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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="rlvect_1.html#V4" title="RLVECT_1:attr.4">add-associative</a> <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">right_zeroed</a> <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">right_complementable</a> <a href="group_1.html#V4" title="GROUP_1:attr.4">associative</a> <a href="group_1.html#V6" title="GROUP_1:attr.6">commutative</a> <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">well-unital</a> <a href="vectsp_1.html#V8" title="VECTSP_1:attr.8">distributive</a> <a href="vectsp_1.html#NV11" title="VECTSP_1:NV.11">Field-like</a>  <a href="vectsp_1.html#L3" title="VECTSP_1:struct.3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon" title="c2">p</font> be   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c1">L</font>;<br/><b>let </b><font color="Maroon" title="c3">v</font>, <font color="Maroon" title="c4">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="c5">F1</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="E2:37"/><i><font color="Green" title="E28">A1</font></i>: 
 <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <font color="Maroon" title="c5">F1</font>
 <b>and </b><br/><a NAME="E3:37"/><i><font color="Green" title="E29">A2</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c2">p</font>
 <b>and </b><br/><a NAME="E4:37"/><i><font color="Green" title="E30">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="c5">F1</font> holds <br/><font color="Maroon" title="c5">F1</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"><font color="Maroon" title="c2">p</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</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="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K5" title="GROUP_1:func.5">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="c4">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/>
<b>consider </b><font color="Maroon" title="c6">F2</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="E6:37"/><i><font color="Green" title="E31">A4</font></i>: 
 <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <font color="Maroon" title="c6">F2</font>
 <b>and </b><br/><a NAME="E7:37"/><i><font color="Green" title="E32">A5</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">F2</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">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c2">p</font></span>)</span>
 <b>and </b><br/><a NAME="E8:37"/><i><font color="Green" title="E33">A6</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="c6">F2</font> holds <br/><font color="Maroon" title="c6">F2</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">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</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="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K5" title="GROUP_1:func.5">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="c4">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/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_1"/>
( <font color="Maroon" title="c3">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">L</font> or <font color="Maroon" title="c3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">L</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1"><b>suppose </b></a><a NAME="E1:37_1_1"/>
<font color="Maroon" title="c3">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">L</font>
 ;<br/><div class="add"><a NAME="E2:37_1_1"/><b>then </b>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">F2</font>
 <b>by </b><i><a class="txt" href="polynom5.html#E3:37"><i><font color="Green" title="E29">A2</font></i></a>, <a class="txt" href="polynom5.html#E7:37"><i><font color="Green" title="E32">A5</font></i></a>, <a class="ref" href="polynom5.html#T26" target="_self" title="POLYNOM5:th.26">Th26</a></i>;<br/><a NAME="E3:37_1_1"/><b>then </b><i><font color="Green" title="E34">A7</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c5">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c6">F2</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:37_1_1_1"/><i><font color="Green" title="E35">A8</font></i>: 
<font color="Maroon" title="c7">i</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="c5">F1</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c8">i1</font> = <font color="Maroon" title="c7">i</font> as    <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>  <b>by </b><i><a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></i>;<br/><i><font color="Green" title="E36">A9</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K5" title="GROUP_1:func.5">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="c4">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> = 
<font color="Maroon" title="c5">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font>
<b>by </b><i><a class="txt" href="polynom5.html#E4:37"><i><font color="Green" title="E30">A3</font></i></a>, <a class="txt" href="polynom5.html#E1:37_1_1_1"><i><font color="Green" title="E35">A8</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c5">F1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">i</font>
<b>by </b><i><a class="txt" href="polynom5.html#E1:37_1_1_1"><i><font color="Green" title="E35">A8</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
;<br/><b>thus </b><font color="Maroon" title="c6">F2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">i</font> = 
<font color="Maroon" title="c6">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font>
<b>by </b><i><a class="txt" href="polynom5.html#E3:37_1_1"><i><font color="Green" title="E34">A7</font></i></a>, <a class="txt" href="polynom5.html#E1:37_1_1_1"><i><font color="Green" title="E35">A8</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K5" title="GROUP_1:func.5">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="c4">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">i1</font> <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#E8:37"><i><font color="Green" title="E33">A6</font></i></a>, <a class="txt" href="polynom5.html#E3:37_1_1"><i><font color="Green" title="E34">A7</font></i></a>, <a class="txt" href="polynom5.html#E1:37_1_1_1"><i><font color="Green" title="E35">A8</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K5" title="GROUP_1:func.5">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="c4">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom5.html#D3" target="_self" title="POLYNOM5:def.3">Def3</a></i>
<br/>.= 
<font color="Maroon" title="c3">v</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">i</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom5.html#E3:37_1_1_1"><i><font color="Green" title="E36">A9</font></i></a>, <a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></i>
;<br/></div><b>end;</b></div><a NAME="E5:37_1_1"/><b>then </b>
<font color="Maroon" title="c6">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v</font> <a href="polynom1.html#K6" title="POLYNOM1:func.6">*</a> <font color="Maroon" title="c5">F1</font>
 <b>by </b><i><a class="txt" href="polynom5.html#E3:37_1_1"><i><font color="Green" title="E34">A7</font></i></a>, <a class="ref" href="polynom1.html#D2" title="POLYNOM1:def.2">POLYNOM1:def 2</a></i>;<br/><b>hence </b><a NAME="E6:37_1_1"/>
 <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">v</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="polynom5.html#E2:37"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="polynom5.html#E6:37"><i><font color="Green" title="E31">A4</font></i></a>, <a class="ref" href="polynom1.html#T28" title="POLYNOM1:th.28">POLYNOM1:28</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_2"><b>suppose </b></a><a NAME="E1:37_1_2"/><i><font color="Green" title="E34">A10</font></i>: 
<font color="Maroon" title="c3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">L</font>
 ;<br/><div class="add"><b>hence </b> <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">v</font> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <font color="Maroon" title="c2">p</font></span>)</span>,<font color="Maroon" title="c4">x</font> = 
 <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <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>,<font color="Maroon" title="c4">x</font>
<b>by </b><i><a class="ref" href="polynom5.html#T27" target="_self" title="POLYNOM5:th.27">Th27</a></i>
<br/>.= 
 <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/>.= 
<font color="Maroon" title="c3">v</font> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c4">x</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom5.html#E1:37_1_2"><i><font color="Green" title="E34">A10</font></i></a>, <a class="ref" href="vectsp_1.html#T39" title="VECTSP_1:th.39">VECTSP_1:39</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div>

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