<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">m</font>, <font color="Maroon" title="c2">k</font> be    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">n</font> being non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 4 holds <br/>( <span class="p1">(<span class="default"><a href="idea_1.html#K12" title="IDEA_1:func.12">IDEAoperationB</a> <font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c2">k</font>,<font color="Olive" title="b1">n</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Olive" title="b1">n</font> &amp; <span class="p1">(<span class="default"><a href="idea_1.html#K12" title="IDEA_1:func.12">IDEAoperationB</a> <font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c2">k</font>,<font color="Olive" title="b1">n</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Olive" title="b1">n</font> &amp; <span class="p1">(<span class="default"><a href="idea_1.html#K12" title="IDEA_1:func.12">IDEAoperationB</a> <font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c2">k</font>,<font color="Olive" title="b1">n</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Olive" title="b1">n</font> &amp; <span class="p1">(<span class="default"><a href="idea_1.html#K12" title="IDEA_1:func.12">IDEAoperationB</a> <font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c2">k</font>,<font color="Olive" title="b1">n</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 4 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Olive" title="b1">n</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">n</font> be  non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 4 implies ( <span class="p1">(<span class="default"><a href="idea_1.html#K12" title="IDEA_1:func.12">IDEAoperationB</a> <font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c3">n</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c3">n</font> &amp; <span class="p1">(<span class="default"><a href="idea_1.html#K12" title="IDEA_1:func.12">IDEAoperationB</a> <font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c3">n</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c3">n</font> &amp; <span class="p1">(<span class="default"><a href="idea_1.html#K12" title="IDEA_1:func.12">IDEAoperationB</a> <font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c3">n</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c3">n</font> &amp; <span class="p1">(<span class="default"><a href="idea_1.html#K12" title="IDEA_1:func.12">IDEAoperationB</a> <font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c3">n</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 4 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c3">n</font> ) )</span><br/>





<span class="kw">assume </span><a NAME="E1:42"/><span class="lab"><font color="Green" title="E33">A1</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 4
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <span class="p1">(<span class="default"><a href="idea_1.html#K12" title="IDEA_1:func.12">IDEAoperationB</a> <font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c3">n</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c3">n</font> &amp; <span class="p1">(<span class="default"><a href="idea_1.html#K12" title="IDEA_1:func.12">IDEAoperationB</a> <font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c3">n</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c3">n</font> &amp; <span class="p1">(<span class="default"><a href="idea_1.html#K12" title="IDEA_1:func.12">IDEAoperationB</a> <font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c3">n</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c3">n</font> &amp; <span class="p1">(<span class="default"><a href="idea_1.html#K12" title="IDEA_1:func.12">IDEAoperationB</a> <font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c3">n</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 4 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c3">n</font> )</span><br/>

<a NAME="E2:42"/><span class="kw">then </span>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E3:42"/><span class="kw">then </span>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">m</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E4:42"/><span class="kw">then </span>
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E5:42"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="idea_1.html#K12" title="IDEA_1:func.12">IDEAoperationB</a> <font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c3">n</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1</span>)</span></span>)</span> <a href="idea_1.html#K3" title="IDEA_1:func.3">'xor'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <span class="p3">(<span class="default"><a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> <span class="p4">(<span class="default"><a href="idea_1.html#K4" title="IDEA_1:func.4">ADD_MOD</a> <span class="p5">(<span class="default"><a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> <span class="p0">(<span class="default"><a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1</span>)</span></span>)</span> <a href="idea_1.html#K3" title="IDEA_1:func.3">'xor'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p0">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 5</span>)</span>,<font color="Maroon" title="c3">n</font></span>)</span>,<span class="p5">(<span class="default"><a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2</span>)</span></span>)</span> <a href="idea_1.html#K3" title="IDEA_1:func.3">'xor'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 4</span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c3">n</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 6</span>)</span>,<font color="Maroon" title="c3">n</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="idea_1.html#D12" target="_self" title="IDEA_1:def.12">Def12</a></span>;<br/>
<a NAME="E6:42"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A2</font></span>: 
<span class="p1">(<span class="default"><a href="idea_1.html#K12" title="IDEA_1:func.12">IDEAoperationB</a> <font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c3">n</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c3">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="binari_3.html#T1" title="BINARI_3:th.1">BINARI_3:1</a></span>;<br/>
<a NAME="E7:42"/>
3 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42"><span class="lab"><font color="Green" title="E33">A1</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E8:42"/><span class="kw">then </span>
3 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">m</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E9:42"/><span class="kw">then </span>
3 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E10:42"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="idea_1.html#K12" title="IDEA_1:func.12">IDEAoperationB</a> <font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c3">n</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3</span>)</span></span>)</span> <a href="idea_1.html#K3" title="IDEA_1:func.3">'xor'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <span class="p3">(<span class="default"><a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> <span class="p4">(<span class="default"><a href="idea_1.html#K4" title="IDEA_1:func.4">ADD_MOD</a> <span class="p5">(<span class="default"><a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> <span class="p0">(<span class="default"><a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1</span>)</span></span>)</span> <a href="idea_1.html#K3" title="IDEA_1:func.3">'xor'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p0">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 5</span>)</span>,<font color="Maroon" title="c3">n</font></span>)</span>,<span class="p5">(<span class="default"><a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2</span>)</span></span>)</span> <a href="idea_1.html#K3" title="IDEA_1:func.3">'xor'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 4</span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c3">n</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 6</span>)</span>,<font color="Maroon" title="c3">n</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="idea_1.html#D12" target="_self" title="IDEA_1:def.12">Def12</a></span>;<br/>
<a NAME="E11:42"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A3</font></span>: 
<span class="p1">(<span class="default"><a href="idea_1.html#K12" title="IDEA_1:func.12">IDEAoperationB</a> <font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c3">n</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c3">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="binari_3.html#T1" title="BINARI_3:th.1">BINARI_3:1</a></span>;<br/>
<a NAME="E12:42"/>
2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42"><span class="lab"><font color="Green" title="E33">A1</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E13:42"/><span class="kw">then </span>
2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">m</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E14:42"/><span class="kw">then </span>
2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E15:42"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="idea_1.html#K12" title="IDEA_1:func.12">IDEAoperationB</a> <font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c3">n</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2</span>)</span></span>)</span> <a href="idea_1.html#K3" title="IDEA_1:func.3">'xor'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <span class="p3">(<span class="default"><a href="idea_1.html#K4" title="IDEA_1:func.4">ADD_MOD</a> <span class="p4">(<span class="default"><a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> <span class="p5">(<span class="default"><a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1</span>)</span></span>)</span> <a href="idea_1.html#K3" title="IDEA_1:func.3">'xor'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 5</span>)</span>,<font color="Maroon" title="c3">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> <span class="p5">(<span class="default"><a href="idea_1.html#K4" title="IDEA_1:func.4">ADD_MOD</a> <span class="p0">(<span class="default"><a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> <span class="p1">(<span class="default"><a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1</span>)</span></span>)</span> <a href="idea_1.html#K3" title="IDEA_1:func.3">'xor'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 5</span>)</span>,<font color="Maroon" title="c3">n</font></span>)</span>,<span class="p0">(<span class="default"><a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2</span>)</span></span>)</span> <a href="idea_1.html#K3" title="IDEA_1:func.3">'xor'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 4</span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c3">n</font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 6</span>)</span>,<font color="Maroon" title="c3">n</font></span>)</span>,<font color="Maroon" title="c3">n</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="idea_1.html#D12" target="_self" title="IDEA_1:def.12">Def12</a></span>;<br/>
<a NAME="E16:42"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A4</font></span>: 
<span class="p1">(<span class="default"><a href="idea_1.html#K12" title="IDEA_1:func.12">IDEAoperationB</a> <font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c3">n</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c3">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="binari_3.html#T1" title="BINARI_3:th.1">BINARI_3:1</a></span>;<br/>
<a NAME="E17:42"/>
4 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">m</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:42"><span class="lab"><font color="Green" title="E33">A1</font></span></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></span>;<br/>
<a NAME="E18:42"/><span class="kw">then </span>
4 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c1">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></span>;<br/>
<a NAME="E19:42"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="idea_1.html#K12" title="IDEA_1:func.12">IDEAoperationB</a> <font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c3">n</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 4 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 4</span>)</span></span>)</span> <a href="idea_1.html#K3" title="IDEA_1:func.3">'xor'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <span class="p3">(<span class="default"><a href="idea_1.html#K4" title="IDEA_1:func.4">ADD_MOD</a> <span class="p4">(<span class="default"><a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> <span class="p5">(<span class="default"><a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1</span>)</span></span>)</span> <a href="idea_1.html#K3" title="IDEA_1:func.3">'xor'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 5</span>)</span>,<font color="Maroon" title="c3">n</font></span>)</span>,<span class="p4">(<span class="default"><a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> <span class="p5">(<span class="default"><a href="idea_1.html#K4" title="IDEA_1:func.4">ADD_MOD</a> <span class="p0">(<span class="default"><a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> <span class="p1">(<span class="default"><a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1</span>)</span></span>)</span> <a href="idea_1.html#K3" title="IDEA_1:func.3">'xor'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3</span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 5</span>)</span>,<font color="Maroon" title="c3">n</font></span>)</span>,<span class="p0">(<span class="default"><a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2</span>)</span></span>)</span> <a href="idea_1.html#K3" title="IDEA_1:func.3">'xor'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 4</span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c3">n</font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 6</span>)</span>,<font color="Maroon" title="c3">n</font></span>)</span>,<font color="Maroon" title="c3">n</font></span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="idea_1.html#D12" target="_self" title="IDEA_1:def.12">Def12</a></span>;<br/>
<a NAME="E20:42"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="idea_1.html#K12" title="IDEA_1:func.12">IDEAoperationB</a> <font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c3">n</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 4 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c3">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="binari_3.html#T1" title="BINARI_3:th.1">BINARI_3:1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E21:42"/>
( <span class="p1">(<span class="default"><a href="idea_1.html#K12" title="IDEA_1:func.12">IDEAoperationB</a> <font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c3">n</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c3">n</font> &amp; <span class="p1">(<span class="default"><a href="idea_1.html#K12" title="IDEA_1:func.12">IDEAoperationB</a> <font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c3">n</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c3">n</font> &amp; <span class="p1">(<span class="default"><a href="idea_1.html#K12" title="IDEA_1:func.12">IDEAoperationB</a> <font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c3">n</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c3">n</font> &amp; <span class="p1">(<span class="default"><a href="idea_1.html#K12" title="IDEA_1:func.12">IDEAoperationB</a> <font color="Maroon" title="c1">m</font>,<font color="Maroon" title="c2">k</font>,<font color="Maroon" title="c3">n</font></span>)</span> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 4 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c3">n</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:42"><span class="lab"><font color="Green" title="E34">A2</font></span></a>, <a class="txt" href="#E16:42"><span class="lab"><font color="Green" title="E36">A4</font></span></a>, <a class="txt" href="#E11:42"><span class="lab"><font color="Green" title="E35">A3</font></span></a>, <a class="ref" href="idea_1.html#D3" target="_self" title="IDEA_1:def.3">Def3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
