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<b>let </b><font color="Maroon" title="c1">n</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <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> ;<br/><b>let </b><font color="Maroon" title="c2">v</font> be    <a href="complfld.html#M1" title="COMPLFLD:mode.1">CRoot</a> of <font color="Maroon" title="c1">n</font>, <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> ;<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">(<span class="default"><a href="group_1.html#K5" title="GROUP_1:func.5">power</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c2">v</font>,$1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> ;<br/>
<a NAME="E1:79"/><i><font color="Green" title="E16">A1</font></i>: 
 ex <font color="Olive" title="b1">n</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <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>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon" title="c1">n</font>
;<br/>

<b>thus </b><a NAME="E1:79_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c1">n</font>]
 <b>by </b><i><a class="ref" href="complfld.html#D2" target="_self" title="COMPLFLD:def.2">Def2</a></i>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green" title="E17">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="79_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">k</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <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> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:79_2"/><i><font color="Green" title="E18">A3</font></i>: 
<font color="Maroon" title="c3">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 <b>and </b><br/><a NAME="E2:79_2"/><i><font color="Green" title="E19">A4</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">k</font>]
 ;<br/><b>consider </b><font color="Maroon" title="c4">t</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E4:79_2"/><i><font color="Green" title="E20">A5</font></i>: 
<font color="Maroon" title="c3">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">t</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T6" title="NAT_1:th.6">NAT_1:6</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c5">t</font> = <font color="Maroon" title="c4">t</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="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c6">t</font> = <font color="Maroon" title="c5">t</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <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="txt" href="complfld.html#E1:79_2"><i><font color="Green" title="E18">A3</font></i></a>, <a class="txt" href="complfld.html#E4:79_2"><i><font color="Green" title="E20">A5</font></i></a></i>;<br/><b>take </b><font color="Maroon" title="c7">t</font> = <font color="Maroon" title="c6">t</font>;<br/><b>thus </b><a NAME="E7:79_2"/>
<font color="Maroon" title="c7">t</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">k</font>
 <b>by </b><i><a class="txt" href="complfld.html#E4:79_2"><i><font color="Green" title="E20">A5</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E8:79_2"/><i><font color="Green" title="E21">A6</font></i>: 
<font color="Maroon" title="c7">t</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="ref" href="nat_1.html#T3" title="NAT_1:th.3">NAT_1:3</a></i>;<br/><a NAME="E9:79_2"/>
<span class="p1">(<span class="default"><a href="group_1.html#K5" title="GROUP_1:func.5">power</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">k</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="group_1.html#K5" title="GROUP_1:func.5">power</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c7">t</font></span>)</span> <a href="group_1.html#K10" title="GROUP_1:func.10">*</a> <font color="Maroon" title="c2">v</font>
 <b>by </b><i><a class="txt" href="complfld.html#E4:79_2"><i><font color="Green" title="E20">A5</font></i></a>, <a class="ref" href="group_1.html#D8" title="GROUP_1:def.8">GROUP_1:def 8</a></i>;<br/><a NAME="E10:79_2"/><b>then </b>
( <span class="p1">(<span class="default"><a href="group_1.html#K5" title="GROUP_1:func.5">power</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2" title="BINOP_1:func.2">.</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c7">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>  or <font color="Maroon" title="c2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a>  )
 <b>by </b><i><a class="txt" href="complfld.html#E2:79_2"><i><font color="Green" title="E19">A4</font></i></a>, <a class="ref" href="vectsp_1.html#T44" title="VECTSP_1:th.44">VECTSP_1:44</a></i>;<br/><b>hence </b><a NAME="E11:79_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">t</font>]
 <b>by </b><i><a class="txt" href="complfld.html#E8:79_2"><i><font color="Green" title="E21">A6</font></i></a>, <a class="ref" href="vectsp_1.html#T95" title="VECTSP_1:th.95">VECTSP_1:95</a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:79"/>
<font color="Maroon">S<sub>1</sub></font>[1]
 
<b>from </b><i><a class="ref" href="complfld.html#S1" target="_self" title="COMPLFLD:sch.1">COMPLFLD:sch 1</a>(<a class="txt" href="complfld.html#E1:79"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="complfld.html#E2:79"><i><font color="Green" title="E17">A2</font></i></a>);<br/></i>
<b>hence </b><a NAME="E4:79"/>
<font color="Maroon" title="c2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">0.</a> <a href="complfld.html#K1" title="COMPLFLD:func.1">F_Complex</a> 
 <b>by </b><i><a class="ref" href="group_1.html#T98" title="GROUP_1:th.98">GROUP_1:98</a></i>;<br/>


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