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<b>let </b><font color="Maroon" title="c3">a1</font>, <font color="Maroon" title="c4">a2</font> be    <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> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:32_1_2"/><i><font color="Green" title="E25">A18</font></i>: 
(  <a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> <font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c3">a1</font> <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> )
 <b>and </b><br/><a NAME="E2:32_1_2"/><i><font color="Green" title="E26">A19</font></i>: 
(  <a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> <font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c4">a2</font>,<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c4">a2</font> <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> )
 ;<br/>

<b>consider </b><font color="Maroon" title="c5">k</font> being    <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> <b> such that </b><br/><a NAME="E4:32_1_2"/><i><font color="Green" title="E27">A20</font></i>: 
<font color="Maroon" title="c5">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> <font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c1">n</font>
 ;<br/>
<a NAME="E5:32_1_2"/><i><font color="Green" title="E28">A21</font></i>: 
( <font color="Maroon" title="c5">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c5">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="txt" href="idea_1.html#E1:32"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="idea_1.html#E4:32_1_2"><i><font color="Green" title="E27">A20</font></i></a>, <a class="ref" href="idea_1.html#T18" target="_self" title="IDEA_1:th.18">Th18</a></i>;<br/>
<a NAME="E6:32_1_2"/><b>then </b><i><font color="Green" title="E29">A22</font></i>: 
<font color="Maroon" title="c5">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">b1</font> being    <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> <b> such that </b><br/><a NAME="E8:32_1_2"/><i><font color="Green" title="E30">A23</font></i>: 
<font color="Maroon" title="c6">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> <font color="Maroon" title="c3">a1</font>,<font color="Maroon" title="c1">n</font>
 ;<br/>
<a NAME="E9:32_1_2"/><i><font color="Green" title="E31">A24</font></i>: 
( <font color="Maroon" title="c6">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c6">b1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="txt" href="idea_1.html#E1:32_1_2"><i><font color="Green" title="E25">A18</font></i></a>, <a class="txt" href="idea_1.html#E8:32_1_2"><i><font color="Green" title="E30">A23</font></i></a>, <a class="ref" href="idea_1.html#T18" target="_self" title="IDEA_1:th.18">Th18</a></i>;<br/>
<a NAME="E10:32_1_2"/><b>then </b><i><font color="Green" title="E32">A25</font></i>: 
<font color="Maroon" title="c6">b1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">b2</font> being    <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> <b> such that </b><br/><a NAME="E12:32_1_2"/><i><font color="Green" title="E33">A26</font></i>: 
<font color="Maroon" title="c7">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> <font color="Maroon" title="c4">a2</font>,<font color="Maroon" title="c1">n</font>
 ;<br/>
<a NAME="E13:32_1_2"/><i><font color="Green" title="E34">A27</font></i>: 
( <font color="Maroon" title="c7">b2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c7">b2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 
<b>by </b><i><a class="txt" href="idea_1.html#E2:32_1_2"><i><font color="Green" title="E26">A19</font></i></a>, <a class="txt" href="idea_1.html#E12:32_1_2"><i><font color="Green" title="E33">A26</font></i></a>, <a class="ref" href="idea_1.html#T18" target="_self" title="IDEA_1:th.18">Th18</a></i>;<br/>
<a NAME="E14:32_1_2"/><b>then </b><i><font color="Green" title="E35">A28</font></i>: 
<font color="Maroon" title="c7">b2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E15:32_1_2"/><i><font color="Green" title="E36">A29</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c6">b1</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="idea_1.html#E1:32"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="idea_1.html#E5:32_1_2"><i><font color="Green" title="E28">A21</font></i></a>, <a class="txt" href="idea_1.html#E6:32_1_2"><i><font color="Green" title="E29">A22</font></i></a>, <a class="txt" href="idea_1.html#E9:32_1_2"><i><font color="Green" title="E31">A24</font></i></a>, <a class="txt" href="idea_1.html#E10:32_1_2"><i><font color="Green" title="E32">A25</font></i></a>, <a class="ref" href="idea_1.html#T5" target="_self" title="IDEA_1:th.5">Th5</a></i>;<br/>
<a NAME="E16:32_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c7">b2</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="idea_1.html#E1:32"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="idea_1.html#E5:32_1_2"><i><font color="Green" title="E28">A21</font></i></a>, <a class="txt" href="idea_1.html#E6:32_1_2"><i><font color="Green" title="E29">A22</font></i></a>, <a class="txt" href="idea_1.html#E13:32_1_2"><i><font color="Green" title="E34">A27</font></i></a>, <a class="txt" href="idea_1.html#E14:32_1_2"><i><font color="Green" title="E35">A28</font></i></a>, <a class="ref" href="idea_1.html#T5" target="_self" title="IDEA_1:th.5">Th5</a></i>;<br/>
<a NAME="E17:32_1_2"/><b>then </b><i><font color="Green" title="E37">A30</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c6">b1</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c7">b2</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="idea_1.html#E1:32_1_2"><i><font color="Green" title="E25">A18</font></i></a>, <a class="txt" href="idea_1.html#E2:32_1_2"><i><font color="Green" title="E26">A19</font></i></a>, <a class="txt" href="idea_1.html#E4:32_1_2"><i><font color="Green" title="E27">A20</font></i></a>, <a class="txt" href="idea_1.html#E8:32_1_2"><i><font color="Green" title="E30">A23</font></i></a>, <a class="txt" href="idea_1.html#E12:32_1_2"><i><font color="Green" title="E33">A26</font></i></a>, <a class="txt" href="idea_1.html#E15:32_1_2"><i><font color="Green" title="E36">A29</font></i></a>, <a class="ref" href="idea_1.html#T21" target="_self" title="IDEA_1:th.21">Th21</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:32_1_2_1_1"/>
( <font color="Maroon" title="c6">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">b2</font> or <font color="Maroon" title="c7">b2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:32_1_2_1_1_1"/><i><font color="Green" title="E38">A31</font></i>: 
<font color="Maroon" title="c6">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">b2</font>
 ;<br/><div class="add"><a NAME="E2:32_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c6">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c7">b2</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T4" title="NAT_1:th.4">NAT_1:4</a></i>;<br/><b>then consider </b><font color="Maroon" title="c8">t</font> being    <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> <b> such that </b><br/><a NAME="E4:32_1_2_1_1_1"/><i><font color="Green" title="E39">A32</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c7">b2</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c6">b1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c8">t</font>
 <b>by </b><i><a class="txt" href="idea_1.html#E17:32_1_2"><i><font color="Green" title="E37">A30</font></i></a>, <a class="ref" href="idea_1.html#T2" target="_self" title="IDEA_1:th.2">Th2</a></i>;<br/><b>consider </b><font color="Maroon" title="c9">b</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><b> such that </b><br/><a NAME="E6:32_1_2_1_1_1"/><i><font color="Green" title="E40">A33</font></i>: 
<font color="Maroon" title="c9">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c6">b1</font>
 ;<br/><a NAME="E7:32_1_2_1_1_1"/>
<font color="Maroon" title="c6">b1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c6">b1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">b2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c6">b1</font>
 <b>by </b><i><a class="txt" href="idea_1.html#E1:32_1_2_1_1_1"><i><font color="Green" title="E38">A31</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c10">b</font> = <font color="Maroon" title="c9">b</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>  <b>by </b><i><a class="txt" href="idea_1.html#E6:32_1_2_1_1_1"><i><font color="Green" title="E40">A33</font></i></a>, <a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/><a NAME="E9:32_1_2_1_1_1"/>
<span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c5">k</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c10">b</font>
 <b>by </b><i><a class="txt" href="idea_1.html#E4:32_1_2_1_1_1"><i><font color="Green" title="E39">A32</font></i></a>, <a class="txt" href="idea_1.html#E6:32_1_2_1_1_1"><i><font color="Green" title="E40">A33</font></i></a>, <a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/><a NAME="E10:32_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E41">A34</font></i>: 
( <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c5">k</font> or <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c10">b</font> )
 <b>by </b><i><a class="txt" href="idea_1.html#E1:32"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="newton.html#T98" title="NEWTON:th.98">NEWTON:98</a></i>;<br/><a NAME="E11:32_1_2_1_1_1"/><i><font color="Green" title="E42">A35</font></i>: 
not <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k</font>
 <b>by </b><i><a class="txt" href="idea_1.html#E5:32_1_2"><i><font color="Green" title="E28">A21</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E12:32_1_2_1_1_1"/>
<font color="Maroon" title="c10">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="txt" href="idea_1.html#E13:32_1_2"><i><font color="Green" title="E34">A27</font></i></a>, <a class="txt" href="idea_1.html#E6:32_1_2_1_1_1"><i><font color="Green" title="E40">A33</font></i></a>, <a class="ref" href="idea_1.html#T4" target="_self" title="IDEA_1:th.4">Th4</a></i>;<br/><a NAME="E13:32_1_2_1_1_1"/><b>then </b>
not <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">b</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E14:32_1_2_1_1_1"/><b>then </b>
not  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">b</font>
 <b>by </b><i><a class="txt" href="idea_1.html#E5:32_1_2"><i><font color="Green" title="E28">A21</font></i></a>, <a class="txt" href="idea_1.html#E10:32_1_2_1_1_1"><i><font color="Green" title="E41">A34</font></i></a>, <a class="txt" href="idea_1.html#E11:32_1_2_1_1_1"><i><font color="Green" title="E42">A35</font></i></a>, <a class="ref" href="nat_d.html#T7" title="NAT_D:th.7">NAT_D:7</a></i>;<br/><a NAME="E15:32_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">b2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c6">b1</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c6">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c6">b1</font>
 <b>by </b><i><a class="txt" href="idea_1.html#E6:32_1_2_1_1_1"><i><font color="Green" title="E40">A33</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><b>hence </b><a NAME="E16:32_1_2_1_1_1"/>
<font color="Maroon" title="c6">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">b2</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="32_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:32_1_2_1_1_2"/><i><font color="Green" title="E38">A36</font></i>: 
<font color="Maroon" title="c7">b2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b1</font>
 ;<br/><div class="add"><a NAME="E2:32_1_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c7">b2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c6">b1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T4" title="NAT_1:th.4">NAT_1:4</a></i>;<br/><b>then consider </b><font color="Maroon" title="c8">t</font> being    <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> <b> such that </b><br/><a NAME="E4:32_1_2_1_1_2"/><i><font color="Green" title="E39">A37</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c6">b1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">k</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c7">b2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c8">t</font>
 <b>by </b><i><a class="txt" href="idea_1.html#E17:32_1_2"><i><font color="Green" title="E37">A30</font></i></a>, <a class="ref" href="idea_1.html#T2" target="_self" title="IDEA_1:th.2">Th2</a></i>;<br/><b>consider </b><font color="Maroon" title="c9">b</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><b> such that </b><br/><a NAME="E6:32_1_2_1_1_2"/><i><font color="Green" title="E40">A38</font></i>: 
<font color="Maroon" title="c9">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">b2</font>
 ;<br/><a NAME="E7:32_1_2_1_1_2"/>
<font color="Maroon" title="c7">b2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">b2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">b2</font>
 <b>by </b><i><a class="txt" href="idea_1.html#E1:32_1_2_1_1_2"><i><font color="Green" title="E38">A36</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c10">b</font> = <font color="Maroon" title="c9">b</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>  <b>by </b><i><a class="txt" href="idea_1.html#E6:32_1_2_1_1_2"><i><font color="Green" title="E40">A38</font></i></a>, <a class="ref" href="int_1.html#T16" title="INT_1:th.16">INT_1:16</a></i>;<br/><a NAME="E9:32_1_2_1_1_2"/>
<span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c5">k</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c10">b</font>
 <b>by </b><i><a class="txt" href="idea_1.html#E4:32_1_2_1_1_2"><i><font color="Green" title="E39">A37</font></i></a>, <a class="txt" href="idea_1.html#E6:32_1_2_1_1_2"><i><font color="Green" title="E40">A38</font></i></a>, <a class="ref" href="nat_d.html#D3" title="NAT_D:def.3">NAT_D:def 3</a></i>;<br/><a NAME="E10:32_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E41">A39</font></i>: 
( <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c5">k</font> or <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c10">b</font> )
 <b>by </b><i><a class="txt" href="idea_1.html#E1:32"><i><font color="Green" title="E24">A1</font></i></a>, <a class="ref" href="newton.html#T98" title="NEWTON:th.98">NEWTON:98</a></i>;<br/><a NAME="E11:32_1_2_1_1_2"/><i><font color="Green" title="E42">A40</font></i>: 
not <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">k</font>
 <b>by </b><i><a class="txt" href="idea_1.html#E5:32_1_2"><i><font color="Green" title="E28">A21</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E12:32_1_2_1_1_2"/>
<font color="Maroon" title="c10">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="txt" href="idea_1.html#E9:32_1_2"><i><font color="Green" title="E31">A24</font></i></a>, <a class="txt" href="idea_1.html#E6:32_1_2_1_1_2"><i><font color="Green" title="E40">A38</font></i></a>, <a class="ref" href="idea_1.html#T4" target="_self" title="IDEA_1:th.4">Th4</a></i>;<br/><a NAME="E13:32_1_2_1_1_2"/><b>then </b>
not <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">b</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E14:32_1_2_1_1_2"/><b>then </b>
not  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">b</font>
 <b>by </b><i><a class="txt" href="idea_1.html#E5:32_1_2"><i><font color="Green" title="E28">A21</font></i></a>, <a class="txt" href="idea_1.html#E10:32_1_2_1_1_2"><i><font color="Green" title="E41">A39</font></i></a>, <a class="txt" href="idea_1.html#E11:32_1_2_1_1_2"><i><font color="Green" title="E42">A40</font></i></a>, <a class="ref" href="nat_d.html#T7" title="NAT_D:th.7">NAT_D:7</a></i>;<br/><a NAME="E15:32_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c6">b1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c7">b2</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c7">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c7">b2</font>
 <b>by </b><i><a class="txt" href="idea_1.html#E6:32_1_2_1_1_2"><i><font color="Green" title="E40">A38</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><b>hence </b><a NAME="E16:32_1_2_1_1_2"/>
<font color="Maroon" title="c7">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b1</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E19:32_1_2"/>
<font color="Maroon" title="c3">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a2</font>
 <b>by </b><i><a class="txt" href="idea_1.html#E1:32_1_2"><i><font color="Green" title="E25">A18</font></i></a>, <a class="txt" href="idea_1.html#E2:32_1_2"><i><font color="Green" title="E26">A19</font></i></a>, <a class="txt" href="idea_1.html#E8:32_1_2"><i><font color="Green" title="E30">A23</font></i></a>, <a class="txt" href="idea_1.html#E12:32_1_2"><i><font color="Green" title="E33">A26</font></i></a>, <a class="ref" href="idea_1.html#T19" target="_self" title="IDEA_1:th.19">Th19</a></i>;<br/>


</div>
