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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="binarith.html#NM1">Tuple</a> of <font color="Maroon">c<sub>1</sub></font>,<a href="margrel1.html#K6">BOOLEAN</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="margrel1.html#K6">BOOLEAN</a> ;<br/>





<a NAME="E1:47"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binarith.html#K8" target="_self">Binary</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binarith.html#K12" target="_self">^</a> <span class="p2"><a href="binarith.html#K13" target="_self">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="cqc_lang.html#K2">IFEQ</a> <font color="Maroon">c<sub>3</sub></font>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p4">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="binarith.html#K13" target="_self">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="cqc_lang.html#K2">IFEQ</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binarith.html#K12" target="_self">^</a> <span class="p3"><a href="binarith.html#K13" target="_self">&lt;*</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="binarith.html#K13" target="_self">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="binarith.html#K5" target="_self">-'</a> 1</span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:47_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
 ;<br/>

<a NAME="E2:47_1"/><i><font color="Green">E33</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="setwiseo.html#K2">{.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span><a href="setwiseo.html#K2">.}</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T11">FINSEQ_1:11</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:47_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="setwiseo.html#K2">{.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span><a href="setwiseo.html#K2">.}</a></span> )
 </span><b>by </b><i><a class="txt" href="binarith.html#E1:47_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="binarith.html#E2:47_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_1_1"><b>suppose </b></a><a NAME="E1:47_1_1_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binarith.html#K8" target="_self">Binary</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binarith.html#K12" target="_self">^</a> <span class="p2"><a href="binarith.html#K13" target="_self">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="cqc_lang.html#K2">IFEQ</a> <font color="Maroon">c<sub>3</sub></font>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p4">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="binarith.html#K13" target="_self">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>4</sub></font> = 
<span class="p1">(<span class="default"><a href="binarith.html#K8" target="_self">Binary</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>4</sub></font>
<b>by </b><i><a class="ref" href="binarith.html#T2" target="_self">Th2</a></i>
<br/>.= 
 <a href="cqc_lang.html#K2">IFEQ</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binarith.html#K5" target="_self">-'</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="binarith.html#E1:47_1_1_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="binarith.html#D6" target="_self">Def6</a></i>
<br/>.= 
 <a href="cqc_lang.html#K2">IFEQ</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binarith.html#K12" target="_self">^</a> <span class="p3"><a href="binarith.html#K13" target="_self">&lt;*</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="binarith.html#K13" target="_self">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binarith.html#K5" target="_self">-'</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="binarith.html#E1:47_1_1_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="binarith.html#T2" target="_self">Th2</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_1_2"><b>suppose </b></a><a NAME="E1:47_1_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="setwiseo.html#K2">{.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span><a href="setwiseo.html#K2">.}</a></span>
 ;<br/><div class="add"><a NAME="E2:47_1_1_2"/><b>then </b><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="binop_2.html#K23">+</a> 1
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binarith.html#K8" target="_self">Binary</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binarith.html#K12" target="_self">^</a> <span class="p2"><a href="binarith.html#K13" target="_self">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="cqc_lang.html#K2">IFEQ</a> <font color="Maroon">c<sub>3</sub></font>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p4">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="binarith.html#K13" target="_self">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>4</sub></font> = 
 <a href="cqc_lang.html#K2">IFEQ</a> <font color="Maroon">c<sub>3</sub></font>,<a href="margrel1.html#K7">FALSE</a> ,0,<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>
<b>by </b><i><a class="ref" href="binarith.html#T3" target="_self">Th3</a></i>
<br/>.= 
 <a href="cqc_lang.html#K2">IFEQ</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binarith.html#K12" target="_self">^</a> <span class="p3"><a href="binarith.html#K13" target="_self">&lt;*</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="binarith.html#K13" target="_self">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<a href="margrel1.html#K7">FALSE</a> ,0,<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>
<b>by </b><i><a class="txt" href="binarith.html#E2:47_1_1_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="binarith.html#T3" target="_self">Th3</a></i>
<br/>.= 
 <a href="cqc_lang.html#K2">IFEQ</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binarith.html#K12" target="_self">^</a> <span class="p3"><a href="binarith.html#K13" target="_self">&lt;*</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="binarith.html#K13" target="_self">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p1">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="binarith.html#K5" target="_self">-'</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="binarith.html#E2:47_1_1_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="binarith.html#T39" target="_self">Th39</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>thus </b> <a href="binarith.html#K9" target="_self">Absval</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binarith.html#K12" target="_self">^</a> <span class="p2"><a href="binarith.html#K13" target="_self">&lt;*</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="binarith.html#K13" target="_self">*&gt;</a></span></span>)</span> = 
<a href="binop_2.html#K47">addnat</a>  <a href="finsop_1.html#K2">$$</a> <span class="p1">(<span class="default"><a href="binarith.html#K8" target="_self">Binary</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="binarith.html#K12" target="_self">^</a> <span class="p3"><a href="binarith.html#K13" target="_self">&lt;*</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="binarith.html#K13" target="_self">*&gt;</a></span></span>)</span></span>)</span>

<br/>.= 
<a href="binop_2.html#K47">addnat</a>  <a href="finsop_1.html#K2">$$</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binarith.html#K8" target="_self">Binary</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binarith.html#K12" target="_self">^</a> <span class="p2"><a href="binarith.html#K13" target="_self">&lt;*</a><span class="default"><span class="p3">(<span class="default"><a href="cqc_lang.html#K2">IFEQ</a> <font color="Maroon">c<sub>3</sub></font>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p4">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span><a href="binarith.html#K13" target="_self">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="txt" href="binarith.html#E1:47"><i><font color="Green">E31</font></i></a>, <a class="ref" href="binarith.html#D6" target="_self">Def6</a></i>
<br/>.= 
<a href="binop_2.html#K47">addnat</a>  <a href="binop_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="binop_2.html#K47">addnat</a>  <a href="finsop_1.html#K2">$$</a> <span class="p2">(<span class="default"><a href="binarith.html#K8" target="_self">Binary</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="cqc_lang.html#K2">IFEQ</a> <font color="Maroon">c<sub>3</sub></font>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finsop_1.html#T5">FINSOP_1:5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="binop_2.html#K47">addnat</a>  <a href="finsop_1.html#K2">$$</a> <span class="p2">(<span class="default"><a href="binarith.html#K8" target="_self">Binary</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K2">IFEQ</a> <font color="Maroon">c<sub>3</sub></font>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="binarith.html#E1"><i><font color="Green">Lemma1</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="binarith.html#K9" target="_self">Absval</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binop_2.html#K23">+</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K2">IFEQ</a> <font color="Maroon">c<sub>3</sub></font>,<a href="margrel1.html#K7">FALSE</a> ,0,<span class="p2">(<span class="default">2 <a href="series_1.html#K3">to_power</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>

;<br/>


</div>
