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<a NAME="E1:19_1_1"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> implies <span class="p1">(<span class="default"><a href="group_1.html#K5">power</a> <font color="Maroon">L</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">i</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:19_1_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">i'</font> = <font color="Maroon">i</font> as    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i><a class="ref" href="int_1.html#T16">INT_1:16</a></i>;<br/>
<a NAME="E3:19_1_1_1"/>
<span class="p1">(<span class="default"><a href="group_1.html#K5">power</a> <font color="Maroon">L</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<font color="Maroon">i'</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>
 
;<br/>
<b>hence </b><a NAME="E4:19_1_1_1"/>
<span class="p1">(<span class="default"><a href="group_1.html#K5">power</a> <font color="Maroon">L</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">i</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:19_1_1"/>
( ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> implies <span class="p1">(<span class="default"><a href="group_1.html#K5">power</a> <font color="Maroon">L</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">i</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ) &amp; ( not 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="group_1.html#K5">power</a> <font color="Maroon">L</font></span>)</span> <a href="binop_1.html#K2">.</a> <font color="Maroon">a</font>,<span class="p2">(<span class="default"><a href="prepower.html#K1">abs</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="vectsp_1.html#K4">"</a>  is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ) )
 ;<br/>


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