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<b>let </b><font color="Maroon" title="c1">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> ;<br/><b>let </b><font color="Maroon" title="c2">m</font>, <font color="Maroon" title="c3">k1</font>, <font color="Maroon" title="c4">k2</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> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:48"/><i><font color="Green" title="E39">A1</font></i>: 
( <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 is <a href="int_2.html#V1" title="INT_2:attr.1">prime</a> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">m</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 4 &amp; <font color="Maroon" title="c2">m</font> <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="c1">n</font> &amp; <font color="Maroon" title="c2">m</font> <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="c1">n</font> &amp; <font color="Maroon" title="c2">m</font> <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="c1">n</font> &amp; <font color="Maroon" title="c2">m</font> <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="c1">n</font> &amp; <font color="Maroon" title="c3">k1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 5 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c3">k1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 6 <a href="idea_1.html#R1" title="IDEA_1:pred.1">is_expressible_by</a> <font color="Maroon" title="c1">n</font> )
 <b>and </b><br/><a NAME="E2:48"/><i><font color="Green" title="E40">A2</font></i>: 
<font color="Maroon" title="c4">k2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 5 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 5
 <b>and </b><br/><a NAME="E3:48"/><i><font color="Green" title="E41">A3</font></i>: 
<font color="Maroon" title="c4">k2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 6 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 6
 ;<br/>

<b>consider </b><font color="Maroon" title="c5">t10</font> being    <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> <b> such that </b><br/><a NAME="E5:48"/><i><font color="Green" title="E42">A4</font></i>: 
<font color="Maroon" title="c5">t10</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> <span class="p1">(<span class="default"><a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">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="c2">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="c1">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="c2">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="c3">k1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 5</span>)</span>,<font color="Maroon" title="c1">n</font>
 ;<br/>
<b>consider </b><font color="Maroon" title="c6">t11</font> being    <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> <b> such that </b><br/><a NAME="E7:48"/><i><font color="Green" title="E43">A5</font></i>: 
<font color="Maroon" title="c6">t11</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> <span class="p1">(<span class="default"><a href="idea_1.html#K4" title="IDEA_1:func.4">ADD_MOD</a> <font color="Maroon" title="c5">t10</font>,<span class="p2">(<span class="default"><a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">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="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">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="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">k1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 6</span>)</span>,<font color="Maroon" title="c1">n</font>
 ;<br/>
<b>consider </b><font color="Maroon" title="c7">t12</font> being    <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> <b> such that </b><br/><a NAME="E9:48"/><i><font color="Green" title="E44">A6</font></i>: 
<font color="Maroon" title="c7">t12</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K4" title="IDEA_1:func.4">ADD_MOD</a> <font color="Maroon" title="c5">t10</font>,<font color="Maroon" title="c6">t11</font>,<font color="Maroon" title="c1">n</font>
 ;<br/>
<b>consider </b><font color="Maroon" title="c8">I1</font> being    <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> <b> such that </b><br/><a NAME="E11:48"/><i><font color="Green" title="E45">A7</font></i>: 
<font color="Maroon" title="c8">I1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K12" title="IDEA_1:func.12">IDEAoperationB</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k1</font>,<font color="Maroon" title="c1">n</font>
 ;<br/>
<b>consider </b><font color="Maroon" title="c9">t20</font> being    <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> <b> such that </b><br/><a NAME="E13:48"/><i><font color="Green" title="E46">A8</font></i>: 
<font color="Maroon" title="c9">t20</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> <span class="p1">(<span class="default"><a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">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="c8">I1</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="c1">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="c8">I1</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="c4">k2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 5</span>)</span>,<font color="Maroon" title="c1">n</font>
 ;<br/>
<b>consider </b><font color="Maroon" title="c10">t21</font> being    <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> <b> such that </b><br/><a NAME="E15:48"/><i><font color="Green" title="E47">A9</font></i>: 
<font color="Maroon" title="c10">t21</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> <span class="p1">(<span class="default"><a href="idea_1.html#K4" title="IDEA_1:func.4">ADD_MOD</a> <font color="Maroon" title="c9">t20</font>,<span class="p2">(<span class="default"><a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">I1</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="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <span class="p5">(<span class="default"><font color="Maroon" title="c8">I1</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="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">k2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 6</span>)</span>,<font color="Maroon" title="c1">n</font>
 ;<br/>
<b>consider </b><font color="Maroon" title="c11">I2</font> being    <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> <b> such that </b><br/><a NAME="E17:48"/><i><font color="Green" title="E48">A10</font></i>: 
<font color="Maroon" title="c11">I2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="idea_1.html#K12" title="IDEA_1:func.12">IDEAoperationB</a> <font color="Maroon" title="c8">I1</font>,<font color="Maroon" title="c4">k2</font>,<font color="Maroon" title="c1">n</font>
 ;<br/>
<a NAME="E18:48"/><i><font color="Green" title="E49">A11</font></i>: 
 <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="c2">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">m</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<i><font color="Green" title="E50">A12</font></i>:  <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="c2">m</font></span>)</span> = 
 <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="c8">I1</font></span>)</span>
<b>by </b><i><a class="txt" href="idea_1.html#E11:48"><i><font color="Green" title="E45">A7</font></i></a>, <a class="ref" href="idea_1.html#D12" target="_self" title="IDEA_1:def.12">Def12</a></i>
<br/>.= 
 <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="c11">I2</font></span>)</span>
<b>by </b><i><a class="txt" href="idea_1.html#E17:48"><i><font color="Green" title="E48">A10</font></i></a>, <a class="ref" href="idea_1.html#D12" target="_self" title="IDEA_1:def.12">Def12</a></i>
<br/>.= 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c11">I2</font>
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>
;<br/>
<a NAME="E20:48"/>
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="c2">m</font></span>)</span>
 
<b>by </b><i><a class="txt" href="idea_1.html#E1:48"><i><font color="Green" title="E39">A1</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E21:48"/><b>then </b>
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="c2">m</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E22:48"/><b>then </b><i><font color="Green" title="E51">A13</font></i>: 
<font color="Maroon" title="c8">I1</font> <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#K7" title="BINARITH:func.7">Absval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">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="c2">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="c1">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c7">t12</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="idea_1.html#E5:48"><i><font color="Green" title="E42">A4</font></i></a>, <a class="txt" href="idea_1.html#E7:48"><i><font color="Green" title="E43">A5</font></i></a>, <a class="txt" href="idea_1.html#E9:48"><i><font color="Green" title="E44">A6</font></i></a>, <a class="txt" href="idea_1.html#E11:48"><i><font color="Green" title="E45">A7</font></i></a>, <a class="ref" href="idea_1.html#D12" target="_self" title="IDEA_1:def.12">Def12</a></i>;<br/>
<a NAME="E23:48"/>
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="c2">m</font>
 
<b>by </b><i><a class="txt" href="idea_1.html#E1:48"><i><font color="Green" title="E39">A1</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E24:48"/><b>then </b>
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="c2">m</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E25:48"/><b>then </b>
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="c2">m</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E26:48"/><b>then </b><i><font color="Green" title="E52">A14</font></i>: 
<font color="Maroon" title="c8">I1</font> <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#K7" title="BINARITH:func.7">Absval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">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="c2">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="c1">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c6">t11</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="idea_1.html#E5:48"><i><font color="Green" title="E42">A4</font></i></a>, <a class="txt" href="idea_1.html#E7:48"><i><font color="Green" title="E43">A5</font></i></a>, <a class="txt" href="idea_1.html#E11:48"><i><font color="Green" title="E45">A7</font></i></a>, <a class="ref" href="idea_1.html#D12" target="_self" title="IDEA_1:def.12">Def12</a></i>;<br/>
<a NAME="E27:48"/>
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="c2">m</font>
 
<b>by </b><i><a class="txt" href="idea_1.html#E1:48"><i><font color="Green" title="E39">A1</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E28:48"/><b>then </b>
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="c2">m</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E29:48"/><b>then </b>
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="c2">m</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E30:48"/><b>then </b><i><font color="Green" title="E53">A15</font></i>: 
<font color="Maroon" title="c8">I1</font> <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#K7" title="BINARITH:func.7">Absval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">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="c2">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="c1">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c7">t12</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="idea_1.html#E5:48"><i><font color="Green" title="E42">A4</font></i></a>, <a class="txt" href="idea_1.html#E7:48"><i><font color="Green" title="E43">A5</font></i></a>, <a class="txt" href="idea_1.html#E9:48"><i><font color="Green" title="E44">A6</font></i></a>, <a class="txt" href="idea_1.html#E11:48"><i><font color="Green" title="E45">A7</font></i></a>, <a class="ref" href="idea_1.html#D12" target="_self" title="IDEA_1:def.12">Def12</a></i>;<br/>
<a NAME="E31:48"/>
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="c2">m</font>
 
<b>by </b><i><a class="txt" href="idea_1.html#E1:48"><i><font color="Green" title="E39">A1</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E32:48"/><b>then </b>
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="c2">m</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E33:48"/><b>then </b>
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="c2">m</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E34:48"/><b>then </b><i><font color="Green" title="E54">A16</font></i>: 
<font color="Maroon" title="c8">I1</font> <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#K7" title="BINARITH:func.7">Absval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">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="c2">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="c1">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c6">t11</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="idea_1.html#E5:48"><i><font color="Green" title="E42">A4</font></i></a>, <a class="txt" href="idea_1.html#E7:48"><i><font color="Green" title="E43">A5</font></i></a>, <a class="txt" href="idea_1.html#E11:48"><i><font color="Green" title="E45">A7</font></i></a>, <a class="ref" href="idea_1.html#D12" target="_self" title="IDEA_1:def.12">Def12</a></i>;<br/>
<b>then </b><i><font color="Green" title="E55">A17</font></i>: <font color="Maroon" title="c9">t20</font> = 
 <a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> <span class="p1">(<span class="default"><a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">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="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c6">t11</font></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="c1">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <span class="p4">(<span class="default"><a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">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="p0">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c6">t11</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">k2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 5</span>)</span>,<font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="txt" href="idea_1.html#E13:48"><i><font color="Green" title="E46">A8</font></i></a>, <a class="txt" href="idea_1.html#E26:48"><i><font color="Green" title="E52">A14</font></i></a>, <a class="ref" href="binari_3.html#T37" title="BINARI_3:th.37">BINARI_3:37</a></i>
<br/>.= 
 <a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> <span class="p1">(<span class="default"><a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">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="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c6">t11</font></span>)</span></span>)</span> <a href="idea_1.html#K3" title="IDEA_1:func.3">'xor'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">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="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c6">t11</font></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">k2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 5</span>)</span>,<font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="ref" href="binari_3.html#T37" title="BINARI_3:th.37">BINARI_3:37</a></i>
<br/>.= 
 <a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> <span class="p1">(<span class="default"><a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">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="c2">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"><span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c6">t11</font></span>)</span> <a href="idea_1.html#K3" title="IDEA_1:func.3">'xor'</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c6">t11</font></span>)</span> <a href="idea_1.html#K3" title="IDEA_1:func.3">'xor'</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <span class="p0">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">k2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 5</span>)</span>,<font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="ref" href="idea_1.html#T10" target="_self" title="IDEA_1:th.10">Th10</a></i>
<br/>.= 
 <a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> <span class="p1">(<span class="default"><a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">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="c2">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"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c6">t11</font></span>)</span> <a href="idea_1.html#K3" title="IDEA_1:func.3">'xor'</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c6">t11</font></span>)</span></span>)</span> <a href="idea_1.html#K3" title="IDEA_1:func.3">'xor'</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">k2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 5</span>)</span>,<font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="ref" href="idea_1.html#T10" target="_self" title="IDEA_1:th.10">Th10</a></i>
<br/>.= 
 <a href="idea_1.html#K9" title="IDEA_1:func.9">MUL_MOD</a> <span class="p1">(<span class="default"><a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">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="c2">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"><span class="p4">(<span class="default"><a href="idea_1.html#K1" title="IDEA_1:func.1">ZERO</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="idea_1.html#K3" title="IDEA_1:func.3">'xor'</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">k2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 5</span>)</span>,<font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="ref" href="idea_1.html#T7" target="_self" title="IDEA_1:th.7">Th7</a></i>
<br/>.= 
<font color="Maroon" title="c5">t10</font>
<b>by </b><i><a class="txt" href="idea_1.html#E2:48"><i><font color="Green" title="E40">A2</font></i></a>, <a class="txt" href="idea_1.html#E5:48"><i><font color="Green" title="E42">A4</font></i></a>, <a class="ref" href="idea_1.html#T9" target="_self" title="IDEA_1:th.9">Th9</a></i>
;<br/>
<b>then </b><i><font color="Green" title="E56">A18</font></i>: <font color="Maroon" title="c10">t21</font> = 
 <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#K4" title="IDEA_1:func.4">ADD_MOD</a> <font color="Maroon" title="c5">t10</font>,<span class="p2">(<span class="default"><a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <span class="p0">(<span class="default"><font color="Maroon" title="c2">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="p5">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c7">t12</font></span>)</span></span>)</span> <a href="idea_1.html#K3" title="IDEA_1:func.3">'xor'</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <span class="p5">(<span class="default"><a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p0">(<span class="default"><span class="p1">(<span class="default"><font color="Maroon" title="c1">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="c2">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="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c7">t12</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">k2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 6</span>)</span>,<font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="txt" href="idea_1.html#E15:48"><i><font color="Green" title="E47">A9</font></i></a>, <a class="txt" href="idea_1.html#E22:48"><i><font color="Green" title="E51">A13</font></i></a>, <a class="txt" href="idea_1.html#E30:48"><i><font color="Green" title="E53">A15</font></i></a>, <a class="ref" href="binari_3.html#T37" title="BINARI_3:th.37">BINARI_3:37</a></i>
<br/>.= 
 <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#K4" title="IDEA_1:func.4">ADD_MOD</a> <font color="Maroon" title="c5">t10</font>,<span class="p2">(<span class="default"><a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <span class="p0">(<span class="default"><font color="Maroon" title="c2">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="p5">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c7">t12</font></span>)</span></span>)</span> <a href="idea_1.html#K3" title="IDEA_1:func.3">'xor'</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <span class="p0">(<span class="default"><font color="Maroon" title="c2">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="p5">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c7">t12</font></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">k2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 6</span>)</span>,<font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="ref" href="binari_3.html#T37" title="BINARI_3:th.37">BINARI_3:37</a></i>
<br/>.= 
 <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#K4" title="IDEA_1:func.4">ADD_MOD</a> <font color="Maroon" title="c5">t10</font>,<span class="p2">(<span class="default"><a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">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="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c7">t12</font></span>)</span> <a href="idea_1.html#K3" title="IDEA_1:func.3">'xor'</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c7">t12</font></span>)</span> <a href="idea_1.html#K3" title="IDEA_1:func.3">'xor'</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 4</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">k2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 6</span>)</span>,<font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="ref" href="idea_1.html#T10" target="_self" title="IDEA_1:th.10">Th10</a></i>
<br/>.= 
 <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#K4" title="IDEA_1:func.4">ADD_MOD</a> <font color="Maroon" title="c5">t10</font>,<span class="p2">(<span class="default"><a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">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="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c7">t12</font></span>)</span> <a href="idea_1.html#K3" title="IDEA_1:func.3">'xor'</a> <span class="p0">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c7">t12</font></span>)</span></span>)</span> <a href="idea_1.html#K3" title="IDEA_1:func.3">'xor'</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <span class="p0">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 4</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">k2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 6</span>)</span>,<font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="ref" href="idea_1.html#T10" target="_self" title="IDEA_1:th.10">Th10</a></i>
<br/>.= 
 <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#K4" title="IDEA_1:func.4">ADD_MOD</a> <font color="Maroon" title="c5">t10</font>,<span class="p2">(<span class="default"><a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">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="p4">(<span class="default"><span class="p5">(<span class="default"><a href="idea_1.html#K1" title="IDEA_1:func.1">ZERO</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="idea_1.html#K3" title="IDEA_1:func.3">'xor'</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <span class="p0">(<span class="default"><font color="Maroon" title="c2">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 4</span>)</span></span>)</span></span>)</span></span>)</span></span>)</span>,<font color="Maroon" title="c1">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">k2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 6</span>)</span>,<font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="ref" href="idea_1.html#T7" target="_self" title="IDEA_1:th.7">Th7</a></i>
<br/>.= 
<font color="Maroon" title="c6">t11</font>
<b>by </b><i><a class="txt" href="idea_1.html#E3:48"><i><font color="Green" title="E41">A3</font></i></a>, <a class="txt" href="idea_1.html#E7:48"><i><font color="Green" title="E43">A5</font></i></a>, <a class="ref" href="idea_1.html#T9" target="_self" title="IDEA_1:th.9">Th9</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c12">j</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/><b>assume </b><a NAME="E1:48_4"/><i><font color="Green" title="E57">A19</font></i>: 
<font color="Maroon" title="c12">j</font> <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="c2">m</font></span>)</span>
 ;<br/><a NAME="E2:48_4"/><b>then </b>
<font color="Maroon" title="c12">j</font> <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="c8">I1</font></span>)</span>
 <b>by </b><i><a class="txt" href="idea_1.html#E11:48"><i><font color="Green" title="E45">A7</font></i></a>, <a class="ref" href="idea_1.html#D12" target="_self" title="IDEA_1:def.12">Def12</a></i>;<br/><a NAME="E3:48_4"/><b>then </b><i><font color="Green" title="E58">A20</font></i>: 
<font color="Maroon" title="c12">j</font> <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="c8">I1</font>
 <b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E4:48_4"/><i><font color="Green" title="E59">A21</font></i>: 
<font color="Maroon" title="c12">j</font> <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="c2">m</font>
 <b>by </b><i><a class="txt" href="idea_1.html#E1:48_4"><i><font color="Green" title="E57">A19</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:48_4_1_1"/>
( <font color="Maroon" title="c12">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c12">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or <font color="Maroon" title="c12">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 or <font color="Maroon" title="c12">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 or ( <font color="Maroon" title="c12">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 &amp; <font color="Maroon" title="c12">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 2 &amp; <font color="Maroon" title="c12">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 3 &amp; <font color="Maroon" title="c12">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 4 ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_4_1_1_1"><b>suppose </b></a><a NAME="E1:48_4_1_1_1"/><i><font color="Green" title="E60">A22</font></i>: 
<font color="Maroon" title="c12">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:48_4_1_1_1"/><i><font color="Green" title="E61">A23</font></i>: 
<font color="Maroon" title="c2">m</font> <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="c1">n</font>
 <b>by </b><i><a class="txt" href="idea_1.html#E1:48"><i><font color="Green" title="E39">A1</font></i></a>, <a class="ref" href="idea_1.html#D3" target="_self" title="IDEA_1:def.3">Def3</a></i>;<br/><b>thus </b><font color="Maroon" title="c11">I2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">j</font> = 
 <a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">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="c8">I1</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="c1">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c6">t11</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="idea_1.html#E13:48"><i><font color="Green" title="E46">A8</font></i></a>, <a class="txt" href="idea_1.html#E15:48"><i><font color="Green" title="E47">A9</font></i></a>, <a class="txt" href="idea_1.html#E17:48"><i><font color="Green" title="E48">A10</font></i></a>, <a class="txt" href="idea_1.html#E36:48"><i><font color="Green" title="E56">A18</font></i></a>, <a class="txt" href="idea_1.html#E3:48_4"><i><font color="Green" title="E58">A20</font></i></a>, <a class="txt" href="idea_1.html#E1:48_4_1_1_1"><i><font color="Green" title="E60">A22</font></i></a>, <a class="ref" href="idea_1.html#D12" target="_self" title="IDEA_1:def.12">Def12</a></i>
<br/>.= 
 <a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">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="c2">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="c1">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c6">t11</font></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="c1">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c6">t11</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="idea_1.html#E34:48"><i><font color="Green" title="E54">A16</font></i></a>, <a class="ref" href="binari_3.html#T37" title="BINARI_3:th.37">BINARI_3:37</a></i>
<br/>.= 
 <a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">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="c2">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"><span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c6">t11</font></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="c1">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c6">t11</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="idea_1.html#T10" target="_self" title="IDEA_1:th.10">Th10</a></i>
<br/>.= 
 <a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="idea_1.html#K1" title="IDEA_1:func.1">ZERO</a> <font color="Maroon" title="c1">n</font></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="c1">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="c2">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="idea_1.html#T7" target="_self" title="IDEA_1:th.7">Th7</a></i>
<br/>.= 
 <a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">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="c2">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="idea_1.html#T9" target="_self" title="IDEA_1:th.9">Th9</a></i>
<br/>.= 
<font color="Maroon" title="c2">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">j</font>
<b>by </b><i><a class="txt" href="idea_1.html#E1:48_4_1_1_1"><i><font color="Green" title="E60">A22</font></i></a>, <a class="txt" href="idea_1.html#E2:48_4_1_1_1"><i><font color="Green" title="E61">A23</font></i></a>, <a class="ref" href="binari_3.html#T36" title="BINARI_3:th.36">BINARI_3:36</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_4_1_1_2"><b>suppose </b></a><a NAME="E1:48_4_1_1_2"/><i><font color="Green" title="E60">A24</font></i>: 
<font color="Maroon" title="c12">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ;<br/><div class="add"><a NAME="E2:48_4_1_1_2"/><i><font color="Green" title="E61">A25</font></i>: 
<font color="Maroon" title="c2">m</font> <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="c1">n</font>
 <b>by </b><i><a class="txt" href="idea_1.html#E1:48"><i><font color="Green" title="E39">A1</font></i></a>, <a class="ref" href="idea_1.html#D3" target="_self" title="IDEA_1:def.3">Def3</a></i>;<br/><b>thus </b><font color="Maroon" title="c11">I2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">j</font> = 
 <a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">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="c8">I1</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="c1">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c7">t12</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="idea_1.html#E9:48"><i><font color="Green" title="E44">A6</font></i></a>, <a class="txt" href="idea_1.html#E13:48"><i><font color="Green" title="E46">A8</font></i></a>, <a class="txt" href="idea_1.html#E15:48"><i><font color="Green" title="E47">A9</font></i></a>, <a class="txt" href="idea_1.html#E17:48"><i><font color="Green" title="E48">A10</font></i></a>, <a class="txt" href="idea_1.html#E35:48"><i><font color="Green" title="E55">A17</font></i></a>, <a class="txt" href="idea_1.html#E36:48"><i><font color="Green" title="E56">A18</font></i></a>, <a class="txt" href="idea_1.html#E3:48_4"><i><font color="Green" title="E58">A20</font></i></a>, <a class="txt" href="idea_1.html#E1:48_4_1_1_2"><i><font color="Green" title="E60">A24</font></i></a>, <a class="ref" href="idea_1.html#D12" target="_self" title="IDEA_1:def.12">Def12</a></i>
<br/>.= 
 <a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">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="c2">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="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c7">t12</font></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="c1">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c7">t12</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="idea_1.html#E30:48"><i><font color="Green" title="E53">A15</font></i></a>, <a class="ref" href="binari_3.html#T37" title="BINARI_3:th.37">BINARI_3:37</a></i>
<br/>.= 
 <a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">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="c2">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"><span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c7">t12</font></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="c1">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c7">t12</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="idea_1.html#T10" target="_self" title="IDEA_1:th.10">Th10</a></i>
<br/>.= 
 <a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="idea_1.html#K1" title="IDEA_1:func.1">ZERO</a> <font color="Maroon" title="c1">n</font></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="c1">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="c2">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="idea_1.html#T7" target="_self" title="IDEA_1:th.7">Th7</a></i>
<br/>.= 
 <a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">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="c2">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 2</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="idea_1.html#T9" target="_self" title="IDEA_1:th.9">Th9</a></i>
<br/>.= 
<font color="Maroon" title="c2">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">j</font>
<b>by </b><i><a class="txt" href="idea_1.html#E1:48_4_1_1_2"><i><font color="Green" title="E60">A24</font></i></a>, <a class="txt" href="idea_1.html#E2:48_4_1_1_2"><i><font color="Green" title="E61">A25</font></i></a>, <a class="ref" href="binari_3.html#T36" title="BINARI_3:th.36">BINARI_3:36</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_4_1_1_3"><b>suppose </b></a><a NAME="E1:48_4_1_1_3"/><i><font color="Green" title="E60">A26</font></i>: 
<font color="Maroon" title="c12">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 ;<br/><div class="add"><a NAME="E2:48_4_1_1_3"/><i><font color="Green" title="E61">A27</font></i>: 
<font color="Maroon" title="c2">m</font> <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="c1">n</font>
 <b>by </b><i><a class="txt" href="idea_1.html#E1:48"><i><font color="Green" title="E39">A1</font></i></a>, <a class="ref" href="idea_1.html#D3" target="_self" title="IDEA_1:def.3">Def3</a></i>;<br/><b>thus </b><font color="Maroon" title="c11">I2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">j</font> = 
 <a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">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="c8">I1</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="c1">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c6">t11</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="idea_1.html#E13:48"><i><font color="Green" title="E46">A8</font></i></a>, <a class="txt" href="idea_1.html#E15:48"><i><font color="Green" title="E47">A9</font></i></a>, <a class="txt" href="idea_1.html#E17:48"><i><font color="Green" title="E48">A10</font></i></a>, <a class="txt" href="idea_1.html#E36:48"><i><font color="Green" title="E56">A18</font></i></a>, <a class="txt" href="idea_1.html#E3:48_4"><i><font color="Green" title="E58">A20</font></i></a>, <a class="txt" href="idea_1.html#E1:48_4_1_1_3"><i><font color="Green" title="E60">A26</font></i></a>, <a class="ref" href="idea_1.html#D12" target="_self" title="IDEA_1:def.12">Def12</a></i>
<br/>.= 
 <a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">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="c2">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="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c6">t11</font></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="c1">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c6">t11</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="idea_1.html#E26:48"><i><font color="Green" title="E52">A14</font></i></a>, <a class="ref" href="binari_3.html#T37" title="BINARI_3:th.37">BINARI_3:37</a></i>
<br/>.= 
 <a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">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="c2">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"><span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c6">t11</font></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="c1">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c6">t11</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="idea_1.html#T10" target="_self" title="IDEA_1:th.10">Th10</a></i>
<br/>.= 
 <a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="idea_1.html#K1" title="IDEA_1:func.1">ZERO</a> <font color="Maroon" title="c1">n</font></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="c1">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="c2">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="idea_1.html#T7" target="_self" title="IDEA_1:th.7">Th7</a></i>
<br/>.= 
 <a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">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="c2">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 3</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="idea_1.html#T9" target="_self" title="IDEA_1:th.9">Th9</a></i>
<br/>.= 
<font color="Maroon" title="c2">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">j</font>
<b>by </b><i><a class="txt" href="idea_1.html#E1:48_4_1_1_3"><i><font color="Green" title="E60">A26</font></i></a>, <a class="txt" href="idea_1.html#E2:48_4_1_1_3"><i><font color="Green" title="E61">A27</font></i></a>, <a class="ref" href="binari_3.html#T36" title="BINARI_3:th.36">BINARI_3:36</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_4_1_1_4"><b>suppose </b></a><a NAME="E1:48_4_1_1_4"/><i><font color="Green" title="E60">A28</font></i>: 
<font color="Maroon" title="c12">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4
 ;<br/><div class="add"><a NAME="E2:48_4_1_1_4"/><i><font color="Green" title="E61">A29</font></i>: 
<font color="Maroon" title="c2">m</font> <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="c1">n</font>
 <b>by </b><i><a class="txt" href="idea_1.html#E1:48"><i><font color="Green" title="E39">A1</font></i></a>, <a class="ref" href="idea_1.html#D3" target="_self" title="IDEA_1:def.3">Def3</a></i>;<br/><b>thus </b><font color="Maroon" title="c11">I2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">j</font> = 
 <a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">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="c8">I1</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="c1">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c7">t12</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="idea_1.html#E9:48"><i><font color="Green" title="E44">A6</font></i></a>, <a class="txt" href="idea_1.html#E13:48"><i><font color="Green" title="E46">A8</font></i></a>, <a class="txt" href="idea_1.html#E15:48"><i><font color="Green" title="E47">A9</font></i></a>, <a class="txt" href="idea_1.html#E17:48"><i><font color="Green" title="E48">A10</font></i></a>, <a class="txt" href="idea_1.html#E35:48"><i><font color="Green" title="E55">A17</font></i></a>, <a class="txt" href="idea_1.html#E36:48"><i><font color="Green" title="E56">A18</font></i></a>, <a class="txt" href="idea_1.html#E3:48_4"><i><font color="Green" title="E58">A20</font></i></a>, <a class="txt" href="idea_1.html#E1:48_4_1_1_4"><i><font color="Green" title="E60">A28</font></i></a>, <a class="ref" href="idea_1.html#D12" target="_self" title="IDEA_1:def.12">Def12</a></i>
<br/>.= 
 <a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">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="c2">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="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c7">t12</font></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="c1">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c7">t12</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="idea_1.html#E22:48"><i><font color="Green" title="E51">A13</font></i></a>, <a class="ref" href="binari_3.html#T37" title="BINARI_3:th.37">BINARI_3:37</a></i>
<br/>.= 
 <a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">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="c2">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"><span class="p3">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c7">t12</font></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="c1">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c7">t12</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="idea_1.html#T10" target="_self" title="IDEA_1:th.10">Th10</a></i>
<br/>.= 
 <a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="idea_1.html#K1" title="IDEA_1:func.1">ZERO</a> <font color="Maroon" title="c1">n</font></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="c1">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="c2">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 4</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="idea_1.html#T7" target="_self" title="IDEA_1:th.7">Th7</a></i>
<br/>.= 
 <a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">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="c2">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> 4</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="idea_1.html#T9" target="_self" title="IDEA_1:th.9">Th9</a></i>
<br/>.= 
<font color="Maroon" title="c2">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">j</font>
<b>by </b><i><a class="txt" href="idea_1.html#E1:48_4_1_1_4"><i><font color="Green" title="E60">A28</font></i></a>, <a class="txt" href="idea_1.html#E2:48_4_1_1_4"><i><font color="Green" title="E61">A29</font></i></a>, <a class="ref" href="binari_3.html#T36" title="BINARI_3:th.36">BINARI_3:36</a></i>
;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_4_1_1_5"><b>suppose </b></a><a NAME="E1:48_4_1_1_5"/><i><font color="Green" title="E60">A30</font></i>: 
( <font color="Maroon" title="c12">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 &amp; <font color="Maroon" title="c12">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 2 &amp; <font color="Maroon" title="c12">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 3 &amp; <font color="Maroon" title="c12">j</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 4 )
 ;<br/><div class="add"><b>hence </b><font color="Maroon" title="c11">I2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">j</font> = 
<font color="Maroon" title="c8">I1</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">j</font>
<b>by </b><i><a class="txt" href="idea_1.html#E17:48"><i><font color="Green" title="E48">A10</font></i></a>, <a class="txt" href="idea_1.html#E3:48_4"><i><font color="Green" title="E58">A20</font></i></a>, <a class="ref" href="idea_1.html#D12" target="_self" title="IDEA_1:def.12">Def12</a></i>
<br/>.= 
<font color="Maroon" title="c2">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">j</font>
<b>by </b><i><a class="txt" href="idea_1.html#E11:48"><i><font color="Green" title="E45">A7</font></i></a>, <a class="txt" href="idea_1.html#E4:48_4"><i><font color="Green" title="E59">A21</font></i></a>, <a class="txt" href="idea_1.html#E1:48_4_1_1_5"><i><font color="Green" title="E60">A30</font></i></a>, <a class="ref" href="idea_1.html#D12" target="_self" title="IDEA_1:def.12">Def12</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:48_4"/>
<font color="Maroon" title="c11">I2</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font> <a href="recdef_1.html#K1" title="RECDEF_1:func.1">.</a> <font color="Maroon" title="c12">j</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E38:48"/>
 <a href="idea_1.html#K12" title="IDEA_1:func.12">IDEAoperationB</a> <span class="p1">(<span class="default"><a href="idea_1.html#K12" title="IDEA_1:func.12">IDEAoperationB</a> <font color="Maroon" title="c2">m</font>,<font color="Maroon" title="c3">k1</font>,<font color="Maroon" title="c1">n</font></span>)</span>,<font color="Maroon" title="c4">k2</font>,<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">m</font>
 <b>by </b><i><a class="txt" href="idea_1.html#E11:48"><i><font color="Green" title="E45">A7</font></i></a>, <a class="txt" href="idea_1.html#E17:48"><i><font color="Green" title="E48">A10</font></i></a>, <a class="txt" href="idea_1.html#E18:48"><i><font color="Green" title="E49">A11</font></i></a>, <a class="txt" href="idea_1.html#E19:48"><i><font color="Green" title="E50">A12</font></i></a>, <a class="ref" href="finseq_1.html#T17" title="FINSEQ_1:th.17">FINSEQ_1:17</a></i>;<br/>


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