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<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">L</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</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>  st <font color="Olive" title="b1">x</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> holds <br/> for <font color="Olive" title="b2">i</font>, <font color="Olive" title="b3">j</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="Olive" title="b1">x</font>,<span class="p1">(<span class="default"><font color="Olive" title="b2">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b3">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> (<span class="p1">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> (<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">i</font>)</span>)</span>,<font color="Olive" title="b3">j</font>)</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"> ( <font color="Maroon" title="c2">x</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> implies  for <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</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="c2">x</font>,<span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b2">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> (<span class="p1">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> (<font color="Maroon" title="c2">x</font>,<font color="Olive" title="b1">i</font>)</span>)</span>,<font color="Olive" title="b2">j</font>) )</span><br/>



<span class="kw">assume </span><a NAME="E1:40"/><span class="lab"><font color="Green" title="E37">A1</font></span>: 
<font color="Maroon" title="c2">x</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>
 ; <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">i</font>, <font color="Olive" title="b2">j</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="c2">x</font>,<span class="p1">(<span class="default"><font color="Olive" title="b1">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b2">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> (<span class="p1">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> (<font color="Maroon" title="c2">x</font>,<font color="Olive" title="b1">i</font>)</span>)</span>,<font color="Olive" title="b2">j</font>)</span><br/>

<span class="kw">let </span><font color="Maroon" title="c3">i</font>, <font color="Maroon" title="c4">j</font> be   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> (<font color="Maroon" title="c2">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">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> (<span class="p1">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> (<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">i</font>)</span>)</span>,<font color="Maroon" title="c4">j</font>)</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:40_1"/>
( ( <font color="Maroon" title="c3">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c4">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) or ( <font color="Maroon" title="c3">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c4">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) or ( <font color="Maroon" title="c3">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c4">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) or ( <font color="Maroon" title="c4">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c3">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_1"><span class="kw">suppose </span></a><a NAME="E1:40_1_1"/>
( <font color="Maroon" title="c3">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c4">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> (<font color="Maroon" title="c2">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">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> (<span class="p1">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> (<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">i</font>)</span>)</span>,<font color="Maroon" title="c4">j</font>)</span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">m</font> = <font color="Maroon" title="c3">i</font>, <font color="Maroon" title="c6">n</font> = <font color="Maroon" title="c4">j</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T3" title="INT_1:th.3">INT_1:3</a></span>;<br/><span class="kw">thus </span> <a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> (<font color="Maroon" title="c2">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">j</font></span>)</span>) = 
 <a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> (<span class="p1">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> (<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c5">m</font>)</span>)</span>,<font color="Maroon" title="c6">n</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom8.html#T24" target="_self" title="POLYNOM8:th.24">Th24</a></span>
<br/>.= 
 <a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> (<span class="p1">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> (<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">i</font>)</span>)</span>,<font color="Maroon" title="c4">j</font>)

; <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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_2"><span class="kw">suppose </span></a><a NAME="E1:40_1_2"/><span class="lab"><font color="Green" title="E38">A2</font></span>: 
( <font color="Maroon" title="c3">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c4">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> (<font color="Maroon" title="c2">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">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> (<span class="p1">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> (<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">i</font>)</span>)</span>,<font color="Maroon" title="c4">j</font>)</span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">m</font> = <font color="Maroon" title="c3">i</font>, <font color="Maroon" title="c6">n</font> =  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c4">j</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T3" title="INT_1:th.3">INT_1:3</a></span>;<br/><a NAME="E3:40_1_2"/><span class="lab"><font color="Green" title="E39">A3</font></span>: 
 <a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> (<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c5">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom8.html#D3" target="_self" title="POLYNOM8:def.3">Def3</a></span>;<br/><a NAME="E4:40_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A4</font></span>: 
 <a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> (<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">i</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40"><span class="lab"><font color="Green" title="E37">A1</font></span></a>, <a class="ref" href="polynom8.html#T1" target="_self" title="POLYNOM8:th.1">Th1</a></span>;<br/><a NAME="E5:40_1_2"/><span class="lab"><font color="Green" title="E41">A5</font></span>: 
 <a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> (<span class="p1">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> (<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">i</font>)</span>)</span>,<font color="Maroon" title="c4">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c2">x</font>,<font color="Maroon" title="c3">i</font>)</span>)</span> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <span class="p2">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c4">j</font></span>)</span></span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_1_2"><span class="lab"><font color="Green" title="E38">A2</font></span></a>, <a class="ref" href="polynom8.html#D3" target="_self" title="POLYNOM8:def.3">Def3</a></span>;<br/><a NAME="E6:40_1_2"/>
<font color="Maroon" title="c3">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">n</font></span>)</span>
 ;<br/><span class="kw">hence </span> <a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> (<font color="Maroon" title="c2">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">j</font></span>)</span>) = 
<span class="p1">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> (<font color="Maroon" title="c2">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">n</font></span>)</span>)</span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40"><span class="lab"><font color="Green" title="E37">A1</font></span></a>, <a class="ref" href="polynom8.html#T18" target="_self" title="POLYNOM8:th.18">Th18</a></span>
<br/>.= 
 <a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">n</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40"><span class="lab"><font color="Green" title="E37">A1</font></span></a>, <a class="ref" href="polynom8.html#T25" target="_self" title="POLYNOM8:th.25">Th25</a></span>
<br/>.= 
 <a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> (<span class="p1">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span>,<font color="Maroon" title="c5">m</font>)</span>)</span>,<font color="Maroon" title="c6">n</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom8.html#T24" target="_self" title="POLYNOM8:th.24">Th24</a></span>
<br/>.= 
 <a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</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="c2">x</font>,<font color="Maroon" title="c3">i</font>)</span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span>,<font color="Maroon" title="c6">n</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40"><span class="lab"><font color="Green" title="E37">A1</font></span></a>, <a class="ref" href="polynom8.html#T25" target="_self" title="POLYNOM8:th.25">Th25</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> (<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">i</font>)</span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span>,<font color="Maroon" title="c4">j</font>)</span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:40_1_2"><span class="lab"><font color="Green" title="E40">A4</font></span></a>, <a class="ref" href="polynom8.html#T18" target="_self" title="POLYNOM8:th.18">Th18</a>, <a class="ref" href="vectsp_1.html#T25" title="VECTSP_1:th.25">VECTSP_1:25</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> (<span class="p3">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> (<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">i</font>)</span>)</span>,<font color="Maroon" title="c4">j</font>)</span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40"><span class="lab"><font color="Green" title="E37">A1</font></span></a>, <a class="txt" href="#E3:40_1_2"><span class="lab"><font color="Green" title="E39">A3</font></span></a>, <a class="ref" href="polynom8.html#T1" target="_self" title="POLYNOM8:th.1">Th1</a>, <a class="ref" href="polynom8.html#T25" target="_self" title="POLYNOM8:th.25">Th25</a></span>
<br/>.= 
 <a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> (<span class="p1">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> (<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">i</font>)</span>)</span>,<font color="Maroon" title="c4">j</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:40_1_2"><span class="lab"><font color="Green" title="E40">A4</font></span></a>, <a class="txt" href="#E5:40_1_2"><span class="lab"><font color="Green" title="E41">A5</font></span></a>, <a class="ref" href="polynom8.html#T1" target="_self" title="POLYNOM8:th.1">Th1</a>, <a class="ref" href="vectsp_1.html#T24" title="VECTSP_1:th.24">VECTSP_1:24</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_3"><span class="kw">suppose </span></a><a NAME="E1:40_1_3"/><span class="lab"><font color="Green" title="E38">A6</font></span>: 
( <font color="Maroon" title="c3">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c4">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> (<font color="Maroon" title="c2">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">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> (<span class="p1">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> (<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">i</font>)</span>)</span>,<font color="Maroon" title="c4">j</font>)</span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">m</font> =  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c3">i</font>, <font color="Maroon" title="c6">n</font> = <font color="Maroon" title="c4">j</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T3" title="INT_1:th.3">INT_1:3</a></span>;<br/><a NAME="E3:40_1_3"/><span class="lab"><font color="Green" title="E39">A7</font></span>: 
 <a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> (<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <span class="p2">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c3">i</font></span>)</span></span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_1_3"><span class="lab"><font color="Green" title="E38">A6</font></span></a>, <a class="ref" href="polynom8.html#D3" target="_self" title="POLYNOM8:def.3">Def3</a></span>;<br/><a NAME="E4:40_1_3"/>
<font color="Maroon" title="c3">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">j</font></span>)</span>
 ;<br/><span class="kw">hence </span> <a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> (<font color="Maroon" title="c2">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">j</font></span>)</span>) = 
<span class="p1">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> (<font color="Maroon" title="c2">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">j</font></span>)</span>)</span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40"><span class="lab"><font color="Green" title="E37">A1</font></span></a>, <a class="ref" href="polynom8.html#T18" target="_self" title="POLYNOM8:th.18">Th18</a></span>
<br/>.= 
 <a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">m</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">j</font></span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40"><span class="lab"><font color="Green" title="E37">A1</font></span></a>, <a class="ref" href="polynom8.html#T25" target="_self" title="POLYNOM8:th.25">Th25</a></span>
<br/>.= 
 <a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> (<span class="p1">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span>,<font color="Maroon" title="c5">m</font>)</span>)</span>,<font color="Maroon" title="c6">n</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom8.html#T24" target="_self" title="POLYNOM8:th.24">Th24</a></span>
<br/>.= 
 <a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span>,<font color="Maroon" title="c3">i</font>)</span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span>,<font color="Maroon" title="c6">n</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40"><span class="lab"><font color="Green" title="E37">A1</font></span></a>, <a class="ref" href="polynom8.html#T18" target="_self" title="POLYNOM8:th.18">Th18</a>, <a class="ref" href="vectsp_1.html#T25" title="VECTSP_1:th.25">VECTSP_1:25</a></span>
<br/>.= 
 <a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> (<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">i</font>)</span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span>,<font color="Maroon" title="c4">j</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40"><span class="lab"><font color="Green" title="E37">A1</font></span></a>, <a class="ref" href="polynom8.html#T25" target="_self" title="POLYNOM8:th.25">Th25</a></span>
<br/>.= 
 <a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> (<span class="p1">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> (<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">i</font>)</span>)</span>,<font color="Maroon" title="c4">j</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40"><span class="lab"><font color="Green" title="E37">A1</font></span></a>, <a class="txt" href="#E3:40_1_3"><span class="lab"><font color="Green" title="E39">A7</font></span></a>, <a class="ref" href="polynom8.html#T1" target="_self" title="POLYNOM8:th.1">Th1</a>, <a class="ref" href="vectsp_1.html#T24" title="VECTSP_1:th.24">VECTSP_1:24</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="40_1_4"><span class="kw">suppose </span></a><a NAME="E1:40_1_4"/><span class="lab"><font color="Green" title="E38">A8</font></span>: 
( <font color="Maroon" title="c4">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c3">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> (<font color="Maroon" title="c2">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">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> (<span class="p1">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> (<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">i</font>)</span>)</span>,<font color="Maroon" title="c4">j</font>)</span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">m</font> =  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c3">i</font>, <font color="Maroon" title="c6">n</font> =  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c4">j</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="int_1.html#T3" title="INT_1:th.3">INT_1:3</a></span>;<br/><a NAME="E3:40_1_4"/><span class="lab"><font color="Green" title="E39">A9</font></span>: 
 <a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> (<font color="Maroon" title="c2">x</font>,<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c3">i</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <font color="Maroon" title="c5">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom8.html#D3" target="_self" title="POLYNOM8:def.3">Def3</a></span>;<br/><a NAME="E4:40_1_4"/>
<font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a>  <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40"><span class="lab"><font color="Green" title="E37">A1</font></span></a>, <a class="ref" href="vectsp_1.html#T25" title="VECTSP_1:th.25">VECTSP_1:25</a></span>;<br/><a NAME="E5:40_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A10</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c3">i</font></span>)</span> <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom8.html#T1" target="_self" title="POLYNOM8:th.1">Th1</a></span>;<br/><a NAME="E6:40_1_4"/><span class="lab"><font color="Green" title="E41">A11</font></span>: 
 <a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span>,<font color="Maroon" title="c3">i</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="c2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span> <a href="binom.html#K2" title="BINOM:func.2">|^</a> <span class="p2">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <font color="Maroon" title="c3">i</font></span>)</span></span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40_1_4"><span class="lab"><font color="Green" title="E38">A8</font></span></a>, <a class="ref" href="polynom8.html#D3" target="_self" title="POLYNOM8:def.3">Def3</a></span>;<br/><a NAME="E7:40_1_4"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">j</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">m</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c6">n</font>
 ;<br/><span class="kw">hence </span> <a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> (<font color="Maroon" title="c2">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c4">j</font></span>)</span>) = 
 <a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> (<span class="p1">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> (<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c5">m</font>)</span>)</span>,<font color="Maroon" title="c6">n</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom8.html#T24" target="_self" title="POLYNOM8:th.24">Th24</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> (<span class="p2">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> (<font color="Maroon" title="c2">x</font>,<span class="p3">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <font color="Maroon" title="c3">i</font></span>)</span>)</span>)</span>,<font color="Maroon" title="c4">j</font>)</span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40"><span class="lab"><font color="Green" title="E37">A1</font></span></a>, <a class="txt" href="#E3:40_1_4"><span class="lab"><font color="Green" title="E39">A9</font></span></a>, <a class="ref" href="polynom8.html#T1" target="_self" title="POLYNOM8:th.1">Th1</a>, <a class="ref" href="polynom8.html#T18" target="_self" title="POLYNOM8:th.18">Th18</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> (<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">i</font>)</span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span>,<font color="Maroon" title="c4">j</font>)</span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40"><span class="lab"><font color="Green" title="E37">A1</font></span></a>, <a class="ref" href="polynom8.html#T18" target="_self" title="POLYNOM8:th.18">Th18</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> (<span class="p2">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span>,<font color="Maroon" title="c3">i</font>)</span>)</span>,<font color="Maroon" title="c4">j</font>)</span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40"><span class="lab"><font color="Green" title="E37">A1</font></span></a>, <a class="ref" href="polynom8.html#T25" target="_self" title="POLYNOM8:th.25">Th25</a></span>
<br/>.= 
 <a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span>,<font color="Maroon" title="c3">i</font>)</span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span>,<font color="Maroon" title="c4">j</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:40_1_4"><span class="lab"><font color="Green" title="E41">A11</font></span></a>, <a class="txt" href="#E5:40_1_4"><span class="lab"><font color="Green" title="E40">A10</font></span></a>, <a class="ref" href="polynom8.html#T25" target="_self" title="POLYNOM8:th.25">Th25</a>, <a class="ref" href="vectsp_1.html#T25" title="VECTSP_1:th.25">VECTSP_1:25</a></span>
<br/>.= 
 <a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> (<span class="p1">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a></span>)</span>,<font color="Maroon" title="c3">i</font>)</span>)</span>,<font color="Maroon" title="c4">j</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40"><span class="lab"><font color="Green" title="E37">A1</font></span></a>, <a class="ref" href="polynom8.html#T25" target="_self" title="POLYNOM8:th.25">Th25</a>, <a class="ref" href="vectsp_1.html#T25" title="VECTSP_1:th.25">VECTSP_1:25</a></span>
<br/>.= 
 <a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> (<span class="p1">(<span class="default"><a href="polynom8.html#K3" title="POLYNOM8:func.3">pow</a> (<font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">i</font>)</span>)</span>,<font color="Maroon" title="c4">j</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:40"><span class="lab"><font color="Green" title="E37">A1</font></span></a>, <a class="ref" href="vectsp_1.html#T24" title="VECTSP_1:th.24">VECTSP_1:24</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></div><span class="kw">end;</span></div>

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