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

<span class="kw">let </span><font color="Maroon" title="c1">T</font> be   <a href="bintree2.html#V1" title="BINTREE2:attr.1">full</a>  <a href="trees_1.html#NM1" title="TREES_1:NM.1">Tree</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">n</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   <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="bintree2.html#K1" title="BINTREE2:func.1">NumberOnLevel</a> (<font color="Olive" title="b1">n</font>,<font color="Maroon" title="c1">T</font>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">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">  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</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="c2">n</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="bintree2.html#K1" title="BINTREE2:func.1">NumberOnLevel</a> (<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">T</font>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c3">y</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">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <font color="Maroon" title="c2">n</font></span>)</span> or <font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="bintree2.html#K1" title="BINTREE2:func.1">NumberOnLevel</a> (<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">T</font>)</span>)</span> )</span><br/>





<span class="kw">assume </span><a NAME="E1:21"/>
<font color="Maroon" title="c3">y</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">2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <font color="Maroon" title="c2">n</font></span>)</span>
 ; <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">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="bintree2.html#K1" title="BINTREE2:func.1">NumberOnLevel</a> (<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">T</font>)</span>)</span></span><br/>

<a NAME="E2:21"/><span class="kw">then </span>
<font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">k</font> where <font color="Olive" title="b1">k</font> is    <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>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <font color="Maroon" title="c2">n</font> ) </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D1" title="FINSEQ_1:def.1">FINSEQ_1:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">k</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> <span class="kw"> such that </span><br/><a NAME="E4:21"/><span class="lab"><font color="Green" title="E16">A1</font></span>: 
<font color="Maroon" title="c4">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">y</font>
 <span class="kw">and </span><br/><a NAME="E5:21"/><span class="lab"><font color="Green" title="E17">A2</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">k</font>
 <span class="kw">and </span><br/><a NAME="E6:21"/><span class="lab"><font color="Green" title="E18">A3</font></span>: 
<font color="Maroon" title="c4">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <font color="Maroon" title="c2">n</font>
 ;<br/>
<a NAME="E7:21"/><span class="lab"><font color="Green" title="E19">A4</font></span>: 
<font color="Maroon" title="c4">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <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="#E5:21"><span class="lab"><font color="Green" title="E17">A2</font></span></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">t</font> =  <a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">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="c4">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>;<br/>
<span class="lab"><font color="Green" title="E20">A5</font></span>:  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">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="c4">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">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="c4">k</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="ref" href="finseq_5.html#D3" title="FINSEQ_5:def.3">FINSEQ_5:def 3</a></span>
<br/>.= 
<font color="Maroon" title="c2">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="E9:21"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p2">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">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="c4">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#D3" title="FINSEQ_5:def.3">FINSEQ_5:def 3</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">F</font> =  <a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">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="c4">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <font color="Maroon" title="c2">n</font> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T92" title="FINSEQ_2:th.92">FINSEQ_2:92</a></span>;<br/>
<a NAME="E11:21"/>
<font color="Maroon" title="c1">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="bintree2.html#D2" target="_self" title="BINTREE2:def.2">Def2</a></span>;<br/>
<a NAME="E12:21"/><span class="kw">then </span>
 <a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">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="c4">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">T</font>
 
<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/>
<a NAME="E13:21"/><span class="kw">then </span>
 <a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">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="c4">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is    <a href="bintree2.html#M1" title="BINTREE2:mode.1">Element</a> of <font color="Maroon" title="c1">T</font> :  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">n</font> </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:21"><span class="lab"><font color="Green" title="E20">A5</font></span></a></span>;<br/>
<a NAME="E14:21"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A6</font></span>: 
 <a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">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="c4">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">T</font> <a href="trees_2.html#K2" title="TREES_2:func.2">-level</a> <font color="Maroon" title="c2">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="trees_2.html#D6" title="TREES_2:def.6">TREES_2:def 6</a></span>;<br/>
<a NAME="E15:21"/><span class="kw">then </span><span class="lab"><font color="Green" title="E22">A7</font></span>: 
 <a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">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="c4">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="bintree2.html#K1" title="BINTREE2:func.1">NumberOnLevel</a> (<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">T</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E16:21"/>
<font color="Maroon" title="c4">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="c2">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:21"><span class="lab"><font color="Green" title="E18">A3</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E17:21"/><span class="kw">then </span>
<font color="Maroon" title="c4">k</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> 2 <a href="series_1.html#K4" title="SERIES_1:func.4">to_power</a> <font color="Maroon" title="c2">n</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="E18:21"/><span class="kw">then </span><span class="lab"><font color="Green" title="E23">A8</font></span>: 
<font color="Maroon" title="c4">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <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="c2">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:21"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="ref" href="xreal_0.html#D2" title="XREAL_0:def.2">XREAL_0:def 2</a></span>;<br/>
<span class="p1">(<span class="default"><a href="bintree2.html#K1" title="BINTREE2:func.1">NumberOnLevel</a> (<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">T</font>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_5.html#K4" title="FINSEQ_5:func.4">Rev</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">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="c4">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:21"><span class="lab"><font color="Green" title="E21">A6</font></span></a>, <a class="ref" href="bintree2.html#D1" target="_self" title="BINTREE2:def.1">Def1</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="binarith.html#K6" title="BINARITH:func.6">Absval</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">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="c4">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_6.html#T25" title="FINSEQ_6:th.25">FINSEQ_6:25</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:21"><span class="lab"><font color="Green" title="E23">A8</font></span></a>, <a class="ref" href="binari_3.html#T35" title="BINARI_3:th.35">BINARI_3:35</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:21"><span class="lab"><font color="Green" title="E19">A4</font></span></a>, <a class="ref" href="xreal_0.html#D2" title="XREAL_0:def.2">XREAL_0:def 2</a></span>
<br/>.= 
<font color="Maroon" title="c3">y</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:21"><span class="lab"><font color="Green" title="E16">A1</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E20:21"/>
<font color="Maroon" title="c3">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="bintree2.html#K1" title="BINTREE2:func.1">NumberOnLevel</a> (<font color="Maroon" title="c2">n</font>,<font color="Maroon" title="c1">T</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:21"><span class="lab"><font color="Green" title="E22">A7</font></span></a>, <a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</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/>


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