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<b>let </b><font color="Maroon">c<sub>1</sub></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#V4">associative</a> <a href="group_1.html#V7">commutative</a> <a href="vectsp_1.html#V7">distributive</a> <a href="vectsp_1.html#V8">left_unital</a> <a href="vectsp_2.html#V2">domRing-like</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<a NAME="E1:43"/><i><font color="Green">E33</font></i>: 
(  <a href="algseq_1.html#K3">len</a> <span class="p1"><a href="algseq_1.html#K5">&lt;%</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="algseq_1.html#K5">%&gt;</a></span> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp;  <a href="algseq_1.html#K3">len</a> <span class="p1"><a href="algseq_1.html#K5">&lt;%</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="algseq_1.html#K5">%&gt;</a></span> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 
<b>by </b><i><a class="ref" href="algseq_1.html#D6">ALGSEQ_1:def 6</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:43_1"/>
( (  <a href="algseq_1.html#K3">len</a> <span class="p1"><a href="algseq_1.html#K5">&lt;%</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="algseq_1.html#K5">%&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp;  <a href="algseq_1.html#K3">len</a> <span class="p1"><a href="algseq_1.html#K5">&lt;%</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="algseq_1.html#K5">%&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 ) or  <a href="algseq_1.html#K3">len</a> <span class="p1"><a href="algseq_1.html#K5">&lt;%</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="algseq_1.html#K5">%&gt;</a></span> <a href="hidden.html#R1">=</a> 0 or  <a href="algseq_1.html#K3">len</a> <span class="p1"><a href="algseq_1.html#K5">&lt;%</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="algseq_1.html#K5">%&gt;</a></span> <a href="hidden.html#R1">=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_1"><b>suppose </b></a><a NAME="E1:43_1_1"/><i><font color="Green">E34</font></i>: 
(  <a href="algseq_1.html#K3">len</a> <span class="p1"><a href="algseq_1.html#K5">&lt;%</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="algseq_1.html#K5">%&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 &amp;  <a href="algseq_1.html#K3">len</a> <span class="p1"><a href="algseq_1.html#K5">&lt;%</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="algseq_1.html#K5">%&gt;</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 ;<br/><div class="add"><a NAME="E2:43_1_1"/><b>then </b>
(  <a href="algseq_1.html#K3">len</a> <span class="p1"><a href="algseq_1.html#K5">&lt;%</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="algseq_1.html#K5">%&gt;</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp;  <a href="algseq_1.html#K3">len</a> <span class="p1"><a href="algseq_1.html#K5">&lt;%</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="algseq_1.html#K5">%&gt;</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 ;<br/><a NAME="E3:43_1_1"/><b>then </b>
(  <a href="algseq_1.html#K3">len</a> <span class="p1"><a href="algseq_1.html#K5">&lt;%</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="algseq_1.html#K5">%&gt;</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1 &amp;  <a href="algseq_1.html#K3">len</a> <span class="p1"><a href="algseq_1.html#K5">&lt;%</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="algseq_1.html#K5">%&gt;</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:43_1_1"/><b>then </b><i><font color="Green">E35</font></i>: 
(  <a href="algseq_1.html#K3">len</a> <span class="p1"><a href="algseq_1.html#K5">&lt;%</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="algseq_1.html#K5">%&gt;</a></span> <a href="hidden.html#R1">=</a> 1 &amp;  <a href="algseq_1.html#K3">len</a> <span class="p1"><a href="algseq_1.html#K5">&lt;%</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="algseq_1.html#K5">%&gt;</a></span> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="txt" href="polynom5.html#E1:43"><i><font color="Green">E33</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E5:43_1_1"/><b>then </b>
( <span class="p1"><a href="algseq_1.html#K5">&lt;%</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="algseq_1.html#K5">%&gt;</a></span> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <span class="p3"><a href="algseq_1.html#K5">&lt;%</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="algseq_1.html#K5">%&gt;</a></span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font> &amp; <span class="p1"><a href="algseq_1.html#K5">&lt;%</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="algseq_1.html#K5">%&gt;</a></span> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <span class="p3"><a href="algseq_1.html#K5">&lt;%</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="algseq_1.html#K5">%&gt;</a></span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="polynom5.html#E27"><i><font color="Green">Lemma18</font></i></a></i>;<br/><a NAME="E6:43_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2"><a href="algseq_1.html#K5">&lt;%</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="algseq_1.html#K5">%&gt;</a></span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algseq_1.html#K3">len</a> <span class="p4"><a href="algseq_1.html#K5">&lt;%</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="algseq_1.html#K5">%&gt;</a></span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="algseq_1.html#K5">&lt;%</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="algseq_1.html#K5">%&gt;</a></span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algseq_1.html#K3">len</a> <span class="p4"><a href="algseq_1.html#K5">&lt;%</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="algseq_1.html#K5">%&gt;</a></span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="vectsp_2.html#D5">VECTSP_2:def 5</a></i>;<br/><a NAME="E7:43_1_1"/><b>then </b><i><font color="Green">E36</font></i>: 
 <a href="algseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2"><a href="algseq_1.html#K5">&lt;%</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="algseq_1.html#K5">%&gt;</a></span> <a href="polynom3.html#K14">*'</a> <span class="p2"><a href="algseq_1.html#K5">&lt;%</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="algseq_1.html#K5">%&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1
 <b>by </b><i><a class="txt" href="polynom5.html#E4:43_1_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="polynom4.html#T13">POLYNOM4:13</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="finseq_1.html#M2">FinSequence</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E9:43_1_1"/><i><font color="Green">E37</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0 <a href="nat_1.html#K1">+</a> 1
 <b>and </b><br/><a NAME="E10:43_1_1"/><i><font color="Green">E38</font></i>: 
<span class="p1">(<span class="default"><span class="p2"><a href="algseq_1.html#K5">&lt;%</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="algseq_1.html#K5">%&gt;</a></span> <a href="polynom3.html#K14">*'</a> <span class="p2"><a href="algseq_1.html#K5">&lt;%</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="algseq_1.html#K5">%&gt;</a></span></span>)</span> <a href="normsp_1.html#K2">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">c<sub>4</sub></font>
 <b>and </b><br/><a NAME="E11:43_1_1"/><i><font color="Green">E39</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="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>4</sub></font> holds <br/><font color="Maroon">c<sub>4</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"><span class="p2"><a href="algseq_1.html#K5">&lt;%</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="algseq_1.html#K5">%&gt;</a></span> <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="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="algseq_1.html#K5">&lt;%</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="algseq_1.html#K5">%&gt;</a></span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K5">-'</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="polynom3.html#D11">POLYNOM3:def 11</a></i>;<br/><a NAME="E12:43_1_1"/>
1 <a href="hidden.html#R2">in</a>  <a href="finsop_1.html#K5">dom</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="polynom5.html#E9:43_1_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>then </b><font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> 1 = 
<span class="p1">(<span class="default"><span class="p2"><a href="algseq_1.html#K5">&lt;%</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="algseq_1.html#K5">%&gt;</a></span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default">1 <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="algseq_1.html#K5">&lt;%</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="algseq_1.html#K5">%&gt;</a></span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="polynom5.html#E11:43_1_1"><i><font color="Green">E39</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="algseq_1.html#K5">&lt;%</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="algseq_1.html#K5">%&gt;</a></span> <a href="normsp_1.html#K2">.</a> 0</span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="algseq_1.html#K5">&lt;%</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="algseq_1.html#K5">%&gt;</a></span> <a href="normsp_1.html#K2">.</a> <span class="p2">(<span class="default">1 <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T51">BINARITH:51</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="algseq_1.html#K5">&lt;%</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="algseq_1.html#K5">%&gt;</a></span> <a href="normsp_1.html#K2">.</a> 0</span>)</span> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="algseq_1.html#K5">&lt;%</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="algseq_1.html#K5">%&gt;</a></span> <a href="normsp_1.html#K2">.</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T51">BINARITH:51</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="algseq_1.html#K5">&lt;%</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="algseq_1.html#K5">%&gt;</a></span> <a href="normsp_1.html#K2">.</a> 0</span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="algseq_1.html#D6">ALGSEQ_1:def 6</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="algseq_1.html#D6">ALGSEQ_1:def 6</a></i>
;<br/><a NAME="E14:43_1_1"/><b>then </b><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="binarith.html#K13">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="polynom5.html#E9:43_1_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/><div><i><font color="Green">E41</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1_2"><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="E1:43_1_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 ;<br/><a NAME="E2:43_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 <a href="nat_1.html#K1">+</a> 1
 ;<br/><a NAME="E3:43_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:43_1_1_2"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><b>hence </b><span class="p1">(<span class="default"><span class="p2"><a href="algseq_1.html#K5">&lt;%</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="algseq_1.html#K5">%&gt;</a></span> <a href="polynom3.html#K14">*'</a> <span class="p2"><a href="algseq_1.html#K5">&lt;%</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="algseq_1.html#K5">%&gt;</a></span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font> = 
<font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="polynom5.html#E10:43_1_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="polynom5.html#E14:43_1_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="rlvect_1.html#T61">RLVECT_1:61</a></i>
<br/>.= 
<span class="p1"><a href="algseq_1.html#K5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="algseq_1.html#K5">%&gt;</a></span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="polynom5.html#E4:43_1_1_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="algseq_1.html#D6">ALGSEQ_1:def 6</a></i>
;<br/><br/></div><b>end;</b></div><a NAME="E16:43_1_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1_1_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:43_1_1_3"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font> or <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/>

<a NAME="E2:43_1_1_3"/><b>then </b>
( <span class="p1"><a href="algseq_1.html#K5">&lt;%</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="algseq_1.html#K5">%&gt;</a></span> <a href="funct_2.html#R2">=</a>  <a href="polynom3.html#K12">0_.</a> <font color="Maroon">c<sub>1</sub></font> or <span class="p1"><a href="algseq_1.html#K5">&lt;%</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="algseq_1.html#K5">%&gt;</a></span> <a href="funct_2.html#R2">=</a>  <a href="polynom3.html#K12">0_.</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="polynom5.html#T35" target="_self">Th35</a></i>;<br/>
<b>hence </b><a NAME="E3:43_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="polynom5.html#E1:43_1_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="polynom4.html#T6">POLYNOM4:6</a></i>;<br/>


</div><b>end;</b></div><a NAME="E17:43_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="vectsp_2.html#D5">VECTSP_2:def 5</a></i>;<br/><a NAME="E18:43_1_1"/><b>then </b>
 <a href="algseq_1.html#K3">len</a> <span class="p1"><a href="algseq_1.html#K5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="algseq_1.html#K5">%&gt;</a></span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="polynom5.html#T34" target="_self">Th34</a></i>;<br/><b>hence </b><a NAME="E19:43_1_1"/>
<span class="p1"><a href="algseq_1.html#K5">&lt;%</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="algseq_1.html#K5">%&gt;</a></span> <a href="polynom3.html#K14">*'</a> <span class="p1"><a href="algseq_1.html#K5">&lt;%</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="algseq_1.html#K5">%&gt;</a></span> <a href="funct_2.html#R2">=</a> <span class="p1"><a href="algseq_1.html#K5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="algseq_1.html#K5">%&gt;</a></span>
 <b>by </b><i><a class="txt" href="polynom5.html#E7:43_1_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="polynom5.html#E15:43_1_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="algseq_1.html#T28">ALGSEQ_1:28</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_2"><b>suppose </b></a><a NAME="E1:43_1_2"/><i><font color="Green">E34</font></i>: 
 <a href="algseq_1.html#K3">len</a> <span class="p1"><a href="algseq_1.html#K5">&lt;%</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="algseq_1.html#K5">%&gt;</a></span> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:43_1_2"/><b>then </b><i><font color="Green">E35</font></i>: 
<span class="p1"><a href="algseq_1.html#K5">&lt;%</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="algseq_1.html#K5">%&gt;</a></span> <a href="funct_2.html#R2">=</a>  <a href="polynom3.html#K12">0_.</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="polynom4.html#T8">POLYNOM4:8</a></i>;<br/><a NAME="E3:43_1_2"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="polynom5.html#E1:43_1_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="polynom5.html#T34" target="_self">Th34</a></i>;<br/><b>thus </b><span class="p1"><a href="algseq_1.html#K5">&lt;%</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="algseq_1.html#K5">%&gt;</a></span> <a href="polynom3.html#K14">*'</a> <span class="p1"><a href="algseq_1.html#K5">&lt;%</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="algseq_1.html#K5">%&gt;</a></span> = 
 <a href="polynom3.html#K12">0_.</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="polynom5.html#E2:43_1_2"><i><font color="Green">E35</font></i></a>, <a class="ref" href="polynom4.html#T5">POLYNOM4:5</a></i>
<br/>.= 
<span class="p1"><a href="algseq_1.html#K5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="algseq_1.html#K5">%&gt;</a></span>
<b>by </b><i><a class="ref" href="polynom5.html#T35" target="_self">Th35</a></i>
<br/>.= 
<span class="p1"><a href="algseq_1.html#K5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="algseq_1.html#K5">%&gt;</a></span>
<b>by </b><i><a class="txt" href="polynom5.html#E3:43_1_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="vectsp_1.html#T39">VECTSP_1:39</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_3"><b>suppose </b></a><a NAME="E1:43_1_3"/><i><font color="Green">E34</font></i>: 
 <a href="algseq_1.html#K3">len</a> <span class="p1"><a href="algseq_1.html#K5">&lt;%</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="algseq_1.html#K5">%&gt;</a></span> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:43_1_3"/><b>then </b><i><font color="Green">E35</font></i>: 
<span class="p1"><a href="algseq_1.html#K5">&lt;%</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="algseq_1.html#K5">%&gt;</a></span> <a href="funct_2.html#R2">=</a>  <a href="polynom3.html#K12">0_.</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="polynom4.html#T8">POLYNOM4:8</a></i>;<br/><a NAME="E3:43_1_3"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="polynom5.html#E1:43_1_3"><i><font color="Green">E34</font></i></a>, <a class="ref" href="polynom5.html#T34" target="_self">Th34</a></i>;<br/><b>thus </b><span class="p1"><a href="algseq_1.html#K5">&lt;%</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="algseq_1.html#K5">%&gt;</a></span> <a href="polynom3.html#K14">*'</a> <span class="p1"><a href="algseq_1.html#K5">&lt;%</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="algseq_1.html#K5">%&gt;</a></span> = 
 <a href="polynom3.html#K12">0_.</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="polynom5.html#E2:43_1_3"><i><font color="Green">E35</font></i></a>, <a class="ref" href="polynom3.html#T35">POLYNOM3:35</a></i>
<br/>.= 
<span class="p1"><a href="algseq_1.html#K5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="algseq_1.html#K5">%&gt;</a></span>
<b>by </b><i><a class="ref" href="polynom5.html#T35" target="_self">Th35</a></i>
<br/>.= 
<span class="p1"><a href="algseq_1.html#K5">&lt;%</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="group_1.html#K10">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="algseq_1.html#K5">%&gt;</a></span>
<b>by </b><i><a class="txt" href="polynom5.html#E3:43_1_3"><i><font color="Green">E36</font></i></a>, <a class="ref" href="vectsp_1.html#T39">VECTSP_1:39</a></i>
;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
