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

<b>let </b><font color="Maroon">n</font> be   <a href="ordinal1.html#NM2">Ordinal</a>;<br/><b>let </b><font color="Maroon">L</font> be  non <a href="struct_0.html#V3">empty</a> <a href="rlvect_1.html#V4">add-associative</a> <a href="rlvect_1.html#V5">right_zeroed</a> <a href="rlvect_1.html#V6">right_complementable</a> <a href="group_1.html#V2">unital</a> <a href="vectsp_1.html#V7">distributive</a> non <a href="realset2.html#V3">trivial</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">p</font> be   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="polynom1.html#NM1">bag</a> of <font color="Maroon">n</font>;<br/>







<b>assume </b><a NAME="E1:44"/><i><font color="Green">E32</font></i>: 
 <a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">b</font></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM5">Function</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font>;<br/>

<b>consider </b><font color="Maroon">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><b> such that </b><br/><a NAME="E3:44"/><i><font color="Green">E34</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">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="polynom2.html#K5" target="_self">eval</a> <font color="Maroon">p</font>,<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">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="Maroon">y</font> holds <br/><font color="Maroon">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/>
<b>reconsider </b><font color="Maroon">sp</font> =  <a href="polynom1.html#K12">Support</a> <font color="Maroon">p</font> as  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font></span>)</span> ;<br/>
<b>set </b><font color="Maroon">sg</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>,<font color="Maroon">sp</font>;<br/>
<a NAME="E5:44"/><i><font color="Green">E35</font></i>: 
 <a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font> <a href="orders_1.html#R3">linearly_orders</a> <font color="Maroon">sp</font>
 
<b>by </b><i/>;<br/>
<a NAME="E6:44"/><b>then </b><i><font color="Green">E36</font></i>: 
(  <a href="relat_1.html#K2">rng</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>,<font color="Maroon">sp</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">b</font></span><a href="tarski.html#K1">}</a></span> &amp; ( for <font color="Olive">l</font>, <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">l</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</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>,<font color="Maroon">sp</font></span>)</span> &amp; <font color="Olive">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</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>,<font color="Maroon">sp</font></span>)</span> &amp; <font color="Olive">l</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">m</font> holds <br/>( <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>,<font color="Maroon">sp</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">l</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</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>,<font color="Maroon">sp</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">m</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><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>,<font color="Maroon">sp</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">l</font></span>)</span>,<span class="p2">(<span class="default"><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>,<font color="Maroon">sp</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">m</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K17">BagOrder</a> <font color="Maroon">n</font> ) ) )
 
<b>by </b><i>, <a class="ref" href="triang_1.html#D2">TRIANG_1:def 2</a></i>;<br/>
<a NAME="E7:44"/><b>then </b>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</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>,<font color="Maroon">sp</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E8:44"/><b>then </b><i><font color="Green">E37</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</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>,<font color="Maroon">sp</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_3.html#T33">FINSEQ_3:33</a></i>;<br/>
<a NAME="E9:44"/><b>then </b><i><font color="Green">E39</font></i>: 
for <font color="Olive">u</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span> holds <br/><font color="Olive">u</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</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>,<font color="Maroon">sp</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E10:44"/>
for <font color="Olive">u</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">u</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</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>,<font color="Maroon">sp</font></span>)</span> holds <br/><font color="Olive">u</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:44_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</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>,<font color="Maroon">sp</font></span>)</span>
 ;<br/>

<b>assume </b><a NAME="E2:44_1"/><i><font color="Green">E45</font></i>: 
not <font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default">1</span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<b>reconsider </b><font color="Maroon">u</font> = <font color="Maroon">u</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="txt" href="polynom2.html#E8:44"><i><font color="Green">E37</font></i></a></i>;<br/>
<a NAME="E4:44_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">u</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 
<b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E5:44_1"/><i><font color="Green">E56</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">u</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E2:44_1_1"/><i><font color="Green">E57</font></i>: 
 <a href="finseq_1.html#K4">dom</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>,<font color="Maroon">sp</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D2">FINSEQ_1:def 2</a></i>;<br/>
<a NAME="E3:44_1_1"/>
 <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">l</font> where <font color="Olive">l</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">l</font> &amp; <font color="Olive">l</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#D1">FINSEQ_1:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon">m'</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E5:44_1_1"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">m'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> &amp; 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> <font color="Maroon">k</font> )
 <b>by </b><i><a class="txt" href="polynom2.html#E8:44"><i><font color="Green">E37</font></i></a>, </i>;<br/>
<b>thus </b><a NAME="E6:44_1_1"/>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">u</font>
 <b>by </b><i><a class="txt" href="polynom2.html#E9:44"><i><font color="Green">E39</font></i></a>, <a class="txt" href="polynom2.html#E1:44_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:44_1"/>
( <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>,<font color="Maroon">sp</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</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>,<font color="Maroon">sp</font></span>)</span> <a href="funct_1.html#K1">.</a> 1 &amp; <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>,<font color="Maroon">sp</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">u</font> <a href="hidden.html#R1">=</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>,<font color="Maroon">sp</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">u</font> )
 
<b>by </b><i><a class="txt" href="polynom2.html#E5:44"><i><font color="Green">E35</font></i></a>, <a class="txt" href="polynom2.html#E8:44"><i><font color="Green">E37</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E7:44_1"/><b>then </b><i><font color="Green">E59</font></i>: 
( <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>,<font color="Maroon">sp</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</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>,<font color="Maroon">sp</font></span>)</span> &amp; <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>,<font color="Maroon">sp</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">u</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</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>,<font color="Maroon">sp</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="polynom2.html#E5:44"><i><font color="Green">E35</font></i></a>, <a class="txt" href="polynom2.html#E8:44"><i><font color="Green">E37</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>then </b><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>,<font color="Maroon">sp</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 = 
<font color="Maroon">b</font>
<b>by </b><i><a class="txt" href="polynom2.html#E3:44"><i><font color="Green">E34</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>
<br/>.= 
<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>,<font color="Maroon">sp</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">u</font>
<b>by </b><i><a class="txt" href="polynom2.html#E3:44"><i><font color="Green">E34</font></i></a>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>
;<br/>
<b>hence </b><a NAME="E9:44_1"/>
contradiction
 <b>by </b><i>, <a class="txt" href="polynom2.html#E5:44"><i><font color="Green">E35</font></i></a>, <a class="txt" href="polynom2.html#E8:44"><i><font color="Green">E37</font></i></a>, , <a class="ref" href="triang_1.html#D2">TRIANG_1:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E11:44"/><b>then </b><i><font color="Green">E60</font></i>: 
 <a href="finseq_1.html#K4">dom</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>,<font color="Maroon">sp</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> 1
 
<b>by </b><i><a class="txt" href="polynom2.html#E6:44"><i><font color="Green">E36</font></i></a>, <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>
<a NAME="E12:44"/><b>then </b><i><font color="Green">E61</font></i>: 
 <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>,<font color="Maroon">sp</font></span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E13:44"/><i><font color="Green">E62</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</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>,<font color="Maroon">sp</font></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom2.html#E8:44"><i><font color="Green">E37</font></i></a>, <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E14:44"/>
<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>,<font color="Maroon">sp</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</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>,<font color="Maroon">sp</font></span>)</span> <a href="funct_1.html#K1">.</a> 1
 
<b>by </b><i><a class="txt" href="polynom2.html#E5:44"><i><font color="Green">E35</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E15:44"/><b>then </b>
<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>,<font color="Maroon">sp</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</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>,<font color="Maroon">sp</font></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom2.html#E9:44"><i><font color="Green">E39</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E16:44"/><b>then </b><i><font color="Green">E63</font></i>: 
<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>,<font color="Maroon">sp</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="txt" href="polynom2.html#E3:44"><i><font color="Green">E34</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<i><font color="Green">E64</font></i>: <font color="Maroon">y</font> <a href="funct_1.html#K1">.</a> 1 = 
<font color="Maroon">y</font> <a href="finseq_4.html#K4">/.</a> 1
<b>by </b><i><a class="txt" href="polynom2.html#E1:44"><i><font color="Green">E32</font></i></a>, , <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>
<br/>.= 
<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>,<font color="Maroon">sp</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</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>,<font color="Maroon">sp</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="polynom2.html#K4" target="_self">@</a> </span>)</span>,<font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom2.html#E1:44"><i><font color="Green">E32</font></i></a>, </i>
<br/>.= 
<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>,<font color="Maroon">sp</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" target="_self">eval</a> <font color="Maroon">b</font>,<font color="Maroon">x</font></span>)</span>
<b>by </b><i/>
;<br/>
<a NAME="E18:44"/><i><font color="Green">E65</font></i>: 
<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>,<font color="Maroon">sp</font></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</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>,<font color="Maroon">sp</font></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom2.html#E5:44"><i><font color="Green">E35</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E19:44"/><b>then </b><i><font color="Green">E66</font></i>: 
<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>,<font color="Maroon">sp</font></span>)</span> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="txt" href="polynom2.html#E3:44"><i><font color="Green">E34</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E20:44"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="polynom1.html#D14">POLYNOM1:def 14</a></i>;<br/>
<a NAME="E21:44"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="polynom1.html#K14">Bags</a> <font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E22:44"/><b>then </b>
1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="partfun1.html#K1">*</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>,<font color="Maroon">sp</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="polynom2.html#E5:44"><i><font color="Green">E35</font></i></a>, , , <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="partfun1.html#K1">*</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>,<font color="Maroon">sp</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="partfun1.html#K1">*</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>,<font color="Maroon">sp</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> 1
<b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><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>,<font color="Maroon">sp</font></span>)</span> <a href="funct_1.html#K1">.</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="polynom2.html#E5:44"><i><font color="Green">E35</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">p</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font>
<b>by </b><i><a class="txt" href="polynom2.html#E3:44"><i><font color="Green">E34</font></i></a>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>
;<br/>
<a NAME="E24:44"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font></span>)</span> <a href="group_1.html#K1">*</a> <span class="p3">(<span class="default"><a href="polynom2.html#K3" target="_self">eval</a> <font color="Maroon">b</font>,<font color="Maroon">x</font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="polynom2.html#E1:44"><i><font color="Green">E32</font></i></a>, , , <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/>
<b>hence </b><a NAME="E25:44"/>
 <a href="polynom2.html#K5" target="_self">eval</a> <font color="Maroon">p</font>,<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</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> <font color="Maroon">b</font>,<font color="Maroon">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="polynom2.html#E1:44"><i><font color="Green">E32</font></i></a>, <a class="ref" href="rlvect_1.html#T61">RLVECT_1:61</a></i>;<br/>


</div>
