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<div class="add">

<b>let </b><font color="Maroon">z</font> be   <a href="struct_0.html#NM1">Element</a> of <a href="complfld.html#K1">F_Complex</a> ;<br/>

<b>assume </b><a NAME="E1:8"/>
<font color="Maroon">z</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <a href="complfld.html#K1">F_Complex</a> 
 ;<br/>

<a NAME="E2:8"/><b>then </b><i><font color="Green">E33</font></i>: 
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complfld.html#K3">.|</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 
<b>by </b><i><a class="ref" href="complfld.html#T94">COMPLFLD:94</a></i>;<br/>
<a NAME="E3:8"/><i><font color="Green">E34</font></i>: 
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complfld.html#K3">.|</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">z</font>,<font color="Maroon">a<sub>1</sub></font></span>)</span></span><a href="complfld.html#K3">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complfld.html#K3">.|</a></span> <a href="power.html#K4">to_power</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">z</font>,0</span>)</span></span><a href="complfld.html#K3">.|</a></span> = 
1
<b>by </b><i><a class="ref" href="group_1.html#D8">GROUP_1:def 8</a>, <a class="ref" href="complex1.html#T134">COMPLEX1:134</a>, <a class="ref" href="complfld.html#T10">COMPLFLD:10</a></i>
<br/>.= 
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complfld.html#K3">.|</a></span> <a href="power.html#K4">to_power</a> 0
<b>by </b><i><a class="ref" href="power.html#T29">POWER:29</a></i>
;<br/>
<a NAME="E5:8"/><b>then </b><i><font color="Green">E38</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
;<br/>
<a NAME="E6:8"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:8_2"/><i><font color="Green">E40</font></i>: 
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">z</font>,<font color="Maroon">n</font></span>)</span></span><a href="complfld.html#K3">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complfld.html#K3">.|</a></span> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font>
 ;<br/>

<b>thus </b><span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="group_1.html#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">z</font>,<span class="p3">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="complfld.html#K3">.|</a></span> = 
<span class="p1"><a href="complfld.html#K3">|.</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#K5">power</a> <a href="complfld.html#K1">F_Complex</a> </span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">z</font>,<font color="Maroon">n</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">z</font></span>)</span></span><a href="complfld.html#K3">.|</a></span>
<b>by </b><i><a class="ref" href="group_1.html#D8">GROUP_1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complfld.html#K3">.|</a></span> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complfld.html#K3">.|</a></span>
<b>by </b><i>, <a class="ref" href="complfld.html#T109">COMPLFLD:109</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complfld.html#K3">.|</a></span> <a href="power.html#K4">to_power</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complfld.html#K3">.|</a></span> <a href="power.html#K4">to_power</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="power.html#T30">POWER:30</a></i>
<br/>.= 
<span class="p1"><a href="complfld.html#K3">|.</a><span class="default"><font color="Maroon">z</font></span><a href="complfld.html#K3">.|</a></span> <a href="power.html#K4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i>, , <a class="ref" href="power.html#T32">POWER:32</a></i>
;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E7:8"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );</i><br/>


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