<?xml version="1.0"?>
<div class="add">

<b>set </b><font color="Maroon" title="c5">S</font> =  <a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p1">(<span class="default"><a href="polynom1.html#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c3">p</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c6">l</font> =  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <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> ,  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L</font>] <b>means </b>$2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p4">(<span class="default"><a href="polynom1.html#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> $1</span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">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" title="TRIANG_1:func.2">SgmX</a> <span class="p5">(<span class="default"><a href="polynom1.html#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p5">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> $1</span>)</span> <a href="polynom2.html#K4" title="POLYNOM2:func.4">@</a> </span>)</span>,<font color="Maroon" title="c4">x</font></span>)</span>;<br/>
<a NAME="E1:43_1_1"/><i><font color="Green" title="E29">A1</font></i>: 
for <font color="Olive" title="b1">k</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>   st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> holds <br/> ex <font color="Olive" title="b2">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L</font> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>,<font color="Olive" title="b2">x</font>]
 
;<br/>
<b>consider </b><font color="Maroon" title="c7">q</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font><b> such that </b><br/><a NAME="E3:43_1_1"/><i><font color="Green" title="E30">A2</font></i>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> &amp; ( for <font color="Olive" title="b1">m</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>   st <font color="Olive" title="b1">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p3">(<span class="default"><a href="polynom1.html#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">m</font>,<font color="Maroon" title="c7">q</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b1">m</font>] ) )
 <b>from </b><i><a class="ref" href="polynom2.html#S1" target="_self" title="POLYNOM2:sch.1">POLYNOM2:sch 1</a>(<a class="txt" href="polynom2.html#E1:43_1_1"><i><font color="Green" title="E29">A1</font></i></a>);<br/></i>
<b>take </b>
 <a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <font color="Maroon" title="c7">q</font>
;<br/>

<a NAME="E4:43_1_1"/><i><font color="Green" title="E31">A3</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom2.html#E3:43_1_1"><i><font color="Green" title="E30">A2</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">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" title="c8">m</font> be    <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> ;<br/><b>assume </b><a NAME="E1:43_1_1_1"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">m</font> &amp; <font color="Maroon" title="c8">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">q</font> )
 ;<br/><a NAME="E2:43_1_1_1"/><b>then </b>
<font color="Maroon" title="c8">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">q</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/><b>hence </b><a NAME="E3:43_1_1_1"/>
<font color="Maroon" title="c7">q</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p4">(<span class="default"><a href="polynom1.html#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">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" title="TRIANG_1:func.2">SgmX</a> <span class="p5">(<span class="default"><a href="polynom1.html#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p5">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="polynom2.html#K4" title="POLYNOM2:func.4">@</a> </span>)</span>,<font color="Maroon" title="c4">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="polynom2.html#E3:43_1_1"><i><font color="Green" title="E30">A2</font></i></a>, <a class="txt" href="polynom2.html#E4:43_1_1"><i><font color="Green" title="E31">A3</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:43_1_1"/>
 ex <font color="Olive" title="b1">y</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font> st <br/>(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p2">(<span class="default"><a href="polynom1.html#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> &amp;  <a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <font color="Maroon" title="c7">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <font color="Olive" title="b1">y</font> &amp; ( for <font color="Olive" title="b2">i</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>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">y</font> holds <br/><font color="Olive" title="b1">y</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="triang_1.html#K2" title="TRIANG_1:func.2">SgmX</a> <span class="p4">(<span class="default"><a href="polynom1.html#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b2">i</font></span>)</span> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">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" title="TRIANG_1:func.2">SgmX</a> <span class="p5">(<span class="default"><a href="polynom1.html#K17" title="POLYNOM1:func.17">BagOrder</a> <font color="Maroon" title="c1">n</font></span>)</span>,<span class="p5">(<span class="default"><a href="polynom1.html#K12" title="POLYNOM1:func.12">Support</a> <font color="Maroon" title="c3">p</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Olive" title="b2">i</font></span>)</span> <a href="polynom2.html#K4" title="POLYNOM2:func.4">@</a> </span>)</span>,<font color="Maroon" title="c4">x</font></span>)</span> ) )
 <b>by </b><i><a class="txt" href="polynom2.html#E4:43_1_1"><i><font color="Green" title="E31">A3</font></i></a></i>;<br/>


</div>
