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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b>  <a href="binarith.html#NM1">Tuple</a> of <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<a href="margrel1.html#K6">BOOLEAN</a>  = <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binari_3.html#K1" target="_self">-BinarySequence</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>2</sub></font> = <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>1</sub></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">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span><b>by </b><i><a class="ref" href="nat_1.html#T19">NAT_1:19</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">E28</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 as  non <a href="xboole_0.html#V1">empty</a> <a href="nat_1.html#NM1">Nat</a> ;<br/><a NAME="E3:33_1_1"/>
2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 2
 <b>by </b><i><a class="txt" href="binari_3.html#E1:33_1_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="power.html#T30">POWER:30</a></i>;<br/><a NAME="E4:33_1_1"/><b>then </b><i><font color="Green">E29</font></i>: 
1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1
 ;<br/><i><font color="Green">E30</font></i>:  <a href="euclid.html#K5">0*</a> <font color="Maroon">c<sub>3</sub></font> = 
1 <a href="finseqop.html#K4">|-&gt;</a> 0
<b>by </b><i><a class="txt" href="binari_3.html#E1:33_1_1"><i><font color="Green">E28</font></i></a></i>
<br/>.= 
<span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><a href="margrel1.html#K7">FALSE</a> </span><a href="binarith.html#K13">*&gt;</a></span>
<b>by </b><i><a class="ref" href="finseq_2.html#T73">FINSEQ_2:73</a>, <a class="ref" href="margrel1.html#D13">MARGREL1:def 13</a></i>
;<br/><b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> =  <a href="euclid.html#K5">0*</a> <font color="Maroon">c<sub>3</sub></font> as   <a href="binarith.html#NM1">Tuple</a> of <font color="Maroon">c<sub>3</sub></font>,<a href="margrel1.html#K6">BOOLEAN</a>  <b>by </b><i><a class="txt" href="binari_3.html#E1:33_1_1"><i><font color="Green">E28</font></i></a></i>;<br/><i><font color="Green">E31</font></i>:  <a href="euclid.html#K5">0*</a> <font color="Maroon">c<sub>1</sub></font> = 
0 <a href="finseqop.html#K4">|-&gt;</a> 0
<b>by </b><i><a class="txt" href="binari_3.html#E1:33_1_1"><i><font color="Green">E28</font></i></a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="finseq_2.html#T72">FINSEQ_2:72</a></i>
;<br/><a NAME="E8:33_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="binari_3.html#K1" target="_self">-BinarySequence</a> 1 <a href="hidden.html#R1">=</a>  <a href="binarith.html#K6">'not'</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="binari_3.html#E4:33_1_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="binari_3.html#E33"><i><font color="Green">Lemma26</font></i></a></i>;<br/><b>hence </b><font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>1</sub></font>) = 
<span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><a href="margrel1.html#K8">TRUE</a> </span><a href="binarith.html#K13">*&gt;</a></span>
<b>by </b><i><a class="txt" href="binari_3.html#E1:33_1_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="binari_3.html#E5:33_1_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="binari_3.html#T15" target="_self">Th15</a>, <a class="ref" href="power.html#T29">POWER:29</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default">1</span><a href="binarith.html#K13">*&gt;</a></span>
<b>by </b><i><a class="txt" href="binari_3.html#E7:33_1_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a>, <a class="ref" href="margrel1.html#D14">MARGREL1:def 14</a></i>
;<br/><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">c<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>1</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="nat_1.html#NM1">Nat</a> ;<br/><a NAME="E3:33_1_2"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>1</sub></font>) <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default">1</span><a href="binarith.html#K13">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="binari_3.html#E30"><i><font color="Green">Lemma23</font></i></a>, <a class="ref" href="margrel1.html#D14">MARGREL1:def 14</a></i>;<br/><b>hence </b><a NAME="E4:33_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binari_3.html#K1" target="_self">-BinarySequence</a> <span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default">1</span><a href="binarith.html#K13">*&gt;</a></span>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

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