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<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:20_1_1_1"/>
( 2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j</font> or 2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">j</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_1_1"><b>suppose </b></a><a NAME="E1:20_1_1_1_1"/><i><font color="Green">E39</font></i>: 
2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j</font>
 ;<br/><div class="add"><a NAME="E2:20_1_1_1_1"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 2 <a href="series_1.html#K3">to_power</a> <font color="Olive">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">m</font> holds <br/><font color="Olive">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">m</font>
 ;<br/><b>hence </b><a NAME="E3:20_1_1_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( 2 <a href="series_1.html#K3">to_power</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j</font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">m</font> &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 2 <a href="series_1.html#K3">to_power</a> <font color="Olive">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">m</font> holds <br/><font color="Olive">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1_1_2"><b>suppose </b></a><a NAME="E1:20_1_1_1_2"/><i><font color="Green">E40</font></i>: 
2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">j</font>
 ;<br/><div class="add"><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( 2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">m</font> );<br/><a NAME="E2:20_1_1_1_2"/>
2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">m</font>
 <b>by </b><i/>;<br/><a NAME="E3:20_1_1_1_2"/><b>then </b>
( 2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">m</font> )
 <b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E4:20_1_1_1_2"/><b>then </b><i><font color="Green">E41</font></i>: 
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 ;<br/><a NAME="E5:20_1_1_1_2"/>
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] &amp; ( for <font color="Olive">l</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">l</font>] holds <br/><font color="Olive">l</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">k</font> ) )
 <b>from </b><i><a class="ref" href="nat_1.html#S5">NAT_1:sch 5</a>();<br/></i><b>hence </b><a NAME="E6:20_1_1_1_2"/>
 ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( 2 <a href="series_1.html#K3">to_power</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j</font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">m</font> &amp; ( for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 2 <a href="series_1.html#K3">to_power</a> <font color="Olive">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">j</font> &amp; <font color="Olive">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">m</font> holds <br/><font color="Olive">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">b<sub>1</sub></font> ) )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

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