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<span class="kw">let </span><font color="Maroon" title="c1">i</font>, <font color="Maroon" title="c2">j</font>, <font color="Maroon" title="c3">k</font>, <font color="Maroon" title="c4">n</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default">2 <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  &amp; <font color="Maroon" title="c1">i</font> <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c4">n</font> &amp; <font color="Maroon" title="c2">j</font> <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c4">n</font> &amp; <font color="Maroon" title="c3">k</font> <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c4">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="c1">i</font>,<font color="Maroon" title="c2">j</font>,<font color="Maroon" title="c4">n</font>)</span>)</span>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c4">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="c1">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="c2">j</font>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c4">n</font>)</span>)</span>,<font color="Maroon" title="c4">n</font>) )</span><br/>







<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:30"/><span class="lab"><font color="Green" title="E26">A1</font></span>: 
<span class="p1">(<span class="default">2 <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 is  <a href="int_2.html#V1" title="INT_2:attr.1">prime</a> 
 <span class="kw">and </span><br/><a NAME="E2:30"/><span class="lab"><font color="Green" title="E27">A2</font></span>: 
<font color="Maroon" title="c1">i</font> <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c4">n</font>
 <span class="kw">and </span><br/><a NAME="E3:30"/><span class="lab"><font color="Green" title="E28">A3</font></span>: 
<font color="Maroon" title="c2">j</font> <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c4">n</font>
 <span class="kw">and </span><br/><a NAME="E4:30"/><span class="lab"><font color="Green" title="E29">A4</font></span>: 
<font color="Maroon" title="c3">k</font> <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c4">n</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> (<span class="p1">(<span class="default"><a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font>,<font color="Maroon" title="c4">n</font>)</span>)</span>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c4">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="c1">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="c2">j</font>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c4">n</font>)</span>)</span>,<font color="Maroon" title="c4">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="c2">j</font>,<font color="Maroon" title="c4">n</font>);<br/>
<a NAME="E5:30"/><span class="lab"><font color="Green" title="E30">A5</font></span>: 
 <a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c2">j</font>,<font color="Maroon" title="c4">n</font>) <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:30"><span class="lab"><font color="Green" title="E28">A3</font></span></a>, <a class="ref" href="idea_1.html#T17" target="_self" onmouseover="rs('idea_1/T17')" onmouseout="rh()">Th17</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="c1">i</font>,<font color="Maroon" title="c4">n</font>);<br/>
<a NAME="E6:30"/>
 <a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c4">n</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c4">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="ref" href="idea_1.html#T17" target="_self" onmouseover="rs('idea_1/T17')" onmouseout="rh()">Th17</a></span>;<br/>
<a NAME="E7:30"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A6</font></span>: 
 <a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c4">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#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/>
<a NAME="E8:30"/>
 <a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c2">j</font>,<font color="Maroon" title="c4">n</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c4">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:30"><span class="lab"><font color="Green" title="E28">A3</font></span></a>, <a class="ref" href="idea_1.html#T17" target="_self" onmouseover="rs('idea_1/T17')" onmouseout="rh()">Th17</a></span>;<br/>
<a NAME="E9:30"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A7</font></span>: 
 <a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c2">j</font>,<font color="Maroon" title="c4">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#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/>
<span class="kw">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="c3">k</font>,<font color="Maroon" title="c4">n</font>);<br/>
<a NAME="E10:30"/><span class="lab"><font color="Green" title="E33">A8</font></span>: 
 <a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c4">n</font>) <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:30"><span class="lab"><font color="Green" title="E29">A4</font></span></a>, <a class="ref" href="idea_1.html#T17" target="_self" onmouseover="rs('idea_1/T17')" onmouseout="rh()">Th17</a></span>;<br/>
<a NAME="E11:30"/>
 <a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c4">n</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c4">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:30"><span class="lab"><font color="Green" title="E29">A4</font></span></a>, <a class="ref" href="idea_1.html#T17" target="_self" onmouseover="rs('idea_1/T17')" onmouseout="rh()">Th17</a></span>;<br/>
<a NAME="E12:30"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A9</font></span>: 
 <a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c4">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#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/>
<a NAME="E13:30"/><span class="lab"><font color="Green" title="E35">A10</font></span>: 
 <a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c4">n</font>) <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="ref" href="idea_1.html#T17" target="_self" onmouseover="rs('idea_1/T17')" onmouseout="rh()">Th17</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="30_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> (<span class="p1">(<span class="default"><a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font>,<font color="Maroon" title="c4">n</font>)</span>)</span>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c4">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="c1">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="c2">j</font>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c4">n</font>)</span>)</span>,<font color="Maroon" title="c4">n</font>)</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="c1">i</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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="c2">j</font>,<font color="Maroon" title="c4">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#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>;<br/><a NAME="E1:30_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="c1">i</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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="c2">j</font>,<font color="Maroon" title="c4">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#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T1" onmouseover="rs('nat_d/T1')" onmouseout="rh()">NAT_D:1</a></span>;<br/><a NAME="E2:30_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">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="c1">i</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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="c2">j</font>,<font color="Maroon" title="c4">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#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c4">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E3:30_1"/><span class="lab"><font color="Green" title="E37">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="c1">i</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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="c2">j</font>,<font color="Maroon" title="c4">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#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="txt" href="#E7:30"><span class="lab"><font color="Green" title="E31">A6</font></span></a>, <a class="txt" href="#E13:30"><span class="lab"><font color="Green" title="E35">A10</font></span></a>, <a class="txt" href="#E9:30"><span class="lab"><font color="Green" title="E32">A7</font></span></a>, <a class="txt" href="#E5:30"><span class="lab"><font color="Green" title="E30">A5</font></span></a>, <a class="ref" href="idea_1.html#T5" target="_self" onmouseover="rs('idea_1/T5')" onmouseout="rh()">Th5</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="30_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> (<span class="p1">(<span class="default"><a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font>,<font color="Maroon" title="c4">n</font>)</span>)</span>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c4">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="c1">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="c2">j</font>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c4">n</font>)</span>)</span>,<font color="Maroon" title="c4">n</font>)</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:30_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="c1">i</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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="c2">j</font>,<font color="Maroon" title="c4">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#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c4">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="c1">i</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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="c2">j</font>,<font color="Maroon" title="c4">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#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c4">n</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30_1"><span class="lab"><font color="Green" title="E36">A11</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:30_1_1_1_1"/><span class="lab"><font color="Green" title="E38">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="c1">i</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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="c2">j</font>,<font color="Maroon" title="c4">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#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c4">n</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> (<span class="p1">(<span class="default"><a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font>,<font color="Maroon" title="c4">n</font>)</span>)</span>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c4">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="c1">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="c2">j</font>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c4">n</font>)</span>)</span>,<font color="Maroon" title="c4">n</font>)</span><br/><div class="add"><span class="kw">then </span><span class="lab"><font color="Green" title="E39">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="c1">i</font>,<font color="Maroon" title="c2">j</font>,<font color="Maroon" title="c4">n</font>)</span>)</span>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c4">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#K5" title="NUMBERS:func.5">0</a>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c4">n</font>)</span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c4">n</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="idea_1.html#D8" target="_self" onmouseover="rs('idea_1/D8')" onmouseout="rh()">Def8</a></span>
<br/>.= 
 <a 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="c1">i</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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="c2">j</font>,<font color="Maroon" title="c4">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#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c4">n</font>)</span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c4">n</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1_1_1_1"><span class="lab"><font color="Green" title="E38">A13</font></span></a>, <a class="ref" href="idea_1.html#D7" target="_self" onmouseover="rs('idea_1/D7')" onmouseout="rh()">Def7</a></span>
<br/>.= 
 <a href="idea_1.html#K8" title="IDEA_1:func.8">ChangeVal_2</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><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="c1">i</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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="c2">j</font>,<font color="Maroon" title="c4">n</font>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c4">n</font>)</span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c4">n</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="euler_2.html#T8" onmouseover="rs('euler_2/T8')" onmouseout="rh()">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="c2">j</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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">k</font>,<font color="Maroon" title="c4">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#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>;<br/><a NAME="E3:30_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="c2">j</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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">k</font>,<font color="Maroon" title="c4">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#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T1" onmouseover="rs('nat_d/T1')" onmouseout="rh()">NAT_D:1</a></span>;<br/><a NAME="E4:30_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">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="c2">j</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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">k</font>,<font color="Maroon" title="c4">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#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c4">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E5:30_1_1_1_1"/><span class="lab"><font color="Green" title="E41">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="c2">j</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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">k</font>,<font color="Maroon" title="c4">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#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="txt" href="#E9:30"><span class="lab"><font color="Green" title="E32">A7</font></span></a>, <a class="txt" href="#E5:30"><span class="lab"><font color="Green" title="E30">A5</font></span></a>, <a class="txt" href="#E12:30"><span class="lab"><font color="Green" title="E34">A9</font></span></a>, <a class="txt" href="#E10:30"><span class="lab"><font color="Green" title="E33">A8</font></span></a>, <a class="ref" href="idea_1.html#T5" target="_self" onmouseover="rs('idea_1/T5')" onmouseout="rh()">Th5</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="30_1_1_1_1_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> (<span class="p1">(<span class="default"><a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font>,<font color="Maroon" title="c4">n</font>)</span>)</span>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c4">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="c1">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="c2">j</font>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c4">n</font>)</span>)</span>,<font color="Maroon" title="c4">n</font>)</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_1_1_2_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:30_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="c2">j</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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">k</font>,<font color="Maroon" title="c4">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#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c4">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">j</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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">k</font>,<font color="Maroon" title="c4">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#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c4">n</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:30_1_1_1_1"><span class="lab"><font color="Green" title="E40">A15</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_1_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:30_1_1_1_1_2_1_1"/><span class="lab"><font color="Green" title="E42">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="c2">j</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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">k</font>,<font color="Maroon" title="c4">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#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c4">n</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> (<span class="p1">(<span class="default"><a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font>,<font color="Maroon" title="c4">n</font>)</span>)</span>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c4">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="c1">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="c2">j</font>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c4">n</font>)</span>)</span>,<font color="Maroon" title="c4">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="c1">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="c2">j</font>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c4">n</font>)</span>)</span>,<font color="Maroon" title="c4">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="c1">i</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Maroon" title="c4">n</font>)</span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c4">n</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="idea_1.html#D8" target="_self" onmouseover="rs('idea_1/D8')" onmouseout="rh()">Def8</a></span>
<br/>.= 
 <a 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="c1">i</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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="c2">j</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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">k</font>,<font color="Maroon" title="c4">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#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c4">n</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E42">A17</font></span></a>, <a class="ref" href="idea_1.html#D7" target="_self" onmouseover="rs('idea_1/D7')" onmouseout="rh()">Def7</a></span>
<br/>.= 
 <a href="idea_1.html#K8" title="IDEA_1:func.8">ChangeVal_2</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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="c2">j</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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">k</font>,<font color="Maroon" title="c4">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#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c4">n</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="euler_2.html#T8" onmouseover="rs('euler_2/T8')" onmouseout="rh()">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="c1">i</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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="c2">j</font>,<font color="Maroon" title="c4">n</font>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c4">n</font>)</span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c4">n</font>)

;<br/><span class="kw">hence </span><a NAME="E3:30_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="c1">i</font>,<font color="Maroon" title="c2">j</font>,<font color="Maroon" title="c4">n</font>)</span>)</span>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c4">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="c1">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="c2">j</font>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c4">n</font>)</span>)</span>,<font color="Maroon" title="c4">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30_1_1_1_1"><span class="lab"><font color="Green" title="E39">A14</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_1_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:30_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="c2">j</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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">k</font>,<font color="Maroon" title="c4">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#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c4">n</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> (<span class="p1">(<span class="default"><a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font>,<font color="Maroon" title="c4">n</font>)</span>)</span>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c4">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="c1">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="c2">j</font>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c4">n</font>)</span>)</span>,<font color="Maroon" title="c4">n</font>)</span><br/><div class="add"><a NAME="E2:30_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="c2">j</font>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c4">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="c2">j</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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">k</font>,<font color="Maroon" title="c4">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#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="idea_1.html#D8" target="_self" onmouseover="rs('idea_1/D8')" onmouseout="rh()">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="c1">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="c2">j</font>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c4">n</font>)</span>)</span>,<font color="Maroon" title="c4">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="c1">i</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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="c2">j</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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">k</font>,<font color="Maroon" title="c4">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#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c4">n</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:30_1_1_1_1"><span class="lab"><font color="Green" title="E41">A16</font></span></a>, <a class="ref" href="idea_1.html#D7" target="_self" onmouseover="rs('idea_1/D7')" onmouseout="rh()">Def7</a></span>
<br/>.= 
 <a href="idea_1.html#K8" title="IDEA_1:func.8">ChangeVal_2</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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="c2">j</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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">k</font>,<font color="Maroon" title="c4">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#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c4">n</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="euler_2.html#T8" onmouseover="rs('euler_2/T8')" onmouseout="rh()">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="c1">i</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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="c2">j</font>,<font color="Maroon" title="c4">n</font>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c4">n</font>)</span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c4">n</font>)

;<br/><span class="kw">hence </span><a NAME="E4:30_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="c1">i</font>,<font color="Maroon" title="c2">j</font>,<font color="Maroon" title="c4">n</font>)</span>)</span>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c4">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="c1">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="c2">j</font>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c4">n</font>)</span>)</span>,<font color="Maroon" title="c4">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30_1_1_1_1"><span class="lab"><font color="Green" title="E39">A14</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E7:30_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="c1">i</font>,<font color="Maroon" title="c2">j</font>,<font color="Maroon" title="c4">n</font>)</span>)</span>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c4">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="c1">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="c2">j</font>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c4">n</font>)</span>)</span>,<font color="Maroon" title="c4">n</font>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:30_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="c1">i</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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="c2">j</font>,<font color="Maroon" title="c4">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#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c4">n</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> (<span class="p1">(<span class="default"><a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font>,<font color="Maroon" title="c4">n</font>)</span>)</span>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c4">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="c1">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="c2">j</font>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c4">n</font>)</span>)</span>,<font color="Maroon" title="c4">n</font>)</span><br/><div class="add"><span class="kw">then </span><span class="lab"><font color="Green" title="E38">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="c1">i</font>,<font color="Maroon" title="c2">j</font>,<font color="Maroon" title="c4">n</font>)</span>)</span>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c4">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="c1">i</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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="c2">j</font>,<font color="Maroon" title="c4">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#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c4">n</font>)</span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c4">n</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="idea_1.html#D8" target="_self" onmouseover="rs('idea_1/D8')" onmouseout="rh()">Def8</a></span>
<br/>.= 
 <a 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="c1">i</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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="c2">j</font>,<font color="Maroon" title="c4">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#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c4">n</font>)</span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c4">n</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:30_1"><span class="lab"><font color="Green" title="E37">A12</font></span></a>, <a class="ref" href="idea_1.html#D7" target="_self" onmouseover="rs('idea_1/D7')" onmouseout="rh()">Def7</a></span>
<br/>.= 
 <a href="idea_1.html#K8" title="IDEA_1:func.8">ChangeVal_2</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><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="c1">i</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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="c2">j</font>,<font color="Maroon" title="c4">n</font>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c4">n</font>)</span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c4">n</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="euler_2.html#T8" onmouseover="rs('euler_2/T8')" onmouseout="rh()">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="c2">j</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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">k</font>,<font color="Maroon" title="c4">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#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>;<br/><a NAME="E3:30_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">j</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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">k</font>,<font color="Maroon" title="c4">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#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T1" onmouseover="rs('nat_d/T1')" onmouseout="rh()">NAT_D:1</a></span>;<br/><a NAME="E4:30_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">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="c2">j</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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">k</font>,<font color="Maroon" title="c4">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#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c4">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" onmouseover="rs('nat_1/T13')" onmouseout="rh()">NAT_1:13</a></span>;<br/><a NAME="E5:30_1_1_1_2"/><span class="lab"><font color="Green" title="E40">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="c2">j</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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">k</font>,<font color="Maroon" title="c4">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#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="txt" href="#E9:30"><span class="lab"><font color="Green" title="E32">A7</font></span></a>, <a class="txt" href="#E5:30"><span class="lab"><font color="Green" title="E30">A5</font></span></a>, <a class="txt" href="#E12:30"><span class="lab"><font color="Green" title="E34">A9</font></span></a>, <a class="txt" href="#E10:30"><span class="lab"><font color="Green" title="E33">A8</font></span></a>, <a class="ref" href="idea_1.html#T5" target="_self" onmouseover="rs('idea_1/T5')" onmouseout="rh()">Th5</a></span>;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="30_1_1_1_2_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> (<span class="p1">(<span class="default"><a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font>,<font color="Maroon" title="c4">n</font>)</span>)</span>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c4">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="c1">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="c2">j</font>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c4">n</font>)</span>)</span>,<font color="Maroon" title="c4">n</font>)</span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_1_2_2_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:30_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="c2">j</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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">k</font>,<font color="Maroon" title="c4">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#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c4">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">j</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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">k</font>,<font color="Maroon" title="c4">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#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c4">n</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:30_1_1_1_2"><span class="lab"><font color="Green" title="E39">A19</font></span></a>, <a class="ref" href="xxreal_0.html#T1" onmouseover="rs('xxreal_0/T1')" onmouseout="rh()">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_1_2_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:30_1_1_1_2_2_1_1"/><span class="lab"><font color="Green" title="E41">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="c2">j</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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">k</font>,<font color="Maroon" title="c4">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#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c4">n</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> (<span class="p1">(<span class="default"><a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font>,<font color="Maroon" title="c4">n</font>)</span>)</span>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c4">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="c1">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="c2">j</font>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c4">n</font>)</span>)</span>,<font color="Maroon" title="c4">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="c1">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="c2">j</font>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c4">n</font>)</span>)</span>,<font color="Maroon" title="c4">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="c1">i</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,<font color="Maroon" title="c4">n</font>)</span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c4">n</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="idea_1.html#D8" target="_self" onmouseover="rs('idea_1/D8')" onmouseout="rh()">Def8</a></span>
<br/>.= 
 <a 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="c1">i</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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="c2">j</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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">k</font>,<font color="Maroon" title="c4">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#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c4">n</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30_1_1_1_2_2_1_1"><span class="lab"><font color="Green" title="E41">A21</font></span></a>, <a class="ref" href="idea_1.html#D7" target="_self" onmouseover="rs('idea_1/D7')" onmouseout="rh()">Def7</a></span>
<br/>.= 
 <a href="idea_1.html#K8" title="IDEA_1:func.8">ChangeVal_2</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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="c2">j</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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">k</font>,<font color="Maroon" title="c4">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#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c4">n</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="euler_2.html#T8" onmouseover="rs('euler_2/T8')" onmouseout="rh()">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="c1">i</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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="c2">j</font>,<font color="Maroon" title="c4">n</font>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c4">n</font>)</span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c4">n</font>)

;<br/><span class="kw">hence </span><a NAME="E3:30_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="c1">i</font>,<font color="Maroon" title="c2">j</font>,<font color="Maroon" title="c4">n</font>)</span>)</span>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c4">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="c1">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="c2">j</font>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c4">n</font>)</span>)</span>,<font color="Maroon" title="c4">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30_1_1_1_2"><span class="lab"><font color="Green" title="E38">A18</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1_1_2_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:30_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="c2">j</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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">k</font>,<font color="Maroon" title="c4">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#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c4">n</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> (<span class="p1">(<span class="default"><a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font>,<font color="Maroon" title="c4">n</font>)</span>)</span>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c4">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="c1">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="c2">j</font>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c4">n</font>)</span>)</span>,<font color="Maroon" title="c4">n</font>)</span><br/><div class="add"><a NAME="E2:30_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="c2">j</font>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c4">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="c2">j</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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">k</font>,<font color="Maroon" title="c4">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#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="idea_1.html#D8" target="_self" onmouseover="rs('idea_1/D8')" onmouseout="rh()">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="c1">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="c2">j</font>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c4">n</font>)</span>)</span>,<font color="Maroon" title="c4">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="c1">i</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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="c2">j</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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">k</font>,<font color="Maroon" title="c4">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#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c4">n</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:30_1_1_1_2"><span class="lab"><font color="Green" title="E40">A20</font></span></a>, <a class="ref" href="idea_1.html#D7" target="_self" onmouseover="rs('idea_1/D7')" onmouseout="rh()">Def7</a></span>
<br/>.= 
 <a href="idea_1.html#K8" title="IDEA_1:func.8">ChangeVal_2</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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="c2">j</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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">k</font>,<font color="Maroon" title="c4">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#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c4">n</font>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="euler_2.html#T8" onmouseover="rs('euler_2/T8')" onmouseout="rh()">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="c1">i</font>,<font color="Maroon" title="c4">n</font>)</span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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="c2">j</font>,<font color="Maroon" title="c4">n</font>)</span>)</span></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p3">(<span class="default"><a href="idea_1.html#K7" title="IDEA_1:func.7">ChangeVal_1</a> (<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c4">n</font>)</span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default">2 <a href="series_1.html#K5" title="SERIES_1:func.5">to_power</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c4">n</font>)

;<br/><span class="kw">hence </span><a NAME="E4:30_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="c1">i</font>,<font color="Maroon" title="c2">j</font>,<font color="Maroon" title="c4">n</font>)</span>)</span>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c4">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="c1">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="c2">j</font>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c4">n</font>)</span>)</span>,<font color="Maroon" title="c4">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30_1_1_1_2"><span class="lab"><font color="Green" title="E38">A18</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E7:30_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="c1">i</font>,<font color="Maroon" title="c2">j</font>,<font color="Maroon" title="c4">n</font>)</span>)</span>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c4">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="c1">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="c2">j</font>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c4">n</font>)</span>)</span>,<font color="Maroon" title="c4">n</font>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E5:30_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="c1">i</font>,<font color="Maroon" title="c2">j</font>,<font color="Maroon" title="c4">n</font>)</span>)</span>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c4">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="c1">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="c2">j</font>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c4">n</font>)</span>)</span>,<font color="Maroon" title="c4">n</font>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E15:30"/>
 <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="c1">i</font>,<font color="Maroon" title="c2">j</font>,<font color="Maroon" title="c4">n</font>)</span>)</span>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c4">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="c1">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="c2">j</font>,<font color="Maroon" title="c3">k</font>,<font color="Maroon" title="c4">n</font>)</span>)</span>,<font color="Maroon" title="c4">n</font>)
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
