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<span class="kw">let </span><font color="Maroon" title="c1">n</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <font color="Maroon" title="c1">n</font> holds <br/> for <font color="Olive" title="b2">x</font> being   <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of <font color="Maroon" title="c1">n</font>, <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a>   st <font color="Olive" title="b2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font> holds <br/>( <font color="Maroon" title="c1">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binarith.html#K3" title="BINARITH:func.3">'not'</a> <font color="Olive" title="b2">x</font> iff <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <font color="Maroon" title="c1">n</font> implies  for <font color="Olive" title="b1">x</font> being   <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of <font color="Maroon" title="c1">n</font>, <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font> holds <br/>( <font color="Maroon" title="c1">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binarith.html#K3" title="BINARITH:func.3">'not'</a> <font color="Olive" title="b1">x</font> iff <font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:32"/><span class="lab"><font color="Green" title="E26">A1</font></span>: 
<font color="Maroon" title="c2">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <font color="Maroon" title="c1">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font> being   <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of <font color="Maroon" title="c1">n</font>, <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font> holds <br/>( <font color="Maroon" title="c1">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binarith.html#K3" title="BINARITH:func.3">'not'</a> <font color="Olive" title="b1">x</font> iff <font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c3">x</font> be   <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of <font color="Maroon" title="c1">n</font>, <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font> implies ( <font color="Maroon" title="c1">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binarith.html#K3" title="BINARITH:func.3">'not'</a> <font color="Maroon" title="c3">x</font> iff <font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 ) )</span><br/>

<span class="kw">assume </span><a NAME="E2:32"/><span class="lab"><font color="Green" title="E27">A2</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binarith.html#K3" title="BINARITH:func.3">'not'</a> <font color="Maroon" title="c3">x</font> iff <font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 )</span><br/>

<span class="kw">thus </span><a NAME="E3:32"/>
( <font color="Maroon" title="c1">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binarith.html#K3" title="BINARITH:func.3">'not'</a> <font color="Maroon" title="c3">x</font> implies <font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 implies <font color="Maroon" title="c1">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binarith.html#K3" title="BINARITH:func.3">'not'</a> <font color="Maroon" title="c3">x</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span> for <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> $1 holds <br/> for <font color="Olive" title="b2">y</font> being   <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of $1, <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a>   st <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> $1 &amp; $1 <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binarith.html#K3" title="BINARITH:func.3">'not'</a> <font color="Olive" title="b2">y</font> holds <br/><font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> $1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1;<br/>
<span class="kw">assume </span><a NAME="E1:32_1"/><span class="lab"><font color="Green" title="E28">A3</font></span>: 
<font color="Maroon" title="c1">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binarith.html#K3" title="BINARITH:func.3">'not'</a> <font color="Maroon" title="c3">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span><br/>

<a NAME="E2:32_1"/><span class="lab"><font color="Green" title="E29">A4</font></span>: 
 for <font color="Olive" title="b1">m</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">m</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">m</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">m</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:32_1_1"/><span class="lab"><font color="Green" title="E30">A5</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">m</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<a NAME="E2:32_1_1"/><span class="lab"><font color="Green" title="E31">A6</font></span>: 
<font color="Maroon" title="c4">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 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"><font color="Maroon" title="c4">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a></span>;<br/>
<a NAME="E3:32_1_1"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T2" title="NAT_1:th.2">NAT_1:2</a></span>;<br/>
<a NAME="E4:32_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A7</font></span>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/>
<a NAME="E5:32_1_1"/>
 <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c4">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="binari_3.html#T4" target="_self" title="BINARI_3:th.4">Th5</a></span>;<br/>
<a NAME="E6:32_1_1"/><span class="kw">then </span>
 <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c4">m</font> is    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">y1</font> =  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c4">m</font> as   <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of <font color="Maroon" title="c4">m</font>, <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a>  ;<br/>
<span class="kw">let </span><font color="Maroon" title="c6">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> implies  for <font color="Olive" title="b1">y</font> being   <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of <font color="Maroon" title="c4">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1, <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a>   st <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binarith.html#K3" title="BINARITH:func.3">'not'</a> <font color="Olive" title="b1">y</font> holds <br/><font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 )</span><br/>

<span class="kw">assume </span><a NAME="E8:32_1_1"/><span class="lab"><font color="Green" title="E33">A8</font></span>: 
<font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">y</font> being   <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of <font color="Maroon" title="c4">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1, <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a>   st <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binarith.html#K3" title="BINARITH:func.3">'not'</a> <font color="Olive" title="b1">y</font> holds <br/><font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span><br/>

<span class="kw">let </span><font color="Maroon" title="c7">y</font> be   <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of <font color="Maroon" title="c4">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1, <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binarith.html#K3" title="BINARITH:func.3">'not'</a> <font color="Maroon" title="c7">y</font> implies <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E9:32_1_1"/><span class="lab"><font color="Green" title="E34">A9</font></span>: 
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <span class="kw">and </span><br/><a NAME="E10:32_1_1"/><span class="lab"><font color="Green" title="E35">A10</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binarith.html#K3" title="BINARITH:func.3">'not'</a> <font color="Maroon" title="c7">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span><br/>

<a NAME="E11:32_1_1"/><span class="lab"><font color="Green" title="E36">A11</font></span>: 
<font color="Maroon" title="c7">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:32_1_1"><span class="lab"><font color="Green" title="E34">A9</font></span></a>, <a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="funcop_1.html#T7" title="FUNCOP_1:th.7">FUNCOP_1:7</a></span>;<br/>
<a NAME="E12:32_1_1"/><span class="lab"><font color="Green" title="E37">A12</font></span>: 
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">y1</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:32_1_1"><span class="lab"><font color="Green" title="E34">A9</font></span></a>, <a class="ref" href="finseq_2.html#T60" title="FINSEQ_2:th.60">FINSEQ_2:60</a></span>;<br/>
<a NAME="E13:32_1_1"/><span class="lab"><font color="Green" title="E38">A13</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D7" title="CARD_1:def.7">CARD_1:def 7</a></span>;<br/>
<a NAME="E14:32_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="binarith.html#K3" title="BINARITH:func.3">'not'</a> <font color="Maroon" title="c7">y</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D7" title="CARD_1:def.7">CARD_1:def 7</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E39">A14</font></span>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><a href="binarith.html#K3" title="BINARITH:func.3">'not'</a> <font color="Maroon" title="c7">y</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:32_1_1"><span class="lab"><font color="Green" title="E35">A10</font></span></a>, <a class="txt" href="#E4:32_1_1"><span class="lab"><font color="Green" title="E32">A7</font></span></a>, <a class="ref" href="finseq_4.html#T15" title="FINSEQ_4:th.15">FINSEQ_4:15</a></span>
<br/>.= 
 <a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">y</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:32_1_1"><span class="lab"><font color="Green" title="E31">A6</font></span></a>, <a class="ref" href="binarith.html#D1" title="BINARITH:def.1">BINARITH:def 1</a></span>
<br/>.= 
 <a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:32_1_1"><span class="lab"><font color="Green" title="E38">A13</font></span></a>, <a class="txt" href="#E4:32_1_1"><span class="lab"><font color="Green" title="E32">A7</font></span></a>, <a class="txt" href="#E11:32_1_1"><span class="lab"><font color="Green" title="E36">A11</font></span></a>, <a class="ref" href="finseq_4.html#T15" title="FINSEQ_4:th.15">FINSEQ_4:15</a></span>
<br/>.= 
 <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 

;<br/>
<a NAME="E16:32_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">m</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p3">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p1"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:32_1_1"><span class="lab"><font color="Green" title="E33">A8</font></span></a>, <a class="txt" href="binari_3.html#E31"><span class="lab"><font color="Green" title="E24">Lm3</font></span></a>, <a class="ref" href="binari_3.html#T26" target="_self" title="BINARI_3:th.26">Th27</a></span>;<br/>
<a NAME="E17:32_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">m</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p3">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p1"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="binarith.html#K3" title="BINARITH:func.3">'not'</a> <font color="Maroon" title="c5">y1</font></span>)</span> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p1"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a></span>)</span></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:32_1_1"><span class="lab"><font color="Green" title="E35">A10</font></span></a>, <a class="txt" href="#E12:32_1_1"><span class="lab"><font color="Green" title="E37">A12</font></span></a>, <a class="ref" href="binari_2.html#T9" title="BINARI_2:th.9">BINARI_2:9</a></span>;<br/>
<a NAME="E18:32_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A15</font></span>: 
<font color="Maroon" title="c4">m</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <font color="Maroon" title="c4">m</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binarith.html#K3" title="BINARITH:func.3">'not'</a> <font color="Maroon" title="c5">y1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T17" title="FINSEQ_2:th.17">FINSEQ_2:17</a></span>;<br/>
<a NAME="E19:32_1_1"/>
<font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <font color="Maroon" title="c4">m</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:32_1_1"><span class="lab"><font color="Green" title="E33">A8</font></span></a>, <a class="ref" href="power.html#T27" title="POWER:th.27">POWER:27</a></span>;<br/>
<a NAME="E20:32_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <font color="Maroon" title="c4">m</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T25" title="POWER:th.25">POWER:25</a></span>;<br/>
<a NAME="E21:32_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <font color="Maroon" title="c4">m</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <font color="Maroon" title="c4">m</font></span>)</span>
 
;<br/>
<a NAME="E22:32_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <font color="Maroon" title="c4">m</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <font color="Maroon" title="c4">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T19" title="XREAL_1:th.19">XREAL_1:19</a></span>;<br/>
<a NAME="E23:32_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <font color="Maroon" title="c4">m</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <font color="Maroon" title="c4">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:32_1_1"><span class="lab"><font color="Green" title="E39">A14</font></span></a>, <a class="ref" href="binari_3.html#T26" target="_self" title="BINARI_3:th.26">Th27</a>, <a class="ref" href="xreal_1.html#T233" title="XREAL_1:th.233">XREAL_1:233</a></span>;<br/>
<a NAME="E24:32_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <font color="Maroon" title="c4">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <font color="Maroon" title="c4">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32_1_1"><span class="lab"><font color="Green" title="E30">A5</font></span></a>, <a class="txt" href="#E18:32_1_1"><span class="lab"><font color="Green" title="E40">A15</font></span></a></span>;<br/>
<a NAME="E25:32_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <font color="Maroon" title="c4">m</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <font color="Maroon" title="c4">m</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:32_1_1"><span class="lab"><font color="Green" title="E39">A14</font></span></a>, <a class="ref" href="binari_3.html#T26" target="_self" title="BINARI_3:th.26">Th27</a>, <a class="ref" href="xreal_1.html#T233" title="XREAL_1:th.233">XREAL_1:233</a></span>;<br/>
<span class="kw">hence </span><font color="Maroon" title="c6">k</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <font color="Maroon" title="c4">m</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> 2</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <font color="Maroon" title="c4">m</font></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T25" title="POWER:th.25">POWER:25</a></span>
<br/>.= 
<span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">m</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T27" title="POWER:th.27">POWER:27</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E3:32_1"/><span class="lab"><font color="Green" title="E30">A16</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">k</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> 1 implies  for <font color="Olive" title="b1">y</font> being   <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of 1, <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a>   st <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> 1 &amp; 1 <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binarith.html#K3" title="BINARITH:func.3">'not'</a> <font color="Olive" title="b1">y</font> holds <br/><font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 )</span><br/>

<a NAME="E1:32_1_2"/><span class="lab"><font color="Green" title="E31">A17</font></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> <span class="p1">(<span class="default">1 <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c4">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D7" title="CARD_1:def.7">CARD_1:def 7</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:32_1_2"/>
<font color="Maroon" title="c4">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">y</font> being   <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of 1, <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a>   st <font color="Olive" title="b1">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> 1 &amp; 1 <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binarith.html#K3" title="BINARITH:func.3">'not'</a> <font color="Olive" title="b1">y</font> holds <br/><font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span><br/>

<a NAME="E3:32_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A18</font></span>: 
<font color="Maroon" title="c4">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T25" title="POWER:th.25">POWER:25</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c5">y</font> be   <a href="finseq_2.html#NM4" title="FINSEQ_2:NM.4">Tuple</a> of 1, <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> 1 &amp; 1 <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binarith.html#K3" title="BINARITH:func.3">'not'</a> <font color="Maroon" title="c5">y</font> implies <font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 )</span><br/>

<span class="kw">assume </span><a NAME="E4:32_1_2"/>
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not 1 <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binarith.html#K3" title="BINARITH:func.3">'not'</a> <font color="Maroon" title="c5">y</font> or <font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 )</span><br/>

<a NAME="E5:32_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A19</font></span>: 
<font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T59" title="FINSEQ_2:th.59">FINSEQ_2:59</a></span>;<br/>
<span class="kw">assume </span><a NAME="E6:32_1_2"/><span class="lab"><font color="Green" title="E34">A20</font></span>: 
1 <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binarith.html#K3" title="BINARITH:func.3">'not'</a> <font color="Maroon" title="c5">y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span><br/>

<a NAME="E7:32_1_2"/><span class="lab"><font color="Green" title="E35">A21</font></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> 1
 
<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/>
<span class="lab"><font color="Green" title="E36">A22</font></span>: 1 = 
<span class="p1"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default">1</span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T40" title="FINSEQ_1:th.40">FINSEQ_1:40</a></span>
<br/>.= 
<span class="p1">(<span class="default">1 <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c4">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:32_1_2"><span class="lab"><font color="Green" title="E33">A19</font></span></a>, <a class="txt" href="#E6:32_1_2"><span class="lab"><font color="Green" title="E34">A20</font></span></a>, <a class="txt" href="#E1:32_1_2"><span class="lab"><font color="Green" title="E31">A17</font></span></a>, <a class="ref" href="binari_3.html#T14" target="_self" title="BINARI_3:th.14">Th15</a>, <a class="ref" href="finseq_4.html#T15" title="FINSEQ_4:th.15">FINSEQ_4:15</a></span>
<br/>.= 
 <a href="funcop_1.html#K15" title="FUNCOP_1:func.15">IFEQ</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p3">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <span class="p4">(<span class="default">1 <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a>,<a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:32_1_2"><span class="lab"><font color="Green" title="E35">A21</font></span></a>, <a class="ref" href="binari_3.html#D1" target="_self" title="BINARI_3:def.1">Def1</a></span>
;<br/>
<a NAME="E9:32_1_2"/>
<font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:32_1_2_2"/>
<font color="Maroon" title="c4">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:32_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:32_1_2"><span class="lab"><font color="Green" title="E32">A18</font></span></a>, <a class="ref" href="nat_1.html#T23" title="NAT_1:th.23">NAT_1:23</a></span>;<br/>
<a NAME="E3:32_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">k</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <span class="p2">(<span class="default">1 <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T27" title="NAT_D:th.27">NAT_D:27</a>, <a class="ref" href="power.html#T34" title="POWER:th.34">POWER:34</a></span>;<br/>
<a NAME="E4:32_1_2_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <span class="p3">(<span class="default">1 <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T26" title="NAT_D:th.26">NAT_D:26</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:32_1_2_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:32_1_2"><span class="lab"><font color="Green" title="E36">A22</font></span></a>, <a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><font color="Maroon" title="c4">k</font> = 
<span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1

<br/>.= 
<span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T25" title="POWER:th.25">POWER:25</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E4:32_1"/>
 for <font color="Olive" title="b1">m</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">m</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S10" title="NAT_1:sch.10">NAT_1:sch 10</a>(<a class="txt" href="#E3:32_1"><span class="lab"><font color="Green" title="E30">A16</font></span></a>, <a class="txt" href="#E2:32_1"><span class="lab"><font color="Green" title="E29">A4</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E5:32_1"/>
<font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="txt" href="#E2:32"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="txt" href="#E1:32_1"><span class="lab"><font color="Green" title="E28">A3</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E4:32"/><span class="lab"><font color="Green" title="E28">A23</font></span>: 
<font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binarith.html#K3" title="BINARITH:func.3">'not'</a> <font color="Maroon" title="c3">x</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c4">i</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> <font color="Maroon" title="c1">n</font> implies <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="c2">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="binarith.html#K3" title="BINARITH:func.3">'not'</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> )</span><br/><a NAME="E1:32_2"/><span class="lab"><font color="Green" title="E29">A24</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D7" title="CARD_1:def.7">CARD_1:def 7</a></span>;<br/><a NAME="E2:32_2"/>
2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T34" title="POWER:th.34">POWER:34</a></span>;<br/><a NAME="E3:32_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A25</font></span>: 
2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E4:32_2"/><span class="lab"><font color="Green" title="E31">A26</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="binarith.html#K3" title="BINARITH:func.3">'not'</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#D7" title="CARD_1:def.7">CARD_1:def 7</a></span>;<br/><a NAME="E5:32_2"/><span class="lab"><font color="Green" title="E32">A27</font></span>: 
2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T34" title="POWER:th.34">POWER:34</a></span>;<br/><a NAME="E6:32_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A28</font></span>: 
2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c5">z</font> = <font color="Maroon" title="c4">i</font> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> ;<br/><span class="kw">assume </span><a NAME="E8:32_2"/><span class="lab"><font color="Green" title="E34">A29</font></span>: 
<font color="Maroon" title="c4">i</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> <font color="Maroon" title="c1">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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="c2">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="binarith.html#K3" title="BINARITH:func.3">'not'</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font></span><br/><a NAME="E9:32_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A30</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E10:32_2"/>
<font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:32_2"><span class="lab"><font color="Green" title="E34">A29</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E11:32_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E12:32_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A31</font></span>: 
<font color="Maroon" title="c4">i</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/><a NAME="E13:32_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:32_2"><span class="lab"><font color="Green" title="E34">A29</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E14:32_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A32</font></span>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:32_2"><span class="lab"><font color="Green" title="E36">A31</font></span></a>, <a class="ref" href="xreal_1.html#T233" title="XREAL_1:th.233">XREAL_1:233</a></span>;<br/><a NAME="E15:32_2"/><span class="kw">then </span>
<font color="Maroon" title="c1">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T20" title="XREAL_1:th.20">XREAL_1:20</a></span>;<br/><a NAME="E16:32_2"/><span class="kw">then </span>
<font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:32_2"><span class="lab"><font color="Green" title="E37">A32</font></span></a>, <a class="ref" href="xreal_1.html#T233" title="XREAL_1:th.233">XREAL_1:233</a></span>;<br/><a NAME="E17:32_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A33</font></span>: 
<span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_2.html#T17" title="NAT_2:th.17">NAT_2:17</a></span>;<br/><a NAME="E18:32_2"/>
2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="power.html#T34" title="POWER:th.34">POWER:34</a></span>;<br/><a NAME="E19:32_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A34</font></span>: 
2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="kw">then </span><font color="Maroon" title="c2">k</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:32"><span class="lab"><font color="Green" title="E28">A23</font></span></a>, <a class="ref" href="xreal_1.html#T233" title="XREAL_1:th.233">XREAL_1:233</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:32_2"><span class="lab"><font color="Green" title="E30">A25</font></span></a>, <a class="txt" href="#E14:32_2"><span class="lab"><font color="Green" title="E37">A32</font></span></a>, <a class="txt" href="#E19:32_2"><span class="lab"><font color="Green" title="E39">A34</font></span></a>, <a class="ref" href="nat_2.html#T11" title="NAT_2:th.11">NAT_2:11</a>, <a class="ref" href="nat_2.html#T15" title="NAT_2:th.15">NAT_2:15</a></span>
<br/>.= 
<span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:32_2"><span class="lab"><font color="Green" title="E37">A32</font></span></a>, <a class="ref" href="nat_2.html#T16" title="NAT_2:th.16">NAT_2:16</a></span>
<br/>.= 
<span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:32_2"><span class="lab"><font color="Green" title="E33">A28</font></span></a>, <a class="ref" href="xreal_1.html#T233" title="XREAL_1:th.233">XREAL_1:233</a></span>
;<br/><a NAME="E21:32_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A35</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p2">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:32_2"><span class="lab"><font color="Green" title="E38">A33</font></span></a>, <a class="txt" href="#E5:32_2"><span class="lab"><font color="Green" title="E32">A27</font></span></a>, <a class="ref" href="nat_2.html#T18" title="NAT_2:th.18">NAT_2:18</a></span>;<br/><a NAME="E22:32_2"/><span class="lab"><font color="Green" title="E41">A36</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:32"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="txt" href="#E8:32_2"><span class="lab"><font color="Green" title="E34">A29</font></span></a>, <a class="ref" href="funcop_1.html#T7" title="FUNCOP_1:th.7">FUNCOP_1:7</a></span>;<br/><a NAME="E23:32_2"/><span class="lab"><font color="Green" title="E42">A37</font></span>: 
<font color="Maroon" title="c4">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:32_2"><span class="lab"><font color="Green" title="E34">A29</font></span></a>, <a class="ref" href="finseq_1.html#T1" title="FINSEQ_1:th.1">FINSEQ_1:1</a></span>;<br/><a NAME="E24:32_2"/>
<font color="Maroon" title="c4">i</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> <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="c2">k</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:32_2"><span class="lab"><font color="Green" title="E34">A29</font></span></a>, <a class="ref" href="card_1.html#D7" title="CARD_1:def.7">CARD_1:def 7</a></span>;<br/><a NAME="E25:32_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">i</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> <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="c2">k</font></span>)</span>
 <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/><span class="kw">hence </span><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="c2">k</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font> = 
<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="c2">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">i</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>
<br/>.= 
 <a href="funcop_1.html#K15" title="FUNCOP_1:func.15">IFEQ</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">k</font> <a href="nat_d.html#K3" title="NAT_D:func.3">div</a> <span class="p3">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <span class="p4">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_d.html#K4" title="NAT_D:func.4">mod</a> 2</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a>,<a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:32_2"><span class="lab"><font color="Green" title="E34">A29</font></span></a>, <a class="ref" href="binari_3.html#D1" target="_self" title="BINARI_3:def.1">Def1</a></span>
<br/>.= 
 <a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:32_2"><span class="lab"><font color="Green" title="E40">A35</font></span></a>, <a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></span>
<br/>.= 
 <a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">z</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:32_2"><span class="lab"><font color="Green" title="E29">A24</font></span></a>, <a class="txt" href="#E9:32_2"><span class="lab"><font color="Green" title="E35">A30</font></span></a>, <a class="txt" href="#E23:32_2"><span class="lab"><font color="Green" title="E42">A37</font></span></a>, <a class="txt" href="#E22:32_2"><span class="lab"><font color="Green" title="E41">A36</font></span></a>, <a class="ref" href="finseq_4.html#T15" title="FINSEQ_4:th.15">FINSEQ_4:15</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="binarith.html#K3" title="BINARITH:func.3">'not'</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">z</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:32_2"><span class="lab"><font color="Green" title="E34">A29</font></span></a>, <a class="ref" href="binarith.html#D1" title="BINARITH:def.1">BINARITH:def 1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="binarith.html#K3" title="BINARITH:func.3">'not'</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">i</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:32_2"><span class="lab"><font color="Green" title="E31">A26</font></span></a>, <a class="txt" href="#E9:32_2"><span class="lab"><font color="Green" title="E35">A30</font></span></a>, <a class="txt" href="#E23:32_2"><span class="lab"><font color="Green" title="E42">A37</font></span></a>, <a class="ref" href="finseq_4.html#T15" title="FINSEQ_4:th.15">FINSEQ_4:15</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E6:32"/>
<font color="Maroon" title="c1">n</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> <font color="Maroon" title="c2">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binarith.html#K3" title="BINARITH:func.3">'not'</a> <font color="Maroon" title="c3">x</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T119" title="FINSEQ_2:th.119">FINSEQ_2:119</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
