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<b>let </b><font color="Maroon" title="c1">L</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a> <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a> <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a> <a href="group_1.html#V1" title="GROUP_1:attr.1">unital</a>  <a href="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon" title="c2">z0</font>, <font color="Maroon" title="c3">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>;<br/>



<b>consider </b><font color="Maroon" title="c4">F</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E2:45"/><i><font color="Green" title="E34">A1</font></i>: 
 <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p1"><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font></span><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">%&gt;</a></span>,<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c4">F</font>
 <b>and </b><br/><a NAME="E3:45"/><i><font color="Green" title="E35">A2</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <span class="p1"><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font></span><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">%&gt;</a></span>
 <b>and </b><br/><a NAME="E4:45"/><i><font color="Green" title="E36">A3</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">F</font> holds <br/><font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font></span><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">%&gt;</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">x</font>,<span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="polynom4.html#D2" title="POLYNOM4:def.2">POLYNOM4:def 2</a></i>;<br/>
<a NAME="E5:45"/><i><font color="Green" title="E37">A4</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">F</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 
<b>by </b><i><a class="txt" href="polynom5.html#E3:45"><i><font color="Green" title="E35">A2</font></i></a>, <a class="ref" href="algseq_1.html#D6" title="ALGSEQ_1:def.6">ALGSEQ_1:def 6</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:45_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 </span><b>by </b><i><a class="txt" href="polynom5.html#E5:45"><i><font color="Green" title="E37">A4</font></i></a>, <a class="ref" href="nat_1.html#T26" title="NAT_1:th.26">NAT_1:26</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_1"><b>suppose </b></a><a NAME="E1:45_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><a NAME="E2:45_1_1"/><b>then </b><i><font color="Green" title="E38">A5</font></i>: 
<span class="p1"><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font></span><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">%&gt;</a></span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="polynom3.html#K9" title="POLYNOM3:func.9">0_.</a> <font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="txt" href="polynom5.html#E3:45"><i><font color="Green" title="E35">A2</font></i></a>, <a class="ref" href="polynom4.html#T8" title="POLYNOM4:th.8">POLYNOM4:8</a></i>;<br/><b>hence </b> <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p1"><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font></span><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">%&gt;</a></span>,<font color="Maroon" title="c3">x</font> = 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">L</font>
<b>by </b><i><a class="ref" href="polynom4.html#T20" title="POLYNOM4:th.20">POLYNOM4:20</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom3.html#K9" title="POLYNOM3:func.9">0_.</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
<br/>.= 
<font color="Maroon" title="c2">z0</font>
<b>by </b><i><a class="txt" href="polynom5.html#E2:45_1_1"><i><font color="Green" title="E38">A5</font></i></a>, <a class="ref" href="polynom5.html#T33" target="_self" title="POLYNOM5:th.33">Th33</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="45_1_2"><b>suppose </b></a><a NAME="E1:45_1_2"/><i><font color="Green" title="E38">A6</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:45_1_2"/><b>then </b>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">F</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_1.html#T6" title="FINSEQ_1:th.6">FINSEQ_1:6</a></i>;<br/><a NAME="E3:45_1_2"/><b>then </b>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c4">F</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>then </b><font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 = 
<span class="p1">(<span class="default"><span class="p2"><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font></span><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">%&gt;</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <span class="p2">(<span class="default">1 <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">x</font>,<span class="p2">(<span class="default">1 <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="polynom5.html#E4:45"><i><font color="Green" title="E36">A3</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font></span><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">%&gt;</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">x</font>,<span class="p2">(<span class="default">1 <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T51" title="BINARITH:th.51">BINARITH:51</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font></span><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">%&gt;</a></span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">x</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T51" title="BINARITH:th.51">BINARITH:51</a></i>
<br/>.= 
<font color="Maroon" title="c2">z0</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K4" title="GROUP_1:func.4">power</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c3">x</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
<b>by </b><i><a class="ref" href="polynom5.html#T33" target="_self" title="POLYNOM5:th.33">Th33</a></i>
<br/>.= 
<font color="Maroon" title="c2">z0</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">L</font></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D8" title="GROUP_1:def.8">GROUP_1:def 8</a></i>
<br/>.= 
<font color="Maroon" title="c2">z0</font>
<b>by </b><i><a class="ref" href="group_1.html#D5" title="GROUP_1:def.5">GROUP_1:def 5</a></i>
;<br/><a NAME="E5:45_1_2"/><b>then </b>
<font color="Maroon" title="c4">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><font color="Maroon" title="c2">z0</font></span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="polynom5.html#E1:45_1_2"><i><font color="Green" title="E38">A6</font></i></a>, <a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>;<br/><b>hence </b><a NAME="E6:45_1_2"/>
 <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p1"><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">&lt;%</a><span class="default"><font color="Maroon" title="c2">z0</font></span><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">%&gt;</a></span>,<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">z0</font>
 <b>by </b><i><a class="txt" href="polynom5.html#E2:45"><i><font color="Green" title="E34">A1</font></i></a>, <a class="ref" href="rlvect_1.html#T61" title="RLVECT_1:th.61">RLVECT_1:61</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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