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<div class="add">

<a NAME="E1:28_1_1"/><span class="lab"><font color="Green" title="E25">A3</font></span>: 
2 <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c1">n</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="ref" href="power.html#T34" onmouseover="rs('power/T34')" onmouseout="rh()">POWER:34</a></span>;<br/>
<a NAME="E2:28_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A4</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="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> <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="xreal_1.html#T6" onmouseover="rs('xreal_1/T6')" onmouseout="rh()">XREAL_1:6</a></span>;<br/>
<a NAME="E3:28_1_1"/><span class="kw">then </span>
<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#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</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="E4:28_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<span class="kw">by</span> <span class="lab"><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="c3">n3</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 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="ref" href="int_1.html#T3" onmouseover="rs('int_1/T3')" onmouseout="rh()">INT_1:3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c4">nn</font> = <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 as   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">n2</font> = <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#K6" title="XCMPLX_0:func.6">-</a> 1 as   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> ;<br/>
<a NAME="E8:28_1_1"/><span class="lab"><font color="Green" title="E27">A5</font></span>: 
2 <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T35" onmouseover="rs('power/T35')" onmouseout="rh()">POWER:35</a></span>;<br/>
<a NAME="E9:28_1_1"/>
<font color="Maroon" title="c5">n2</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">n2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">nn</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c3">n3</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E28">A6</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c5">n2</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">nn</font> = 
1 <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">nn</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T21" onmouseover="rs('nat_d/T21')" onmouseout="rh()">NAT_D:21</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:28_1_1"><span class="lab"><font color="Green" title="E26">A4</font></span></a>, <a class="ref" href="nat_d.html#T24" onmouseover="rs('nat_d/T24')" onmouseout="rh()">NAT_D:24</a></span>
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_2"><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_1_2"/>
( <font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  or <font color="Maroon" title="c2">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_2_1"><span class="kw">suppose </span></a><a NAME="E1:28_1_1_2_1"/><span class="lab"><font color="Green" title="E29">A7</font></span>: 
<font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</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">  ex <font color="Olive">b<sub>1</sub></font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>(  <a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> (<font color="Maroon" title="c2">i</font>,<font color="Olive">b<sub>1</sub></font>,<font color="Maroon" title="c1">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Olive">b<sub>1</sub></font> <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> )</span><br/><div class="add"><span class="kw">consider </span><font color="Maroon" title="c6">j</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E3:28_1_1_2_1"/><span class="lab"><font color="Green" title="E30">A8</font></span>: 
<font color="Maroon" title="c6">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 ;<br/><span class="kw">take </span>
<font color="Maroon" title="c6">j</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"> (  <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="c6">j</font>,<font color="Maroon" title="c1">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c6">j</font> <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> )</span><br/><a NAME="E4:28_1_1_2_1"/><span class="lab"><font color="Green" title="E31">A9</font></span>: 
<font color="Maroon" title="c6">j</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">by</span> <span class="lab"><a class="txt" href="#E1:28_1_1"><span class="lab"><font color="Green" title="E25">A3</font></span></a>, <a class="txt" href="#E3:28_1_1_2_1"><span class="lab"><font color="Green" title="E30">A8</font></span></a></span>;<br/> <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="c6">j</font>,<font color="Maroon" title="c1">n</font>) = 
 <a href="idea_1.html#K8" title="IDEA_1:func.8">ChangeVal_2</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">nn</font></span>)</span>,<font color="Maroon" title="c1">n</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28_1_1_2_1"><span class="lab"><font color="Green" title="E29">A7</font></span></a>, <a class="txt" href="#E3:28_1_1_2_1"><span class="lab"><font color="Green" title="E30">A8</font></span></a>, <a class="ref" href="idea_1.html#D7" target="_self" onmouseover="rs('idea_1/D7')" onmouseout="rh()">Def7</a></span>
<br/>.= 
 <a href="idea_1.html#K8" title="IDEA_1:func.8">ChangeVal_2</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">n2</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c5">n2</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <font color="Maroon" title="c4">nn</font></span>)</span>,<font color="Maroon" title="c1">n</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="idea_1.html#D7" target="_self" onmouseover="rs('idea_1/D7')" onmouseout="rh()">Def7</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:28_1_1"><span class="lab"><font color="Green" title="E27">A5</font></span></a>, <a class="txt" href="#E10:28_1_1"><span class="lab"><font color="Green" title="E28">A6</font></span></a>, <a class="ref" href="idea_1.html#D8" target="_self" onmouseover="rs('idea_1/D8')" onmouseout="rh()">Def8</a></span>
;<br/><span class="kw">hence </span><a NAME="E6:28_1_1_2_1"/>
(  <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="c6">j</font>,<font color="Maroon" title="c1">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c6">j</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">by</span> <span class="lab"><a class="txt" href="#E4:28_1_1_2_1"><span class="lab"><font color="Green" title="E31">A9</font></span></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><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_2_2"><span class="kw">suppose </span></a><a NAME="E1:28_1_1_2_2"/><span class="lab"><font color="Green" title="E29">A10</font></span>: 
<font color="Maroon" title="c2">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</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">  ex <font color="Olive">b<sub>1</sub></font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> st <br/>(  <a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> (<font color="Maroon" title="c2">i</font>,<font color="Olive">b<sub>1</sub></font>,<font color="Maroon" title="c1">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Olive">b<sub>1</sub></font> <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> )</span><br/><div class="add"><a NAME="E2:28_1_1_2_2"/>
<font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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></span>;<br/><a NAME="E3:28_1_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c2">i</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">then </span><span class="kw">consider </span><font color="Maroon" title="c6">a</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><span class="kw"> such that </span><br/><a NAME="E5:28_1_1_2_2"/><span class="lab"><font color="Green" title="E30">A11</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">a</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c2">i</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> 1
 <span class="kw">and </span><br/><a NAME="E6:28_1_1_2_2"/><span class="lab"><font color="Green" title="E31">A12</font></span>: 
<font color="Maroon" title="c6">a</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="txt" href="#E2:28"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="txt" href="#E2:28_1_1"><span class="lab"><font color="Green" title="E26">A4</font></span></a>, <a class="txt" href="#E1:28_1_1_2_2"><span class="lab"><font color="Green" title="E29">A10</font></span></a>, <a class="ref" href="idea_1.html#T1" target="_self" onmouseover="rs('idea_1/T1')" onmouseout="rh()">Th1</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c7">k</font> =  <a href="idea_1.html#K8" title="IDEA_1:func.8">ChangeVal_2</a> (<font color="Maroon" title="c6">a</font>,<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"> (  <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="c7">k</font>,<font color="Maroon" title="c1">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c7">k</font> <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> )</span><br/><a NAME="E7:28_1_1_2_2"/><span class="lab"><font color="Green" title="E32">A13</font></span>: 
<font color="Maroon" title="c6">a</font> <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="#E5:28_1_1_2_2"><span class="lab"><font color="Green" title="E30">A11</font></span></a>, <a class="ref" href="nat_d.html#T24" onmouseover="rs('nat_d/T24')" onmouseout="rh()">NAT_D:24</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="28_1_1_2_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"> (  <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="c7">k</font>,<font color="Maroon" title="c1">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c7">k</font> <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> )</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_2_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_1_2_2_1_1"/>
( <font color="Maroon" title="c6">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 2 <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c1">n</font> or <font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c1">n</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_2_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:28_1_1_2_2_1_1_1"/><span class="lab"><font color="Green" title="E33">A14</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 2 <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</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"> (  <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="c7">k</font>,<font color="Maroon" title="c1">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c7">k</font> <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> )</span><br/><div class="add"><a NAME="E2:28_1_1_2_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A15</font></span>: 
<font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">a</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="idea_1.html#D8" target="_self" onmouseover="rs('idea_1/D8')" onmouseout="rh()">Def8</a></span>;<br/><a NAME="E3:28_1_1_2_2_1_1_1"/><span class="kw">then </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="#E6:28_1_1_2_2"><span class="lab"><font color="Green" title="E31">A12</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="E4:28_1_1_2_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&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_1_1_2_2_1_1_1"><span class="lab"><font color="Green" title="E33">A14</font></span></a>, <a class="txt" href="#E2:28_1_1_2_2_1_1_1"><span class="lab"><font color="Green" title="E34">A15</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><a NAME="E5:28_1_1_2_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A16</font></span>: 
<font color="Maroon" title="c7">k</font> <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font>
 ;<br/> <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="c7">k</font>,<font color="Maroon" title="c1">n</font>) = 
 <a href="idea_1.html#K8" title="IDEA_1:func.8">ChangeVal_2</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p2">(<span class="default"><span class="p3">(<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>)</span>,<font color="Maroon" title="c1">n</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28_1_1_2_2"><span class="lab"><font color="Green" title="E29">A10</font></span></a>, <a class="ref" href="idea_1.html#D7" target="_self" onmouseover="rs('idea_1/D7')" onmouseout="rh()">Def7</a></span>
<br/>.= 
 <a href="idea_1.html#K8" title="IDEA_1:func.8">ChangeVal_2</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">a</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p2">(<span class="default"><span class="p3">(<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>)</span>,<font color="Maroon" title="c1">n</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:28_1_1_2_2"><span class="lab"><font color="Green" title="E32">A13</font></span></a>, <a class="txt" href="#E2:28_1_1_2_2_1_1_1"><span class="lab"><font color="Green" title="E34">A15</font></span></a>, <a class="ref" href="idea_1.html#D7" target="_self" onmouseover="rs('idea_1/D7')" onmouseout="rh()">Def7</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:28_1_1"><span class="lab"><font color="Green" title="E27">A5</font></span></a>, <a class="txt" href="#E5:28_1_1_2_2"><span class="lab"><font color="Green" title="E30">A11</font></span></a>, <a class="ref" href="idea_1.html#D8" target="_self" onmouseover="rs('idea_1/D8')" onmouseout="rh()">Def8</a></span>
;<br/><span class="kw">hence </span><a NAME="E7:28_1_1_2_2_1_1_1"/>
(  <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="c7">k</font>,<font color="Maroon" title="c1">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c7">k</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">by</span> <span class="lab"><a class="txt" href="#E5:28_1_1_2_2_1_1_1"><span class="lab"><font color="Green" title="E35">A16</font></span></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><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="28_1_1_2_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:28_1_1_2_2_1_1_2"/><span class="lab"><font color="Green" title="E33">A17</font></span>: 
<font color="Maroon" title="c6">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</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"> (  <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="c7">k</font>,<font color="Maroon" title="c1">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c7">k</font> <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> )</span><br/><div class="add"><a NAME="E2:28_1_1_2_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A18</font></span>: 
<font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="idea_1.html#D8" target="_self" onmouseover="rs('idea_1/D8')" onmouseout="rh()">Def8</a></span>;<br/><a NAME="E3:28_1_1_2_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A19</font></span>: 
<font color="Maroon" title="c7">k</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">by</span> <span class="lab"><a class="txt" href="#E1:28_1_1"><span class="lab"><font color="Green" title="E25">A3</font></span></a></span>;<br/> <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="c7">k</font>,<font color="Maroon" title="c1">n</font>) = 
 <a href="idea_1.html#K8" title="IDEA_1:func.8">ChangeVal_2</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p2">(<span class="default"><span class="p3">(<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>)</span>,<font color="Maroon" title="c1">n</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28_1_1_2_2"><span class="lab"><font color="Green" title="E29">A10</font></span></a>, <a class="ref" href="idea_1.html#D7" target="_self" onmouseover="rs('idea_1/D7')" onmouseout="rh()">Def7</a></span>
<br/>.= 
 <a href="idea_1.html#K8" title="IDEA_1:func.8">ChangeVal_2</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c6">a</font></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p2">(<span class="default"><span class="p3">(<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>)</span>,<font color="Maroon" title="c1">n</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28_1_1_2_2_1_1_2"><span class="lab"><font color="Green" title="E33">A17</font></span></a>, <a class="txt" href="#E2:28_1_1_2_2_1_1_2"><span class="lab"><font color="Green" title="E34">A18</font></span></a>, <a class="ref" href="idea_1.html#D7" target="_self" onmouseover="rs('idea_1/D7')" onmouseout="rh()">Def7</a></span>
<br/>.= 
1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:28_1_1"><span class="lab"><font color="Green" title="E27">A5</font></span></a>, <a class="txt" href="#E5:28_1_1_2_2"><span class="lab"><font color="Green" title="E30">A11</font></span></a>, <a class="ref" href="idea_1.html#D8" target="_self" onmouseover="rs('idea_1/D8')" onmouseout="rh()">Def8</a></span>
;<br/><span class="kw">hence </span><a NAME="E5:28_1_1_2_2_1_1_2"/>
(  <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="c7">k</font>,<font color="Maroon" title="c1">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c7">k</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">by</span> <span class="lab"><a class="txt" href="#E3:28_1_1_2_2_1_1_2"><span class="lab"><font color="Green" title="E35">A19</font></span></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><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="E9:28_1_1_2_2"/>
(  <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="c7">k</font>,<font color="Maroon" title="c1">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c7">k</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"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
