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<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>set </b><font color="Maroon">cS</font> =  <a href="uproots.html#K1" target="_self">canFS</a> <span class="p1">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:19_1_2_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="uproots.html#NK4" target="_self">degree</a> <font color="Maroon">b</font>
 ;<br/><b>consider </b><font color="Maroon">f</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E3:19_1_2_1"/><i><font color="Green">E47</font></i>: 
 <a href="uproots.html#NK4" target="_self">degree</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="rvsum_1.html#K15">Sum</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E4:19_1_2_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E5:19_1_2_1"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:19_1_2_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:19_1_2_1_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E4:19_1_2_1_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E5:19_1_2_1_1"/>
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p3">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/>
<a NAME="E6:19_1_2_1_1"/><b>then </b>
<font color="Maroon">f</font> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> 
 
;<br/>
<b>hence </b><a NAME="E7:19_1_2_1_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/><b>take </b><font color="Maroon">f</font> = <font color="Maroon">f</font>;<br/><b>thus </b><a NAME="E7:19_1_2_1"/>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="wsierp_1.html#K7">Sum</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span> )
 <b>by </b><i>, , </i>;<br/></div>
<b>end;</b></div>

<b>given </b><font color="Maroon">f</font> being    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><a NAME="E2:19_1_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="wsierp_1.html#K7">Sum</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E3:19_1_2"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><a href="uproots.html#K1" target="_self">canFS</a> <span class="p2">(<span class="default"><a href="polynom1.html#K11">support</a> <font color="Maroon">b</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E4:19_1_2"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as    <a href="finseq_1.html#M2">FinSequence</a> of  <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="finseq_1.html#D4">FINSEQ_1:def 4</a></i>;<br/>
<a NAME="E6:19_1_2"/>
<font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font>
 
;<br/>
<b>hence </b><a NAME="E7:19_1_2"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="uproots.html#NK4" target="_self">degree</a> <font color="Maroon">b</font>
 <b>by </b><i>, , </i>;<br/>


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