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<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>



<span class="kw">defpred </span><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> ] <span class="kw">means </span><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/>
<a NAME="E1:44"/><span class="lab"><font color="Green" title="E34">A1</font></span>: 
 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_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">assume </span><a NAME="E1:44_1"/>
<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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/>

<span class="kw">hence </span><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#K13" title="POLYNOM3:func.13">*'</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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom5.html#T19" target="_self" title="POLYNOM5:th.19">Th20</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom5.html#T35" target="_self" title="POLYNOM5:th.35">Th36</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D7" title="GROUP_1:def.7">GROUP_1:def 7</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom5.html#T15" target="_self" title="POLYNOM5:th.15">Th16</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom5.html#T27" target="_self" title="POLYNOM5:th.27">Th28</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom5.html#T29" target="_self" title="POLYNOM5:th.29">Th30</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="group_1.html#D7" title="GROUP_1:def.7">GROUP_1:def 7</a></span>
;<br/>
<a NAME="E3:44"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A2</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
;<br/>
<span class="kw">thus </span><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>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E3:44"><span class="lab"><font color="Green" title="E35">A2</font></span></a>, <a class="txt" href="#E1:44"><span class="lab"><font color="Green" title="E34">A1</font></span></a>);</span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
