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<b>let </b><font color="Maroon">G</font> be   <a href="group_1.html#NM1">Group</a>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">F</font> be    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font>;<br/>





<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font>] <b>means </b> <a href="group_4.html#K3">Product</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="group_5.html#K1" target="_self">|^</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="group_4.html#K3">Product</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="group_3.html#K2">|^</a> <font color="Maroon">a</font>;<br/>
<a NAME="E1:16"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="finseq_1.html#K6">&lt;*&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">p</font> =  <a href="finseq_1.html#K6">&lt;*&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font>;<br/>
<b>thus </b> <a href="group_4.html#K3">Product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K6">&lt;*&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font></span>)</span> <a href="group_5.html#K1" target="_self">|^</a> <font color="Maroon">a</font></span>)</span> = 
 <a href="group_4.html#K3">Product</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K6">&lt;*&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="group_1.html#K2">1.</a> <font color="Maroon">G</font>
<b>by </b><i><a class="ref" href="group_4.html#T11">GROUP_4:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">G</font></span>)</span> <a href="group_3.html#K2">|^</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="group_3.html#T22">GROUP_3:22</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_4.html#K3">Product</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K6">&lt;*&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font></span>)</span></span>)</span> <a href="group_3.html#K2">|^</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="group_4.html#T11">GROUP_4:11</a></i>
;<br/>


</div><b>end;</b></div>
<div><i><font color="Green">E59</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2"><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 the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">G</font>;<br/><b>assume </b><a NAME="E1:16_2"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">F</font>]
 ;<br/><b>thus </b><a NAME="E2:16_2"/>
<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="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">b</font></span><a href="finseq_1.html#K12">*&gt;</a></span>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_1"><b>proof </b></a><div class="add">

<b>thus </b> <a href="group_4.html#K3">Product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">b</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="group_5.html#K1" target="_self">|^</a> <font color="Maroon">a</font></span>)</span> = 
 <a href="group_4.html#K3">Product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="group_5.html#K1" target="_self">|^</a> <font color="Maroon">a</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">b</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="group_5.html#K1" target="_self">|^</a> <font color="Maroon">a</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_4.html#K3">Product</a> <font color="Maroon">F</font></span>)</span> <a href="group_3.html#K2">|^</a> <font color="Maroon">a</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="group_4.html#K3">Product</a> <span class="p2">(<span class="default"><span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">b</font></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="group_5.html#K1" target="_self">|^</a> <font color="Maroon">a</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="group_4.html#T8">GROUP_4:8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_4.html#K3">Product</a> <font color="Maroon">F</font></span>)</span> <a href="group_3.html#K2">|^</a> <font color="Maroon">a</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="group_4.html#K3">Product</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="group_3.html#K2">|^</a> <font color="Maroon">a</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_4.html#K3">Product</a> <font color="Maroon">F</font></span>)</span> <a href="group_3.html#K2">|^</a> <font color="Maroon">a</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="group_3.html#K2">|^</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="ref" href="group_4.html#T12">GROUP_4:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_4.html#K3">Product</a> <font color="Maroon">F</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font></span>)</span> <a href="group_3.html#K2">|^</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="group_3.html#T28">GROUP_3:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_4.html#K3">Product</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="finseq_1.html#K8">^</a> <span class="p3"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><font color="Maroon">b</font></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span></span>)</span> <a href="group_3.html#K2">|^</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="group_4.html#T9">GROUP_4:9</a></i>
;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E3:16"/>
for <font color="Olive">F</font> being   <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">G</font> 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>(, );<br/></i>
<b>hence </b><a NAME="E4:16"/>
 <a href="group_4.html#K3">Product</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="group_5.html#K1" target="_self">|^</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="group_4.html#K3">Product</a> <font color="Maroon">F</font></span>)</span> <a href="group_3.html#K2">|^</a> <font color="Maroon">a</font>
 ;<br/>


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