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<b>let </b><font color="Maroon" title="c1">z1</font> be   <a href="binarith.html#NM1" title="BINARITH:NM.1">Tuple</a> of 2,<a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> </span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span> <a href="binarith.html#K10" title="BINARITH:func.10">^</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> </span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span> implies  <a href="binari_2.html#K3" title="BINARI_2:func.3">Intval</a> <font color="Maroon" title="c1">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/>

<b>assume </b><a NAME="E1:8"/><i><font color="Green" title="E5">A1</font></i>: 
<font color="Maroon" title="c1">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> </span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span> <a href="binarith.html#K10" title="BINARITH:func.10">^</a> <span class="p1"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> </span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="binari_2.html#K3" title="BINARI_2:func.3">Intval</a> <font color="Maroon" title="c1">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/>

<b>consider </b><font color="Maroon" title="c2">k1</font>, <font color="Maroon" title="c3">k2</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> <b> such that </b><br/><a NAME="E3:8"/><i><font color="Green" title="E6">A2</font></i>: 
 <a href="binarith.html#K6" title="BINARITH:func.6">Binary</a> <font color="Maroon" title="c1">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><font color="Maroon" title="c2">k1</font>,<font color="Maroon" title="c3">k2</font></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="finseq_2.html#T120" title="FINSEQ_2:th.120">FINSEQ_2:120</a></i>;<br/>
<a NAME="E4:8"/>
<font color="Maroon" title="c1">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> ,<a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> </span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="binari_2.html#E1:8"><i><font color="Green" title="E5">A1</font></i></a>, <a class="ref" href="finseq_1.html#D9" title="FINSEQ_1:def.9">FINSEQ_1:def 9</a></i>;<br/>
<a NAME="E5:8"/><b>then </b><i><font color="Green" title="E7">A3</font></i>: 
( <font color="Maroon" title="c1">z1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>  &amp; <font color="Maroon" title="c1">z1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a>  )
 
<b>by </b><i><a class="ref" href="finseq_4.html#T26" title="FINSEQ_4:th.26">FINSEQ_4:26</a></i>;<br/>
<a NAME="E6:8"/><i><font color="Green" title="E8">A4</font></i>: 
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
 
<b>by </b><i><a class="ref" href="finseq_1.html#T5" title="FINSEQ_1:th.5">FINSEQ_1:5</a></i>;<br/>
<a NAME="E7:8"/>
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 2
 
<b>by </b><i><a class="ref" href="finseq_1.html#T7" title="FINSEQ_1:th.7">FINSEQ_1:7</a></i>;<br/>
<b>then </b><i><font color="Green" title="E9">A5</font></i>: <span class="p1">(<span class="default"><a href="binarith.html#K6" title="BINARITH:func.6">Binary</a> <font color="Maroon" title="c1">z1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 = 
 <a href="funcop_1.html#K15" title="FUNCOP_1:func.15">IFEQ</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1</span>)</span>,<a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p2">(<span class="default">1 <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="binari_2.html#E6:8"><i><font color="Green" title="E8">A4</font></i></a>, <a class="ref" href="binarith.html#D6" title="BINARITH:def.6">BINARITH:def 6</a></i>
<br/>.= 
2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p1">(<span class="default">1 <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="binari_2.html#E5:8"><i><font color="Green" title="E7">A3</font></i></a>, <a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></i>
;<br/>
<a NAME="E9:8"/>
1 <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
;<br/>
<a NAME="E10:8"/><b>then </b>
1 <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <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="ref" href="binarith.html#D3" title="BINARITH:def.3">BINARITH:def 3</a></i>;<br/>
<a NAME="E11:8"/><b>then </b><i><font color="Green" title="E10">A6</font></i>: 
<span class="p1">(<span class="default"><a href="binarith.html#K6" title="BINARITH:func.6">Binary</a> <font color="Maroon" title="c1">z1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="binari_2.html#E8:8"><i><font color="Green" title="E9">A5</font></i></a>, <a class="ref" href="power.html#T29" title="POWER:th.29">POWER:29</a></i>;<br/>
<a NAME="E12:8"/><i><font color="Green" title="E11">A7</font></i>: 
<span class="p1">(<span class="default"><a href="binarith.html#K6" title="BINARITH:func.6">Binary</a> <font color="Maroon" title="c1">z1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">k1</font>
 
<b>by </b><i><a class="txt" href="binari_2.html#E3:8"><i><font color="Green" title="E6">A2</font></i></a>, <a class="ref" href="finseq_4.html#T26" title="FINSEQ_4:th.26">FINSEQ_4:26</a></i>;<br/>
<a NAME="E13:8"/>
2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 2
 
<b>by </b><i><a class="ref" href="finseq_1.html#T5" title="FINSEQ_1:th.5">FINSEQ_1:5</a></i>;<br/>
<b>then </b><i><font color="Green" title="E12">A8</font></i>: <span class="p1">(<span class="default"><a href="binarith.html#K6" title="BINARITH:func.6">Binary</a> <font color="Maroon" title="c1">z1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 = 
 <a href="funcop_1.html#K15" title="FUNCOP_1:func.15">IFEQ</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">z1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2</span>)</span>,<a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> ,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p1">(<span class="default">2 <a href="series_1.html#K3" title="SERIES_1:func.3">to_power</a> <span class="p2">(<span class="default">2 <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#D6" title="BINARITH:def.6">BINARITH:def 6</a></i>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="txt" href="binari_2.html#E5:8"><i><font color="Green" title="E7">A3</font></i></a>, <a class="ref" href="funcop_1.html#D8" title="FUNCOP_1:def.8">FUNCOP_1:def 8</a></i>
;<br/>
<a NAME="E15:8"/>
<span class="p1">(<span class="default"><a href="binarith.html#K6" title="BINARITH:func.6">Binary</a> <font color="Maroon" title="c1">z1</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">k2</font>
 
<b>by </b><i><a class="txt" href="binari_2.html#E3:8"><i><font color="Green" title="E6">A2</font></i></a>, <a class="ref" href="finseq_4.html#T26" title="FINSEQ_4:th.26">FINSEQ_4:26</a></i>;<br/>
<b>then </b> <a href="binarith.html#K7" title="BINARITH:func.7">Absval</a> <font color="Maroon" title="c1">z1</font> = 
<a href="binop_2.html#K47" title="BINOP_2:func.47">addnat</a>  <a href="finsop_1.html#K2" title="FINSOP_1:func.2">$$</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default">1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>
<b>by </b><i><a class="txt" href="binari_2.html#E3:8"><i><font color="Green" title="E6">A2</font></i></a>, <a class="txt" href="binari_2.html#E11:8"><i><font color="Green" title="E10">A6</font></i></a>, <a class="txt" href="binari_2.html#E12:8"><i><font color="Green" title="E11">A7</font></i></a>, <a class="txt" href="binari_2.html#E14:8"><i><font color="Green" title="E12">A8</font></i></a>, <a class="ref" href="binarith.html#D7" title="BINARITH:def.7">BINARITH:def 7</a></i>
<br/>.= 
<a href="binop_2.html#K47" title="BINOP_2:func.47">addnat</a>  <a href="finsop_1.html#K2" title="FINSOP_1:func.2">$$</a> <span class="p1">(<span class="default"><span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default">1</span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span> <a href="binarith.html#K10" title="BINARITH:func.10">^</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_1.html#D9" title="FINSEQ_1:def.9">FINSEQ_1:def 9</a></i>
<br/>.= 
<a href="binop_2.html#K47" title="BINOP_2:func.47">addnat</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> <span class="p1">(<span class="default"><a href="binop_2.html#K47" title="BINOP_2:func.47">addnat</a>  <a href="finsop_1.html#K2" title="FINSOP_1:func.2">$$</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default">1</span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span>,<span class="p1">(<span class="default"><a href="binop_2.html#K47" title="BINOP_2:func.47">addnat</a>  <a href="finsop_1.html#K2" title="FINSOP_1:func.2">$$</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="finsop_1.html#T6" title="FINSOP_1:th.6">FINSOP_1:6</a></i>
<br/>.= 
<a href="binop_2.html#K47" title="BINOP_2:func.47">addnat</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> 1,<span class="p1">(<span class="default"><a href="binop_2.html#K47" title="BINOP_2:func.47">addnat</a>  <a href="finsop_1.html#K2" title="FINSOP_1:func.2">$$</a> <span class="p2"><a href="binarith.html#K11" title="BINARITH:func.11">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="binarith.html#K11" title="BINARITH:func.11">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="ref" href="finsop_1.html#T12" title="FINSOP_1:th.12">FINSOP_1:12</a></i>
<br/>.= 
<a href="binop_2.html#K47" title="BINOP_2:func.47">addnat</a>  <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> 1,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="ref" href="finsop_1.html#T12" title="FINSOP_1:th.12">FINSOP_1:12</a></i>
<br/>.= 
1 <a href="binop_2.html#K23" title="BINOP_2:func.23">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
<b>by </b><i><a class="ref" href="binop_2.html#D23" title="BINOP_2:def.23">BINOP_2:def 23</a></i>
<br/>.= 
1

;<br/>
<b>hence </b><a NAME="E17:8"/>
 <a href="binari_2.html#K3" title="BINARI_2:func.3">Intval</a> <font color="Maroon" title="c1">z1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="binari_2.html#E5:8"><i><font color="Green" title="E7">A3</font></i></a>, <a class="ref" href="binari_2.html#D3" target="_self" title="BINARI_2:def.3">Def3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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