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<span class="kw">let </span><font color="Maroon" title="c1">n</font>, <font color="Maroon" title="c2">i</font>, <font color="Maroon" title="c3">j</font>, <font color="Maroon" title="c4">k</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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">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 is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  &amp; <font color="Maroon" title="c2">i</font> <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c3">j</font> <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c4">k</font> <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> implies  <a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> (<span class="p1">(<span class="default"><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">j</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c1">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> (<font color="Maroon" title="c2">i</font>,<span class="p1">(<span class="default"><a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> (<font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c1">n</font>) )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:34"/><span class="lab"><font color="Green" title="E27">A1</font></span>: 
<span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">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 is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a> 
 <span class="kw">and </span><br/><a NAME="E2:34"/><span class="lab"><font color="Green" title="E28">A2</font></span>: 
<font color="Maroon" title="c2">i</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="E3:34"/><span class="lab"><font color="Green" title="E29">A3</font></span>: 
<font color="Maroon" title="c3">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">and </span><br/><a NAME="E4:34"/><span class="lab"><font color="Green" title="E30">A4</font></span>: 
<font color="Maroon" title="c4">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" 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> (<span class="p1">(<span class="default"><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">j</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c1">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> (<font color="Maroon" title="c2">i</font>,<span class="p1">(<span class="default"><a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> (<font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c1">n</font>)</span><br/>

<span class="kw">set </span><font color="Maroon" title="c5">J</font> =  <a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c1">n</font>);<br/>
<a NAME="E5:34"/><span class="lab"><font color="Green" title="E31">A5</font></span>: 
 <a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c1">n</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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="ref" href="idea_1.html#T17" target="_self" title="IDEA_1:th.17">Th18</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">I</font> =  <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="E6:34"/>
 <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>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:34"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="ref" href="idea_1.html#T17" target="_self" title="IDEA_1:th.17">Th18</a></span>;<br/>
<a NAME="E7:34"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A6</font></span>: 
 <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>) <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#K4" title="SERIES_1:func.4">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 class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E8:34"/>
 <a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c1">n</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="ref" href="idea_1.html#T17" target="_self" title="IDEA_1:th.17">Th18</a></span>;<br/>
<a NAME="E9:34"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A7</font></span>: 
 <a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c1">n</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#K4" title="SERIES_1:func.4">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 class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">K</font> =  <a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c1">n</font>);<br/>
<a NAME="E10:34"/><span class="lab"><font color="Green" title="E34">A8</font></span>: 
 <a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c1">n</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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:34"><span class="lab"><font color="Green" title="E30">A4</font></span></a>, <a class="ref" href="idea_1.html#T17" target="_self" title="IDEA_1:th.17">Th18</a></span>;<br/>
<a NAME="E11:34"/>
 <a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c1">n</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <font color="Maroon" title="c1">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:34"><span class="lab"><font color="Green" title="E30">A4</font></span></a>, <a class="ref" href="idea_1.html#T17" target="_self" title="IDEA_1:th.17">Th18</a></span>;<br/>
<a NAME="E12:34"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A9</font></span>: 
 <a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c1">n</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#K4" title="SERIES_1:func.4">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 class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E13:34"/><span class="lab"><font color="Green" title="E36">A10</font></span>: 
 <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>) <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:34"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="ref" href="idea_1.html#T17" target="_self" title="IDEA_1:th.17">Th18</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1"><span class="kw">now </span></a><div class="add"><span class="kw">set </span><font color="Maroon" title="c8">CV</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><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>)</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c3">j</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="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">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>;<br/><a NAME="E1:34_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><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>)</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c3">j</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="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">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="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">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 class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T1" title="NAT_D:th.1">NAT_D:1</a></span>;<br/><a NAME="E2:34_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A11</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><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>)</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c3">j</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="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">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="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E3:34_1"/><span class="lab"><font color="Green" title="E38">A12</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><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>)</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c3">j</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="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="txt" href="#E7:34"><span class="lab"><font color="Green" title="E32">A6</font></span></a>, <a class="txt" href="#E13:34"><span class="lab"><font color="Green" title="E36">A10</font></span></a>, <a class="txt" href="#E9:34"><span class="lab"><font color="Green" title="E33">A7</font></span></a>, <a class="txt" href="#E5:34"><span class="lab"><font color="Green" title="E31">A5</font></span></a>, <a class="ref" href="idea_1.html#T5" target="_self" title="IDEA_1:th.5">Th5</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:34_1_1_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><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>)</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c3">j</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="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">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> 2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <font color="Maroon" title="c1">n</font> or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><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>)</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c3">j</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="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">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="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <font color="Maroon" title="c1">n</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:34_1"><span class="lab"><font color="Green" title="E37">A11</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:34_1_1_1_1"/><span class="lab"><font color="Green" title="E39">A13</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><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>)</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c3">j</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="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">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> 2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <font color="Maroon" title="c1">n</font>
 ; <a class="txt" 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> (<span class="p1">(<span class="default"><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">j</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c1">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> (<font color="Maroon" title="c2">i</font>,<span class="p1">(<span class="default"><a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> (<font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c1">n</font>)</span><br/><div class="add"><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A14</font></span>:  <a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> (<span class="p1">(<span class="default"><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">j</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">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"><span class="p3">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</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="c4">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#K4" title="SERIES_1:func.4">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></span>)</span>,<font color="Maroon" title="c1">n</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="idea_1.html#D8" target="_self" title="IDEA_1:def.8">Def8</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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><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>)</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p5">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c3">j</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="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">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></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</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="c4">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#K4" title="SERIES_1:func.4">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></span>)</span>,<font color="Maroon" title="c1">n</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34_1_1_1_1"><span class="lab"><font color="Green" title="E39">A13</font></span></a>, <a class="ref" href="idea_1.html#D7" target="_self" title="IDEA_1:def.7">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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><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>)</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p4">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</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="c4">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#K4" title="SERIES_1:func.4">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></span>)</span>,<font color="Maroon" title="c1">n</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="euler_2.html#T8" title="EULER_2:th.8">EULER_2:8</a></span>
;<br/><span class="kw">set </span><font color="Maroon" title="c9">CV2</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c4">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="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">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>;<br/><a NAME="E3:34_1_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c4">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="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">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="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">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 class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T1" title="NAT_D:th.1">NAT_D:1</a></span>;<br/><a NAME="E4:34_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A15</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c4">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="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">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="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E5:34_1_1_1_1"/><span class="lab"><font color="Green" title="E42">A16</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c4">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="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="txt" href="#E9:34"><span class="lab"><font color="Green" title="E33">A7</font></span></a>, <a class="txt" href="#E5:34"><span class="lab"><font color="Green" title="E31">A5</font></span></a>, <a class="txt" href="#E12:34"><span class="lab"><font color="Green" title="E35">A9</font></span></a>, <a class="txt" href="#E10:34"><span class="lab"><font color="Green" title="E34">A8</font></span></a>, <a class="ref" href="idea_1.html#T5" target="_self" title="IDEA_1:th.5">Th5</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_1_1_2"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_1_1_1_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:34_1_1_1_1_2_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c4">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="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">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> 2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <font color="Maroon" title="c1">n</font> or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c4">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="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">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="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <font color="Maroon" title="c1">n</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:34_1_1_1_1"><span class="lab"><font color="Green" title="E41">A15</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_1_1_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:34_1_1_1_1_2_1_1"/><span class="lab"><font color="Green" title="E43">A17</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c4">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="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">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> 2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <font color="Maroon" title="c1">n</font>
 ; <a class="txt" 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> (<span class="p1">(<span class="default"><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">j</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c1">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> (<font color="Maroon" title="c2">i</font>,<span class="p1">(<span class="default"><a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> (<font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c1">n</font>)</span><br/><div class="add"><span class="kw">then </span> <a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> (<font color="Maroon" title="c2">i</font>,<span class="p1">(<span class="default"><a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> (<font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<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"><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>)</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</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#K6" title="NUMBERS:func.6">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> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">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></span>)</span>,<font color="Maroon" title="c1">n</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="idea_1.html#D8" target="_self" title="IDEA_1:def.8">Def8</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"><span class="p3">(<span class="default"><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>)</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p5">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c4">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="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">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></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#K4" title="SERIES_1:func.4">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></span>)</span>,<font color="Maroon" title="c1">n</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34_1_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E43">A17</font></span></a>, <a class="ref" href="idea_1.html#D7" target="_self" title="IDEA_1:def.7">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"><span class="p3">(<span class="default"><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>)</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p4">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></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#K4" title="SERIES_1:func.4">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></span>)</span>,<font color="Maroon" title="c1">n</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="euler_2.html#T8" title="EULER_2:th.8">EULER_2:8</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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><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>)</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p4">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</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="c4">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#K4" title="SERIES_1:func.4">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></span>)</span>,<font color="Maroon" title="c1">n</font>)

;<br/><span class="kw">hence </span><a NAME="E3:34_1_1_1_1_2_1_1"/>
 <a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> (<span class="p1">(<span class="default"><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">j</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c1">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> (<font color="Maroon" title="c2">i</font>,<span class="p1">(<span class="default"><a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> (<font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c1">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:34_1_1_1_1"><span class="lab"><font color="Green" title="E40">A14</font></span></a></span>; <a class="txt" 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="34_1_1_1_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:34_1_1_1_1_2_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c4">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="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">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="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <font color="Maroon" title="c1">n</font>
 ; <a class="txt" 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> (<span class="p1">(<span class="default"><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">j</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c1">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> (<font color="Maroon" title="c2">i</font>,<span class="p1">(<span class="default"><a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> (<font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c1">n</font>)</span><br/><div class="add"><a NAME="E2:34_1_1_1_1_2_1_2"/><span class="kw">then </span>
 <a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> (<font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c1">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c4">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="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="idea_1.html#D8" target="_self" title="IDEA_1:def.8">Def8</a></span>;<br/><span class="kw">then </span> <a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> (<font color="Maroon" title="c2">i</font>,<span class="p1">(<span class="default"><a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> (<font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<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"><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>)</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p5">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c4">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="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">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></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#K4" title="SERIES_1:func.4">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></span>)</span>,<font color="Maroon" title="c1">n</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:34_1_1_1_1"><span class="lab"><font color="Green" title="E42">A16</font></span></a>, <a class="ref" href="idea_1.html#D7" target="_self" title="IDEA_1:def.7">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"><span class="p3">(<span class="default"><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>)</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p4">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></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#K4" title="SERIES_1:func.4">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></span>)</span>,<font color="Maroon" title="c1">n</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="euler_2.html#T8" title="EULER_2:th.8">EULER_2:8</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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><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>)</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p4">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</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="c4">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#K4" title="SERIES_1:func.4">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></span>)</span>,<font color="Maroon" title="c1">n</font>)

;<br/><span class="kw">hence </span><a NAME="E4:34_1_1_1_1_2_1_2"/>
 <a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> (<span class="p1">(<span class="default"><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">j</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c1">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> (<font color="Maroon" title="c2">i</font>,<span class="p1">(<span class="default"><a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> (<font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c1">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:34_1_1_1_1"><span class="lab"><font color="Green" title="E40">A14</font></span></a></span>; <a class="txt" 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="E7:34_1_1_1_1"/>
 <a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> (<span class="p1">(<span class="default"><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">j</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c1">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> (<font color="Maroon" title="c2">i</font>,<span class="p1">(<span class="default"><a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> (<font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c1">n</font>)
 ; <a class="txt" 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="34_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:34_1_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><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>)</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c3">j</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="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">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="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <font color="Maroon" title="c1">n</font>
 ; <a class="txt" 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> (<span class="p1">(<span class="default"><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">j</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c1">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> (<font color="Maroon" title="c2">i</font>,<span class="p1">(<span class="default"><a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> (<font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c1">n</font>)</span><br/><div class="add"><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A18</font></span>:  <a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> (<span class="p1">(<span class="default"><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">j</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">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"><span class="p3">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><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>)</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p0">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c3">j</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="p5">(<span class="default"><span class="p0">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">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></span>)</span>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</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="c4">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#K4" title="SERIES_1:func.4">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></span>)</span>,<font color="Maroon" title="c1">n</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="idea_1.html#D8" target="_self" title="IDEA_1:def.8">Def8</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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><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>)</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p5">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c3">j</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="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">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></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</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="c4">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#K4" title="SERIES_1:func.4">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></span>)</span>,<font color="Maroon" title="c1">n</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:34_1"><span class="lab"><font color="Green" title="E38">A12</font></span></a>, <a class="ref" href="idea_1.html#D7" target="_self" title="IDEA_1:def.7">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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><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>)</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p4">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</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="c4">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#K4" title="SERIES_1:func.4">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></span>)</span>,<font color="Maroon" title="c1">n</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="euler_2.html#T8" title="EULER_2:th.8">EULER_2:8</a></span>
;<br/><span class="kw">set </span><font color="Maroon" title="c9">CV2</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c4">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="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">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>;<br/><a NAME="E3:34_1_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c4">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="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">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="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">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 class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T1" title="NAT_D:th.1">NAT_D:1</a></span>;<br/><a NAME="E4:34_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A19</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c4">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="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">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="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E5:34_1_1_1_2"/><span class="lab"><font color="Green" title="E41">A20</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c4">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="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="txt" href="#E9:34"><span class="lab"><font color="Green" title="E33">A7</font></span></a>, <a class="txt" href="#E5:34"><span class="lab"><font color="Green" title="E31">A5</font></span></a>, <a class="txt" href="#E12:34"><span class="lab"><font color="Green" title="E35">A9</font></span></a>, <a class="txt" href="#E10:34"><span class="lab"><font color="Green" title="E34">A8</font></span></a>, <a class="ref" href="idea_1.html#T5" target="_self" title="IDEA_1:th.5">Th5</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_1_2_2"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_1_1_2_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:34_1_1_1_2_2_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c4">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="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">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> 2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <font color="Maroon" title="c1">n</font> or <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c4">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="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">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="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <font color="Maroon" title="c1">n</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:34_1_1_1_2"><span class="lab"><font color="Green" title="E40">A19</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_1_1_2_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:34_1_1_1_2_2_1_1"/><span class="lab"><font color="Green" title="E42">A21</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c4">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="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">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> 2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <font color="Maroon" title="c1">n</font>
 ; <a class="txt" 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> (<span class="p1">(<span class="default"><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">j</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c1">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> (<font color="Maroon" title="c2">i</font>,<span class="p1">(<span class="default"><a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> (<font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c1">n</font>)</span><br/><div class="add"><span class="kw">then </span> <a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> (<font color="Maroon" title="c2">i</font>,<span class="p1">(<span class="default"><a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> (<font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<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"><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>)</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</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#K6" title="NUMBERS:func.6">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> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">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></span>)</span>,<font color="Maroon" title="c1">n</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="idea_1.html#D8" target="_self" title="IDEA_1:def.8">Def8</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"><span class="p3">(<span class="default"><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>)</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p5">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c4">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="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">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></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#K4" title="SERIES_1:func.4">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></span>)</span>,<font color="Maroon" title="c1">n</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:34_1_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E42">A21</font></span></a>, <a class="ref" href="idea_1.html#D7" target="_self" title="IDEA_1:def.7">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"><span class="p3">(<span class="default"><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>)</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p4">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></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#K4" title="SERIES_1:func.4">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></span>)</span>,<font color="Maroon" title="c1">n</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="euler_2.html#T8" title="EULER_2:th.8">EULER_2:8</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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><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>)</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p4">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</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="c4">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#K4" title="SERIES_1:func.4">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></span>)</span>,<font color="Maroon" title="c1">n</font>)

;<br/><span class="kw">hence </span><a NAME="E3:34_1_1_1_2_2_1_1"/>
 <a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> (<span class="p1">(<span class="default"><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">j</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c1">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> (<font color="Maroon" title="c2">i</font>,<span class="p1">(<span class="default"><a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> (<font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c1">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:34_1_1_1_2"><span class="lab"><font color="Green" title="E39">A18</font></span></a></span>; <a class="txt" 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="34_1_1_1_2_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:34_1_1_1_2_2_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c4">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="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">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="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <font color="Maroon" title="c1">n</font>
 ; <a class="txt" 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> (<span class="p1">(<span class="default"><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">j</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c1">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> (<font color="Maroon" title="c2">i</font>,<span class="p1">(<span class="default"><a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> (<font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c1">n</font>)</span><br/><div class="add"><a NAME="E2:34_1_1_1_2_2_1_2"/><span class="kw">then </span>
 <a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> (<font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c1">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c4">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="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="idea_1.html#D8" target="_self" title="IDEA_1:def.8">Def8</a></span>;<br/><span class="kw">then </span> <a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> (<font color="Maroon" title="c2">i</font>,<span class="p1">(<span class="default"><a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> (<font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<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"><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>)</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p5">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c4">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="p4">(<span class="default"><span class="p5">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">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></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#K4" title="SERIES_1:func.4">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></span>)</span>,<font color="Maroon" title="c1">n</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:34_1_1_1_2"><span class="lab"><font color="Green" title="E41">A20</font></span></a>, <a class="ref" href="idea_1.html#D7" target="_self" title="IDEA_1:def.7">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"><span class="p3">(<span class="default"><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>)</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c1">n</font>)</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p4">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></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#K4" title="SERIES_1:func.4">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></span>)</span>,<font color="Maroon" title="c1">n</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="euler_2.html#T8" title="EULER_2:th.8">EULER_2:8</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"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><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>)</span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p4">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c1">n</font>)</span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</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="c4">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#K4" title="SERIES_1:func.4">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></span>)</span>,<font color="Maroon" title="c1">n</font>)

;<br/><span class="kw">hence </span><a NAME="E4:34_1_1_1_2_2_1_2"/>
 <a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> (<span class="p1">(<span class="default"><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">j</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c1">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> (<font color="Maroon" title="c2">i</font>,<span class="p1">(<span class="default"><a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> (<font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c1">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:34_1_1_1_2"><span class="lab"><font color="Green" title="E39">A18</font></span></a></span>; <a class="txt" 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="E7:34_1_1_1_2"/>
 <a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> (<span class="p1">(<span class="default"><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">j</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c1">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> (<font color="Maroon" title="c2">i</font>,<span class="p1">(<span class="default"><a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> (<font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c1">n</font>)
 ; <a class="txt" 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="E5:34_1"/>
 <a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> (<span class="p1">(<span class="default"><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">j</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c1">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> (<font color="Maroon" title="c2">i</font>,<span class="p1">(<span class="default"><a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> (<font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c1">n</font>)
 ; <a class="txt" 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>
<span class="kw">hence </span><a NAME="E15:34"/>
 <a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> (<span class="p1">(<span class="default"><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">j</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c1">n</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> (<font color="Maroon" title="c2">i</font>,<span class="p1">(<span class="default"><a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> (<font color="Maroon" title="c3">j</font>,<font color="Maroon" title="c4">k</font>,<font color="Maroon" title="c1">n</font>)</span>)</span>,<font color="Maroon" title="c1">n</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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