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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="algstr_0.html#V33" title="ALGSTR_0:attr.33">almost_left_invertible</a> <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</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#V3" title="GROUP_1:attr.3">associative</a> <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a> <a href="vectsp_1.html#V4" title="VECTSP_1:attr.4">well-unital</a> <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</a>  <a href="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><br/> for <font color="Olive" title="b2">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>  holds  <span class="p1"><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">&lt;%</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">%&gt;</a></span> <a href="polynom5.html#K2" title="POLYNOM5:func.2">`^</a> <font color="Olive" title="b2">n</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1"><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">&lt;%</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="Olive" title="b1">x</font>,<font color="Olive" title="b2">n</font></span>)</span></span><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">%&gt;</a></span></span><br/><b>let </b><font color="Maroon" title="c2">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> 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>  holds  <span class="p1"><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">&lt;%</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">%&gt;</a></span> <a href="polynom5.html#K2" title="POLYNOM5:func.2">`^</a> <font color="Olive" title="b1">n</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1"><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">&lt;%</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="c2">x</font>,<font color="Olive" title="b1">n</font></span>)</span></span><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">%&gt;</a></span></span><br/>



<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <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>means </b><span class="p1"><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">&lt;%</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">%&gt;</a></span> <a href="polynom5.html#K2" title="POLYNOM5:func.2">`^</a> $1 <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1"><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">&lt;%</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="c2">x</font>,$1</span>)</span></span><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">%&gt;</a></span>;<br/>
<span class="p1"><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">&lt;%</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">%&gt;</a></span> <a href="polynom5.html#K2" title="POLYNOM5:func.2">`^</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  = 
 <a href="polynom3.html#K10" title="POLYNOM3:func.10">1_.</a> <font color="Maroon" title="c1">L</font>
<b>by </b><i><a class="ref" href="polynom5.html#T16" target="_self" title="POLYNOM5:th.16">Th16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="polynom5.html#K3" title="POLYNOM5:func.3">*</a> <span class="p1">(<span class="default"><a href="polynom3.html#K10" title="POLYNOM3:func.10">1_.</a> <font color="Maroon" title="c1">L</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom5.html#T28" target="_self" title="POLYNOM5:th.28">Th28</a></i>
<br/>.= 
<span class="p1"><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">&lt;%</a><span class="default"><span class="p2">(<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></span><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">%&gt;</a></span>
<b>by </b><i><a class="ref" href="polynom5.html#T30" target="_self" title="POLYNOM5:th.30">Th30</a></i>
<br/>.= 
<span class="p1"><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">&lt;%</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="c2">x</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">%&gt;</a></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/>
<a NAME="E2:44"/><b>then </b><i><font color="Green" title="E34">A1</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
;<br/>
<a NAME="E3:44"/><i><font color="Green" title="E35">A2</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="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">n</font> be    <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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">n</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<b>assume </b><a NAME="E1:44_2"/>
<span class="p1"><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">&lt;%</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">%&gt;</a></span> <a href="polynom5.html#K2" title="POLYNOM5:func.2">`^</a> <font color="Maroon" title="c3">n</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1"><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">&lt;%</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="c2">x</font>,<font color="Maroon" title="c3">n</font></span>)</span></span><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">%&gt;</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/>

<b>hence </b><span class="p1"><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">&lt;%</a><span class="default"><font color="Maroon" title="c2">x</font></span><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">%&gt;</a></span> <a href="polynom5.html#K2" title="POLYNOM5:func.2">`^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> = 
<span class="p1"><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">&lt;%</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="c2">x</font>,<font color="Maroon" title="c3">n</font></span>)</span></span><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">%&gt;</a></span> <a href="polynom3.html#K12" title="POLYNOM3:func.12">*'</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">x</font></span><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">%&gt;</a></span>
<b>by </b><i><a class="ref" href="polynom5.html#T20" target="_self" title="POLYNOM5:th.20">Th20</a></i>
<br/>.= 
<span class="p1"><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">&lt;%</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<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="c2">x</font>,<font color="Maroon" title="c3">n</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c2">x</font></span>)</span></span><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">%&gt;</a></span>
<b>by </b><i><a class="ref" href="polynom5.html#T36" target="_self" title="POLYNOM5:th.36">Th36</a></i>
<br/>.= 
<span class="p1"><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">&lt;%</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="c2">x</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">%&gt;</a></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/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E4:44"/>
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>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E2:44"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="#E3:44"><i><font color="Green" title="E35">A2</font></i></a>);</i> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
