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<div class="add">

<b>set </b><font color="Maroon">S</font> =  <a href="triang_1.html#K2">SgmX</a> <span class="p1">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">l</font> =  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ,  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>] <b>means </b><font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="partfun1.html#K1">*</a> <span class="p3">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p4">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" target="_self">eval</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p5">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p5">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="polynom2.html#K4" target="_self">@</a> </span>)</span>,<font color="Maroon">x</font></span>)</span>;<br/>
<a NAME="E1:43_1_1"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span> holds <br/> ex <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>,<font color="Olive">x</font>]
 
;<br/>
<b>consider </b><font color="Maroon">q</font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:43_1_1"/><i><font color="Green">E34</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span> &amp; ( for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">m</font>,<font color="Maroon">q</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">m</font>] ) )
 <b>from </b><i><a class="ref" href="polynom2.html#S1" target="_self">POLYNOM2:sch 1</a>();<br/></i>
<b>take </b>
 <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">q</font>
;<br/>

<a NAME="E4:43_1_1"/><i><font color="Green">E35</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:43_1_1_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font> )
 ;<br/><a NAME="E2:43_1_1_1"/><b>then </b>
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/><b>hence </b><a NAME="E3:43_1_1_1"/>
<font color="Maroon">q</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="partfun1.html#K1">*</a> <span class="p3">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p4">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">m</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" target="_self">eval</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p5">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p5">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">m</font></span>)</span> <a href="polynom2.html#K4" target="_self">@</a> </span>)</span>,<font color="Maroon">x</font></span>)</span>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:43_1_1"/>
 ex <font color="Olive">y</font> being   <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> st <br/>(  <a href="finseq_1.html#K3">len</a> <font color="Olive">y</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span> &amp;  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Olive">y</font> &amp; ( for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">y</font> holds <br/><font color="Olive">y</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="partfun1.html#K1">*</a> <span class="p3">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p4">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" target="_self">eval</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="triang_1.html#K2">SgmX</a> <span class="p5">(<span class="default"><a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font></span>)</span>,<span class="p5">(<span class="default"><a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span> <a href="polynom2.html#K4" target="_self">@</a> </span>)</span>,<font color="Maroon">x</font></span>)</span> ) )
 <b>by </b><i/>;<br/>


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