>







<b>assume </b><a NAME="E1:8"/><i><font color="Green" title="E8">A1</font></i>: 
 <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">I</font></span>)</span> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c3">I</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p1">(<span class="default"><a href="gr_cy_1.html#K1" title="GR_CY_1:func.1">Sum</a> <font color="Maroon" title="c3">I</font></span>)</span>
 ;<br/>

<a NAME="E2:8"/><i><font color="Green" title="E9">A2</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">I</font></span>)</span> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">I</font>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T69" title="FINSEQ_2:th.69">FINSEQ_2:69</a></i>;<br/>
<i><font color="Green" title="E10">A3</font></i>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> = 
1
<b>by </b><i><a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>
<br/>.= 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c4">w</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
<b>by </b><i><a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>
;<br/>
 <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p5"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c4">w</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span>)</span> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c4">w</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span>)</span> = 
 <a href="group_4.html#K3" title="GROUP_4:func.3">Product</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="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">I</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c4">w</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_2.html#T19" title="FINSEQ_2:th.19">FINSEQ_2:19</a></i>
<br/>.= 
 <a href="group_4.html#K3" title="GROUP_4:func.3">Product</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="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">I</font></span>)</span> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c4">w</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_2.html#T74" title="FINSEQ_2:th.74">FINSEQ_2:74</a></i>
<br/>.= 
 <a href="group_4.html#K3" title="GROUP_4:func.3">Product</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="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">I</font></span>)</span> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c3">I</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c4">w</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="gr_cy_1.html#E2:8"><i><font color="Green" title="E9">A2</font></i></a>, <a class="txt" href="gr_cy_1.html#E3:8"><i><font color="Green" title="E10">A3</font></i></a>, <a class="ref" href="group_4.html#T25" title="GROUP_4:th.25">GROUP_4:25</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">I</font></span>)</span> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c3">I</font></span>)</span></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c2">a</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p4">(<span class="default"><a href="group_4.html#K2" title="GROUP_4:func.2">@</a> <span class="p5">(<span class="default"><a href="group_4.html#K2" title="GROUP_4:func.2">@</a> <font color="Maroon" title="c4">w</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_4.html#T8" title="GROUP_4:th.8">GROUP_4:8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">I</font></span>)</span> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <font color="Maroon" title="c3">I</font></span>)</span></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p2"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p4">(<span class="default"><a href="group_4.html#K2" title="GROUP_4:func.2">@</a> <font color="Maroon" title="c4">w</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="group_4.html#T28" title="GROUP_4:th.28">GROUP_4:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p2">(<span class="default"><a href="gr_cy_1.html#K1" title="GR_CY_1:func.1">Sum</a> <font color="Maroon" title="c3">I</font></span>)</span></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p2">(<span class="default"><a href="group_4.html#K2" title="GROUP_4:func.2">@</a> <font color="Maroon" title="c4">w</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="gr_cy_1.html#E1:8"><i><font color="Green" title="E8">A1</font></i></a>, <a class="ref" href="finsop_1.html#T12" title="FINSOP_1:th.12">FINSOP_1:12</a></i>
<br/>.= 
<font color="Maroon" title="c2">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="gr_cy_1.html#K1" title="GR_CY_1:func.1">Sum</a> <font color="Maroon" title="c3">I</font></span>)</span> <a href="binop_2.html#K20" title="BINOP_2:func.20">+</a> <span class="p2">(<span class="default"><a href="group_4.html#K2" title="GROUP_4:func.2">@</a> <font color="Maroon" title="c4">w</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#T63" title="GROUP_1:th.63">GROUP_1:63</a></i>
<br/>.= 
<font color="Maroon" title="c2">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p1">(<span class="default"><a href="gr_cy_1.html#K1" title="GR_CY_1:func.1">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c4">w</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rvsum_1.html#T104" title="RVSUM_1:th.104">RVSUM_1:104</a></i>
;<br/>
<b>hence </b><a NAME="E5:8"/>
 <a href="group_4.html#K3" title="GROUP_4:func.3">Product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p5"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c4">w</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span>)</span> <a href="finsop_1.html#K1" title="FINSOP_1:func.1">|-&gt;</a> <font color="Maroon" title="c2">a</font></span>)</span> <a href="group_4.html#K4" title="GROUP_4:func.4">|^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c4">w</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font> <a href="group_1.html#K6" title="GROUP_1:func.6">|^</a> <span class="p1">(<span class="default"><a href="gr_cy_1.html#K1" title="GR_CY_1:func.1">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">I</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><font color="Maroon" title="c4">w</font></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span>)</span></span>)</span>
 ;<br/>


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