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<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="algseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>,   <a href="hidden.html#M1">set</a> ] means <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="polynom5.html#K3" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="polynom5.html#K2" target="_self">`^</a> <span class="p2">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span>;<br/>
<div><i><font color="Green">E41</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="58_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:58_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="polynom5.html#K3" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="polynom5.html#K2" target="_self">`^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <b>by </b><i><a class="ref" href="polynom3.html#D12">POLYNOM3:def 12</a></i>;<br/><b>take </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font>;<br/><b>thus </b><a NAME="E3:58_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>7</sub></font>]
 ;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">c<sub>1</sub></font></span>)</span><b> such that </b><br/><a NAME="E3:58_1_1"/><i><font color="Green">E42</font></i>: 
 <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E4:58_1_1"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="finseq_1.html#S5">FINSEQ_1:sch 5</a>(<a class="txt" href="polynom5.html#E1:58_1_1"><i><font color="Green">E41</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> =  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">c<sub>5</sub></font> as   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="ref" href="polynom3.html#D12">POLYNOM3:def 12</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>6</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>5</sub></font>
;<br/>

<b>thus </b><a NAME="E6:58_1_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">c<sub>5</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="algseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>5</sub></font> holds <br/><font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="polynom5.html#K3" target="_self">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="polynom5.html#K2" target="_self">`^</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> ) )
 <b>by </b><i><a class="txt" href="polynom5.html#E3:58_1_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="polynom5.html#E4:58_1_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>


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