<?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">T</font> be  <a href="relat_2.html#V6">connected</a> <a href="bagorder.html#NM1">TermOrder</a> of <font color="Maroon">n</font>;<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">f</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">P</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span>;<br/><b>let </b><font color="Maroon">A</font> be    <a href="ideal_1.html#M2">LeftLinearCombination</a> of <font color="Maroon">P</font>;<br/>











<b>assume </b><a NAME="E1:49"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">A</font> <a href="groeb_2.html#R2" target="_self">is_MonomialRepresentation_of</a> <font color="Maroon">f</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="E2:49"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">A</font> holds <br/>for <font color="Olive">m</font> being  <a href="polynom7.html#NM1">Monomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font><br/> for <font color="Olive">p</font> being  <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font>  st <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Olive">m</font> <a href="polynom1.html#K28">*'</a> <font color="Olive">p</font> &amp; <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> holds <br/><span class="p1">(<span class="default"><font color="Olive">m</font> <a href="polynom1.html#K28">*'</a> <font color="Olive">p</font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 ;<br/>

<a NAME="E3:49"/><i><font color="Green">E34</font></i>: 
 <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font>
 
<b>by </b><i>, </i>;<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> ] <b>means </b>for <font color="Olive">f</font> being  <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font><br/> for <font color="Olive">A</font> being   <a href="ideal_1.html#M2">LeftLinearCombination</a> of <font color="Maroon">P</font>  st <font color="Olive">A</font> <a href="groeb_2.html#R2" target="_self">is_MonomialRepresentation_of</a> <font color="Olive">f</font> &amp; <font color="Olive">f</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Olive">A</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></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 <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Olive">A</font> holds <br/>for <font color="Olive">m</font> being  <a href="polynom7.html#NM1">Monomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font><br/> for <font color="Olive">p</font> being  <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font>  st <font color="Olive">A</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Olive">m</font> <a href="polynom1.html#K28">*'</a> <font color="Olive">p</font> &amp; <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> holds <br/><span class="p1">(<span class="default"><font color="Olive">m</font> <a href="polynom1.html#K28">*'</a> <font color="Olive">p</font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font> ) holds <br/><font color="Olive">f</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>;<br/>
<a NAME="E4:49"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">f</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">A</font> be    <a href="ideal_1.html#M2">LeftLinearCombination</a> of <font color="Maroon">P</font>;<br/>



<b>assume </b><a NAME="E1:49_1"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">A</font> <a href="groeb_2.html#R2" target="_self">is_MonomialRepresentation_of</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">A</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/>

<b>assume </b><a NAME="E2:49_1"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">A</font> holds <br/>for <font color="Olive">m</font> being  <a href="polynom7.html#NM1">Monomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font><br/> for <font color="Olive">p</font> being  <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font>  st <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Olive">m</font> <a href="polynom1.html#K28">*'</a> <font color="Olive">p</font> &amp; <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> holds <br/><span class="p1">(<span class="default"><font color="Olive">m</font> <a href="polynom1.html#K28">*'</a> <font color="Olive">p</font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 ;<br/>

<font color="Maroon">f</font> = 
 <a href="rlvect_1.html#K9">Sum</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 <span class="p2">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="finseq_1.html#T32">FINSEQ_1:32</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T60">RLVECT_1:60</a></i>
<br/>.= 
 <a href="polynom1.html#K26">0_</a> <font color="Maroon">n</font>,<font color="Maroon">L</font>
<b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>
;<br/>
<b>hence </b><a NAME="E4:49_1"/>
<font color="Maroon">f</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="polynom1.html#T81">POLYNOM1:81</a></i>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green">E39</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="49_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:49_2"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 ;<br/><a NAME="E2:49_2"/>
for <font color="Olive">f</font> being  <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font><br/> for <font color="Olive">A</font> being   <a href="ideal_1.html#M2">LeftLinearCombination</a> of <font color="Maroon">P</font>  st <font color="Olive">A</font> <a href="groeb_2.html#R2" target="_self">is_MonomialRepresentation_of</a> <font color="Olive">f</font> &amp; <font color="Olive">f</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Olive">A</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 &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 <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Olive">A</font> holds <br/>for <font color="Olive">m</font> being  <a href="polynom7.html#NM1">Monomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font><br/> for <font color="Olive">p</font> being  <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font>  st <font color="Olive">A</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Olive">m</font> <a href="polynom1.html#K28">*'</a> <font color="Olive">p</font> &amp; <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> holds <br/><span class="p1">(<span class="default"><font color="Olive">m</font> <a href="polynom1.html#K28">*'</a> <font color="Olive">p</font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font> ) holds <br/><font color="Olive">f</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">f</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">A</font> be    <a href="ideal_1.html#M2">LeftLinearCombination</a> of <font color="Maroon">P</font>;<br/>



<b>assume </b><a NAME="E1:49_2_1"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">A</font> <a href="groeb_2.html#R2" target="_self">is_MonomialRepresentation_of</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">A</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/>

<b>assume </b><a NAME="E2:49_2_1"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">A</font> holds <br/>for <font color="Olive">m</font> being  <a href="polynom7.html#NM1">Monomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font><br/> for <font color="Olive">p</font> being  <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font>  st <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Olive">m</font> <a href="polynom1.html#K28">*'</a> <font color="Olive">p</font> &amp; <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> holds <br/><span class="p1">(<span class="default"><font color="Olive">m</font> <a href="polynom1.html#K28">*'</a> <font color="Olive">p</font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 ;<br/>

<b>set </b><font color="Maroon">B</font> = <font color="Maroon">A</font> <a href="relset_1.html#K8">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon">B</font> = <font color="Maroon">A</font> <a href="relset_1.html#K8">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">k</font></span>)</span> as   <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K30">Polynom-Ring</a> <font color="Maroon">n</font>,<font color="Maroon">L</font></span>)</span> <b>by </b><i><a class="ref" href="finseq_1.html#T23">FINSEQ_1:23</a></i>;<br/>
<b>reconsider </b><font color="Maroon">B</font> = <font color="Maroon">B</font> as    <a href="ideal_1.html#M2">LeftLinearCombination</a> of <font color="Maroon">P</font> <b>by </b><i><a class="ref" href="ideal_1.html#T42">IDEAL_1:42</a></i>;<br/>
<b>set </b><font color="Maroon">g</font> =  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">B</font>;<br/>
<b>reconsider </b><font color="Maroon">g</font> =  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">B</font> as   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font> <b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/>
<a NAME="E6:49_2_1"/><i><font color="Green">E43</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E7:49_2_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E8:49_2_1"/><b>then </b><i><font color="Green">E48</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T21">FINSEQ_1:21</a></i>;<br/>
<a NAME="E9:49_2_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E10:49_2_1"/>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">B</font> <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="txt" href="groeb_2.html#E2:49"><i><font color="Green">E33</font></i></a>, <a class="ref" href="finseq_1.html#T21">FINSEQ_1:21</a></i>;<br/>
<a NAME="E11:49_2_1"/><b>then </b><i><font color="Green">E50</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="txt" href="groeb_2.html#E4:49"><i><font color="Green">E35</font></i></a>, <a class="ref" href="finseq_1.html#T7">FINSEQ_1:7</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:49_2_1_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">B</font>
 ;<br/><b>then consider </b><font color="Maroon">m</font> being   <a href="polynom7.html#NM1">Monomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font>, <font color="Maroon">p</font> being   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:49_2_1_1"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">A</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">p</font> )
 <b>by </b><i>, , </i>;<br/><font color="Maroon">B</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> = 
<font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
<br/>.= 
<font color="Maroon">A</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>
<b>by </b><i>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
;<br/><b>hence </b><a NAME="E5:49_2_1_1"/>
 ex <font color="Olive">m</font> being  <a href="polynom7.html#NM1">Monomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font> ex <font color="Olive">p</font> being  <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font> st <br/>( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">B</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Olive">m</font> <a href="polynom1.html#K28">*'</a> <font color="Olive">p</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<a NAME="E13:49_2_1"/><b>then </b><i><font color="Green">E54</font></i>: 
<font color="Maroon">B</font> <a href="groeb_2.html#R2" target="_self">is_MonomialRepresentation_of</a> <font color="Maroon">g</font>
 
<b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">h</font> = <font color="Maroon">A</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font> <b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>;<br/>
<a NAME="E15:49_2_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">A</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T6">FINSEQ_1:6</a></i>;<br/>
<b>then </b><font color="Maroon">B</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span> = 
<font color="Maroon">B</font> <a href="finseq_1.html#K7">^</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span>
<b>by </b><i><a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<font color="Maroon">A</font>
<b>by </b><i>, <a class="ref" href="finseq_3.html#T61">FINSEQ_3:61</a></i>
;<br/>
<b>then </b><font color="Maroon">f</font> = 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">B</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">A</font> <a href="finseq_4.html#K4">/.</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>
<b>by </b><i>, <a class="ref" href="rlvect_1.html#T58">RLVECT_1:58</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">B</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T61">RLVECT_1:61</a></i>
<br/>.= 
<font color="Maroon">g</font> <a href="polynom1.html#K22">+</a> <font color="Maroon">h</font>
<b>by </b><i><a class="ref" href="polynom1.html#D27">POLYNOM1:def 27</a></i>
;<br/>
<a NAME="E18:49_2_1"/><b>then </b><i><font color="Green">E56</font></i>: 
<font color="Maroon">f</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font></span>)</span> <a href="rlvect_1.html#K2">+</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font></span>)</span>
 
<b>by </b><i><a class="ref" href="polynom1.html#D21">POLYNOM1:def 21</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_2_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:49_2_1_4"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">B</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_2_1_4_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">m</font> be   <a href="polynom7.html#NM1">Monomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font>;<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>assume </b><a NAME="E1:49_2_1_4_1"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">p</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> )
 ;<br/><a NAME="E2:49_2_1_4_1"/><b>then </b>
<font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="txt" href="groeb_2.html#E1:49_2"><i><font color="Green">E40</font></i></a>, <a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>;<br/><b>hence </b><a NAME="E3:49_2_1_4_1"/>
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">p</font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 <b>by </b><i><a class="txt" href="groeb_2.html#E1:49"><i><font color="Green">E31</font></i></a>, , <a class="txt" href="groeb_2.html#E1:49_2"><i><font color="Green">E40</font></i></a>, <a class="txt" href="groeb_2.html#E1:49_2_1"><i><font color="Green">E41</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:49_2_1_4"/>
for <font color="Olive">m</font> being  <a href="polynom7.html#NM1">Monomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font><br/> for <font color="Olive">p</font> being  <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font>  st <font color="Maroon">B</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Olive">m</font> <a href="polynom1.html#K28">*'</a> <font color="Olive">p</font> &amp; <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> holds <br/><span class="p1">(<span class="default"><font color="Olive">m</font> <a href="polynom1.html#K28">*'</a> <font color="Olive">p</font></span>)</span> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 ;<br/></div><b>end;</b></div>
<a NAME="E20:49_2_1"/><b>then </b><i><font color="Green">E60</font></i>: 
<font color="Maroon">g</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 
<b>by </b><i>, <a class="txt" href="groeb_2.html#E3:49"><i><font color="Green">E34</font></i></a>, </i>;<br/>
<b>consider </b><font color="Maroon">m</font> being   <a href="polynom7.html#NM1">Monomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font>, <font color="Maroon">p</font> being   <a href="polynom1.html#NM4">Polynomial</a> of <font color="Maroon">n</font>,<font color="Maroon">L</font><b> such that </b><br/><a NAME="E22:49_2_1"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">A</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="polynom1.html#K28">*'</a> <font color="Maroon">p</font> )
 <b>by </b><i>, , </i>;<br/>
<a NAME="E23:49_2_1"/>
<font color="Maroon">A</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E24:49_2_1"/><b>then </b>
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font>
 
<b>by </b><i><a class="txt" href="groeb_2.html#E1:49"><i><font color="Green">E31</font></i></a>, , <a class="txt" href="groeb_2.html#E1:49_2_1"><i><font color="Green">E41</font></i></a></i>;<br/>
<b>hence </b><a NAME="E25:49_2_1"/>
<font color="Maroon">f</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 <b>by </b><i><a class="txt" href="groeb_2.html#E5:49"><i><font color="Green">E39</font></i></a>, <a class="txt" href="groeb_2.html#E1:49_2"><i><font color="Green">E40</font></i></a>, <a class="ref" href="rlvect_1.html#D7">RLVECT_1:def 7</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:49_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/></div><b>end;</b></div>
<a NAME="E6:49"/><i><font color="Green">E62</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>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<b>consider </b><font color="Maroon">n</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="E8:49"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">A</font>
 ;<br/>
<b>thus </b><a NAME="E9:49"/>
<font color="Maroon">f</font> <a href="polynom1.html#K15">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">L</font>
 <b>by </b><i>, , <a class="ref" href="groeb_2.html#T2" target="_self">Th2</a>, , </i>;<br/>


</div>
