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<span class="kw">let </span><font color="Maroon" title="c3">a1</font>, <font color="Maroon" title="c4">a2</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <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> &amp;  <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> implies <font color="Maroon" title="c3">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a2</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:28_1_2"/><span class="lab"><font color="Green" title="E25">A20</font></span>: 
 <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
 <span class="kw">and </span><br/><a NAME="E2:28_1_2"/><span class="lab"><font color="Green" title="E26">A21</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E3:28_1_2"/><span class="lab"><font color="Green" title="E27">A22</font></span>: 
 <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
 <span class="kw">and </span><br/><a NAME="E4:28_1_2"/><span class="lab"><font color="Green" title="E28">A23</font></span>: 
<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>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a2</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c5">b2</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E6:28_1_2"/><span class="lab"><font color="Green" title="E29">A24</font></span>: 
<font color="Maroon" title="c5">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="E7:28_1_2"/>
<font color="Maroon" title="c5">b2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:28_1_2"><span class="lab"><font color="Green" title="E28">A23</font></span></a>, <a class="txt" href="#E6:28_1_2"><span class="lab"><font color="Green" title="E29">A24</font></span></a>, <a class="ref" href="idea_1.html#T17" target="_self" onmouseover="rs('idea_1/T17')" onmouseout="rh()">Th17</a></span>;<br/>
<a NAME="E8:28_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A25</font></span>: 
<font color="Maroon" title="c5">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#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">b1</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E10:28_1_2"/><span class="lab"><font color="Green" title="E31">A26</font></span>: 
<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="E11:28_1_2"/>
<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#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:28_1_2"><span class="lab"><font color="Green" title="E26">A21</font></span></a>, <a class="txt" href="#E10:28_1_2"><span class="lab"><font color="Green" title="E31">A26</font></span></a>, <a class="ref" href="idea_1.html#T17" target="_self" onmouseover="rs('idea_1/T17')" onmouseout="rh()">Th17</a></span>;<br/>
<a NAME="E12:28_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A27</font></span>: 
<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#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E14:28_1_2"/><span class="lab"><font color="Green" title="E33">A28</font></span>: 
<font color="Maroon" title="c7">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="E15:28_1_2"/><span class="lab"><font color="Green" title="E34">A29</font></span>: 
<font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="txt" href="#E14:28_1_2"><span class="lab"><font color="Green" title="E33">A28</font></span></a>, <a class="ref" href="idea_1.html#T17" target="_self" onmouseover="rs('idea_1/T17')" onmouseout="rh()">Th17</a></span>;<br/>
<a NAME="E16:28_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A30</font></span>: 
<font color="Maroon" title="c7">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#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/>
<a NAME="E17:28_1_2"/><span class="lab"><font color="Green" title="E36">A31</font></span>: 
<font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="txt" href="#E14:28_1_2"><span class="lab"><font color="Green" title="E33">A28</font></span></a>, <a class="ref" href="idea_1.html#T17" target="_self" onmouseover="rs('idea_1/T17')" onmouseout="rh()">Th17</a></span>;<br/>
<a NAME="E18:28_1_2"/>
<font color="Maroon" title="c5">b2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:28_1_2"><span class="lab"><font color="Green" title="E28">A23</font></span></a>, <a class="txt" href="#E6:28_1_2"><span class="lab"><font color="Green" title="E29">A24</font></span></a>, <a class="ref" href="idea_1.html#T17" target="_self" onmouseover="rs('idea_1/T17')" onmouseout="rh()">Th17</a></span>;<br/>
<a NAME="E19:28_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A32</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">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#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:28"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="txt" href="#E17:28_1_2"><span class="lab"><font color="Green" title="E36">A31</font></span></a>, <a class="txt" href="#E16:28_1_2"><span class="lab"><font color="Green" title="E35">A30</font></span></a>, <a class="txt" href="#E8:28_1_2"><span class="lab"><font color="Green" title="E30">A25</font></span></a>, <a class="ref" href="idea_1.html#T5" target="_self" onmouseover="rs('idea_1/T5')" onmouseout="rh()">Th5</a></span>;<br/>
<a NAME="E20:28_1_2"/>
<font color="Maroon" title="c6">b1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:28_1_2"><span class="lab"><font color="Green" title="E26">A21</font></span></a>, <a class="txt" href="#E10:28_1_2"><span class="lab"><font color="Green" title="E31">A26</font></span></a>, <a class="ref" href="idea_1.html#T17" target="_self" onmouseover="rs('idea_1/T17')" onmouseout="rh()">Th17</a></span>;<br/>
<a NAME="E21:28_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:28"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="txt" href="#E17:28_1_2"><span class="lab"><font color="Green" title="E36">A31</font></span></a>, <a class="txt" href="#E16:28_1_2"><span class="lab"><font color="Green" title="E35">A30</font></span></a>, <a class="txt" href="#E12:28_1_2"><span class="lab"><font color="Green" title="E32">A27</font></span></a>, <a class="ref" href="idea_1.html#T5" target="_self" onmouseover="rs('idea_1/T5')" onmouseout="rh()">Th5</a></span>;<br/>
<a NAME="E22:28_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A33</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">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#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28_1_2"><span class="lab"><font color="Green" title="E25">A20</font></span></a>, <a class="txt" href="#E3:28_1_2"><span class="lab"><font color="Green" title="E27">A22</font></span></a>, <a class="txt" href="#E14:28_1_2"><span class="lab"><font color="Green" title="E33">A28</font></span></a>, <a class="txt" href="#E10:28_1_2"><span class="lab"><font color="Green" title="E31">A26</font></span></a>, <a class="txt" href="#E6:28_1_2"><span class="lab"><font color="Green" title="E29">A24</font></span></a>, <a class="txt" href="#E19:28_1_2"><span class="lab"><font color="Green" title="E37">A32</font></span></a>, <a class="ref" href="idea_1.html#T20" target="_self" onmouseover="rs('idea_1/T20')" onmouseout="rh()">Th20</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="28_1_2_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b2</font></span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_2_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:28_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="c5">b2</font> or <font color="Maroon" title="c5">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="28_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:28_1_2_1_1_1"/><span class="lab"><font color="Green" title="E39">A34</font></span>: 
<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">b2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">b1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b2</font></span><br/><div class="add"><span class="kw">consider </span><font color="Maroon" title="c8">b</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><span class="kw"> such that </span><br/><a NAME="E3:28_1_2_1_1_1"/><span class="lab"><font color="Green" title="E40">A35</font></span>: 
<font color="Maroon" title="c8">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">b2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c6">b1</font>
 ;<br/><a NAME="E4:28_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="c5">b2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c6">b1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28_1_2_1_1_1"><span class="lab"><font color="Green" title="E39">A34</font></span></a>, <a class="ref" href="xreal_1.html#T9" onmouseover="rs('xreal_1/T9')" onmouseout="rh()">XREAL_1:9</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">b</font> = <font color="Maroon" title="c8">b</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:28_1_2_1_1_1"><span class="lab"><font color="Green" title="E40">A35</font></span></a>, <a class="ref" href="int_1.html#T3" onmouseover="rs('int_1/T3')" onmouseout="rh()">INT_1:3</a></span>;<br/><a NAME="E6:28_1_2_1_1_1"/>
 ex <font color="Olive" title="b1">t</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">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="c7">k</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">t</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:28_1_2"><span class="lab"><font color="Green" title="E38">A33</font></span></a>, <a class="txt" href="#E1:28_1_2_1_1_1"><span class="lab"><font color="Green" title="E39">A34</font></span></a>, <a class="ref" href="idea_1.html#T2" target="_self" onmouseover="rs('idea_1/T2')" onmouseout="rh()">Th2</a>, <a class="ref" href="nat_1.html#T4" onmouseover="rs('nat_1/T4')" onmouseout="rh()">NAT_1:4</a></span>;<br/><a NAME="E7:28_1_2_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default">2 <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c9">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:28_1_2_1_1_1"><span class="lab"><font color="Green" title="E40">A35</font></span></a>, <a class="ref" href="nat_d.html#D3" onmouseover="rs('nat_d/D3')" onmouseout="rh()">NAT_D:def 3</a></span>;<br/><a NAME="E8:28_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A36</font></span>: 
( <span class="p1">(<span class="default">2 <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c7">k</font> or <span class="p1">(<span class="default">2 <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c9">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:28"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="ref" href="newton.html#T80" onmouseover="rs('newton/T80')" onmouseout="rh()">NEWTON:80</a></span>;<br/><a NAME="E9:28_1_2_1_1_1"/>
<font color="Maroon" title="c9">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:28_1_2"><span class="lab"><font color="Green" title="E28">A23</font></span></a>, <a class="txt" href="#E6:28_1_2"><span class="lab"><font color="Green" title="E29">A24</font></span></a>, <a class="txt" href="#E3:28_1_2_1_1_1"><span class="lab"><font color="Green" title="E40">A35</font></span></a>, <a class="ref" href="idea_1.html#T4" target="_self" onmouseover="rs('idea_1/T4')" onmouseout="rh()">Th4</a>, <a class="ref" href="idea_1.html#T17" target="_self" onmouseover="rs('idea_1/T17')" onmouseout="rh()">Th17</a></span>;<br/><a NAME="E10:28_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A37</font></span>: 
 not <span class="p1">(<span class="default">2 <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E11:28_1_2_1_1_1"/>
 not <span class="p1">(<span class="default">2 <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:28_1_2"><span class="lab"><font color="Green" title="E34">A29</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E12:28_1_2_1_1_1"/><span class="kw">then </span>
 not  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:28_1_2"><span class="lab"><font color="Green" title="E36">A31</font></span></a>, <a class="txt" href="#E8:28_1_2_1_1_1"><span class="lab"><font color="Green" title="E41">A36</font></span></a>, <a class="txt" href="#E10:28_1_2_1_1_1"><span class="lab"><font color="Green" title="E42">A37</font></span></a>, <a class="ref" href="nat_d.html#T7" onmouseover="rs('nat_d/T7')" onmouseout="rh()">NAT_D:7</a></span>;<br/><a NAME="E13:28_1_2_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">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#K5" title="NUMBERS:func.5">0</a> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c6">b1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:28_1_2_1_1_1"><span class="lab"><font color="Green" title="E40">A35</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E14:28_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="c5">b2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:28_1_2_1_1_2"/><span class="lab"><font color="Green" title="E39">A38</font></span>: 
<font color="Maroon" title="c5">b2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">b1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b1</font></span><br/><div class="add"><span class="kw">consider </span><font color="Maroon" title="c8">b</font> being   <a href="int_1.html#NM1" title="INT_1:NM.1">Integer</a><span class="kw"> such that </span><br/><a NAME="E3:28_1_2_1_1_2"/><span class="lab"><font color="Green" title="E40">A39</font></span>: 
<font color="Maroon" title="c8">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="c5">b2</font>
 ;<br/><a NAME="E4:28_1_2_1_1_2"/>
<font color="Maroon" title="c5">b2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c5">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="c5">b2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28_1_2_1_1_2"><span class="lab"><font color="Green" title="E39">A38</font></span></a>, <a class="ref" href="xreal_1.html#T9" onmouseover="rs('xreal_1/T9')" onmouseout="rh()">XREAL_1:9</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">b</font> = <font color="Maroon" title="c8">b</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#NK1" title="NUMBERS:NK.1">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:28_1_2_1_1_2"><span class="lab"><font color="Green" title="E40">A39</font></span></a>, <a class="ref" href="int_1.html#T3" onmouseover="rs('int_1/T3')" onmouseout="rh()">INT_1:3</a></span>;<br/><a NAME="E6:28_1_2_1_1_2"/>
 ex <font color="Olive" title="b1">t</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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="c7">k</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">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#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">t</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:28_1_2"><span class="lab"><font color="Green" title="E38">A33</font></span></a>, <a class="txt" href="#E1:28_1_2_1_1_2"><span class="lab"><font color="Green" title="E39">A38</font></span></a>, <a class="ref" href="idea_1.html#T2" target="_self" onmouseover="rs('idea_1/T2')" onmouseout="rh()">Th2</a>, <a class="ref" href="nat_1.html#T4" onmouseover="rs('nat_1/T4')" onmouseout="rh()">NAT_1:4</a></span>;<br/><a NAME="E7:28_1_2_1_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default">2 <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c7">k</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c9">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:28_1_2_1_1_2"><span class="lab"><font color="Green" title="E40">A39</font></span></a>, <a class="ref" href="nat_d.html#D3" onmouseover="rs('nat_d/D3')" onmouseout="rh()">NAT_D:def 3</a></span>;<br/><a NAME="E8:28_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A40</font></span>: 
( <span class="p1">(<span class="default">2 <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c7">k</font> or <span class="p1">(<span class="default">2 <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="nat_d.html#R1" title="NAT_D:pred.1">divides</a> <font color="Maroon" title="c9">b</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:28"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="ref" href="newton.html#T80" onmouseover="rs('newton/T80')" onmouseout="rh()">NEWTON:80</a></span>;<br/><a NAME="E9:28_1_2_1_1_2"/>
<font color="Maroon" title="c9">b</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:28_1_2"><span class="lab"><font color="Green" title="E26">A21</font></span></a>, <a class="txt" href="#E10:28_1_2"><span class="lab"><font color="Green" title="E31">A26</font></span></a>, <a class="txt" href="#E3:28_1_2_1_1_2"><span class="lab"><font color="Green" title="E40">A39</font></span></a>, <a class="ref" href="idea_1.html#T4" target="_self" onmouseover="rs('idea_1/T4')" onmouseout="rh()">Th4</a>, <a class="ref" href="idea_1.html#T17" target="_self" onmouseover="rs('idea_1/T17')" onmouseout="rh()">Th17</a></span>;<br/><a NAME="E10:28_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A41</font></span>: 
 not <span class="p1">(<span class="default">2 <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">b</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E11:28_1_2_1_1_2"/>
 not <span class="p1">(<span class="default">2 <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:28_1_2"><span class="lab"><font color="Green" title="E34">A29</font></span></a>, <a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E12:28_1_2_1_1_2"/><span class="kw">then </span>
 not  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">b</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:28_1_2"><span class="lab"><font color="Green" title="E36">A31</font></span></a>, <a class="txt" href="#E8:28_1_2_1_1_2"><span class="lab"><font color="Green" title="E41">A40</font></span></a>, <a class="txt" href="#E10:28_1_2_1_1_2"><span class="lab"><font color="Green" title="E42">A41</font></span></a>, <a class="ref" href="nat_d.html#T7" onmouseover="rs('nat_d/T7')" onmouseout="rh()">NAT_D:7</a></span>;<br/><a NAME="E13:28_1_2_1_1_2"/><span class="kw">then </span>
<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="c5">b2</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">b2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:28_1_2_1_1_2"><span class="lab"><font color="Green" title="E40">A39</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E14:28_1_2_1_1_2"/>
<font color="Maroon" title="c5">b2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E24:28_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:28_1_2"><span class="lab"><font color="Green" title="E26">A21</font></span></a>, <a class="txt" href="#E4:28_1_2"><span class="lab"><font color="Green" title="E28">A23</font></span></a>, <a class="txt" href="#E10:28_1_2"><span class="lab"><font color="Green" title="E31">A26</font></span></a>, <a class="txt" href="#E6:28_1_2"><span class="lab"><font color="Green" title="E29">A24</font></span></a>, <a class="ref" href="idea_1.html#T18" target="_self" onmouseover="rs('idea_1/T18')" onmouseout="rh()">Th18</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
