<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">L</font> be   <a href="vectsp_1.html#NM2" title="VECTSP_1:NM.2">Field</a>;<br/><b>let </b><font color="Maroon" title="c2">i</font>, <font color="Maroon" title="c3">j</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>;<br/><b>let </b><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>assume </b><a NAME="E1:33"/><i><font color="Green" title="E30">A1</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <font color="Maroon" title="c1">L</font>
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>] <b>means </b>for <font color="Olive" title="b1">i</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> holds   <a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> $1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> <font color="Maroon" title="c4">x</font>,<font color="Olive" title="b1">i</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> <font color="Maroon" title="c4">x</font>,$1</span>)</span>;<br/>
<a NAME="E2:33"/><i><font color="Green" title="E31">A2</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">i</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>;<br/>

<b>thus </b> <a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 0</span>)</span> = 
<span class="p1">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">i</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1.</a> <font color="Maroon" title="c1">L</font></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D13" title="VECTSP_1:def.13">VECTSP_1:def 13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">i</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> <font color="Maroon" title="c4">x</font>,0</span>)</span>
<b>by </b><i><a class="ref" href="polynom8.html#T13" target="_self" title="POLYNOM8:th.13">Th13</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E3:33"/><i><font color="Green" title="E32">A3</font></i>: 
for <font color="Olive" title="b1">j</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">j</font>] holds <br/>( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">j</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>;<br/>

<b>assume </b><a NAME="E1:33_2"/><i><font color="Green" title="E33">A4</font></i>: 
for <font color="Olive" title="b1">i</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> holds   <a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> <font color="Maroon" title="c4">x</font>,<font color="Olive" title="b1">i</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">j</font></span>)</span>
 ;<br/>

<b>thus </b><a NAME="E2:33_2"/>
for <font color="Olive" title="b1">i</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> holds   <a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> <font color="Maroon" title="c4">x</font>,<font color="Olive" title="b1">i</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> <font color="Maroon" title="c4">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">i</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>;<br/>

<b>thus </b> <a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> = 
 <a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">j</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> <font color="Maroon" title="c4">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">j</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom8.html#E1:33_2"><i><font color="Green" title="E33">A4</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> <font color="Maroon" title="c4">x</font>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</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"><a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">j</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom8.html#E1:33"><i><font color="Green" title="E30">A1</font></i></a>, <a class="ref" href="polynom8.html#T20" target="_self" title="POLYNOM8:th.20">Th20</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c6">i</font></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="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> <font color="Maroon" title="c4">x</font>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> <font color="Maroon" title="c4">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="polynom8.html#E1:33_2"><i><font color="Green" title="E33">A4</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> <font color="Maroon" title="c4">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span></span>)</span>

;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon" title="c6">i</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>;<br/>

<b>thus </b> <a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span> = 
 <a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">j</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> <font color="Maroon" title="c4">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">j</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom8.html#E1:33_2"><i><font color="Green" title="E33">A4</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> <font color="Maroon" title="c4">x</font>,1</span>)</span></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">j</font></span>)</span>
<b>by </b><i><a class="txt" href="polynom8.html#E1:33"><i><font color="Green" title="E30">A1</font></i></a>, <a class="ref" href="polynom8.html#T19" target="_self" title="POLYNOM8:th.19">Th19</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c6">i</font></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="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> <font color="Maroon" title="c4">x</font>,1</span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p2">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c5">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="group_1.html#D4" title="GROUP_1:def.4">GROUP_1:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c6">i</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> <font color="Maroon" title="c4">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">j</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="polynom8.html#E1:33_2"><i><font color="Green" title="E33">A4</font></i></a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E4:33"/>
for <font color="Olive" title="b1">j</font> being  <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">j</font>]
 
<b>from </b><i><a class="ref" href="int_1.html#S4" title="INT_1:sch.4">INT_1:sch 4</a>(<a class="txt" href="polynom8.html#E2:33"><i><font color="Green" title="E31">A2</font></i></a>, <a class="txt" href="polynom8.html#E3:33"><i><font color="Green" title="E32">A3</font></i></a>);<br/></i>
<b>hence </b><a NAME="E5:33"/>
<span class="p1">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c2">i</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <span class="p1">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> <font color="Maroon" title="c4">x</font>,<font color="Maroon" title="c3">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> <font color="Maroon" title="c4">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c3">j</font></span>)</span>
 ;<br/>


</div>
