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<b>let </b><font color="Maroon" title="c1">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</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> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">i</font></span>)</span> <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">1</span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span><br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>) <b>-&gt; </b>  <a href="binarith.html#NM1" title="BINARITH:NM.1">Tuple</a> of <span class="p1">(<span class="default">$1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a>  = <span class="p1">(<span class="default">$1 <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> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> $1</span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c2">Bi</font> = <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">i</font>);<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1"/>
( <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c1">i</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><b>by </b><i><a class="ref" href="nat_1.html#T3" title="NAT_1:th.3">NAT_1:3</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1"><b>suppose </b></a><a NAME="E1:33_1_1"/><i><font color="Green" title="E27">A1</font></i>: 
<font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</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> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">i</font></span>)</span> <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">1</span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span><br/><div class="add"><b>reconsider </b><font color="Maroon" title="c3">i1</font> = <font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 as  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> ;<br/><a NAME="E3:33_1_1"/>
2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c3">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 <b>by </b><i><a class="txt" href="#E1:33_1_1"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="power.html#T30" title="POWER:th.30">POWER:30</a></i>;<br/><a NAME="E4:33_1_1"/><b>then </b><i><font color="Green" title="E28">A2</font></i>: 
1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c3">i1</font></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1
 ;<br/><a NAME="E5:33_1_1"/><i><font color="Green" title="E29">A3</font></i>: 
 <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c3">i1</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>
 <b>by </b><i><a class="txt" href="#E1:33_1_1"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="finseq_2.html#T73" title="FINSEQ_2:th.73">FINSEQ_2:73</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c4">x</font> =  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c3">i1</font> as   <a href="binarith.html#NM1" title="BINARITH:NM.1">Tuple</a> of <font color="Maroon" title="c3">i1</font>,<a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a>  <b>by </b><i><a class="txt" href="#E1:33_1_1"><i><font color="Green" title="E27">A1</font></i></a></i>;<br/><a NAME="E7:33_1_1"/><i><font color="Green" title="E30">A4</font></i>: 
 <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="#E1:33_1_1"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="finseq_2.html#T72" title="FINSEQ_2:th.72">FINSEQ_2:72</a></i>;<br/><a NAME="E8:33_1_1"/>
<font color="Maroon" title="c3">i1</font> <a href="binari_3.html#K1" title="BINARI_3:func.1">-BinarySequence</a> 1 <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="c4">x</font>
 <b>by </b><i><a class="txt" href="#E4:33_1_1"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="binari_3.html#E33"><i><font color="Green" title="E25">Lm4</font></i></a></i>;<br/><b>hence </b><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">i</font>) = 
<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>
<b>by </b><i><a class="txt" href="#E1:33_1_1"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="#E5:33_1_1"><i><font color="Green" title="E29">A3</font></i></a>, <a class="ref" href="binari_3.html#T15" target="_self" title="BINARI_3:th.15">Th15</a>, <a class="ref" href="power.html#T29" title="POWER:th.29">POWER:29</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">i</font></span>)</span> <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">1</span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span>
<b>by </b><i><a class="txt" href="#E7:33_1_1"><i><font color="Green" title="E30">A4</font></i></a>, <a class="ref" href="finseq_1.html#T47" title="FINSEQ_1:th.47">FINSEQ_1:47</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_2"><b>suppose </b></a><a NAME="E1:33_1_2"/>
<font color="Maroon" title="c1">i</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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c1">i</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> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">i</font></span>)</span> <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">1</span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span><br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c3">i'</font> = <font color="Maroon" title="c1">i</font> as  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> ;<br/><a NAME="E3:33_1_2"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c3">i'</font></span>)</span> <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">1</span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="binari_3.html#E30"><i><font color="Green" title="E22">Lm1</font></i></a></i>;<br/><b>hence </b><a NAME="E4:33_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c1">i</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> <span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <font color="Maroon" title="c1">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">i</font></span>)</span> <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">1</span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

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