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

<b>let </b><font color="Maroon" title="c1">n</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <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> ;<br/><b>let </b><font color="Maroon" title="c2">y</font> be   <a href="binarith.html#NM1" title="BINARITH:NM.1">Tuple</a> of <font color="Maroon" title="c1">n</font>,<a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a> ;<br/>



<b>assume </b><a NAME="E1:22"/><i><font color="Green" title="E16">A1</font></i>: 
<font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="euclid.html#K5" title="EUCLID:func.5">0*</a> <font color="Maroon" title="c1">n</font>
 ;<br/>

<a NAME="E2:22"/><i><font color="Green" title="E17">A2</font></i>: 
<font color="Maroon" title="c1">n</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> <font color="Maroon" title="c1">n</font>
 
<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="E3:22"/><i><font color="Green" title="E18">A3</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/>
<a NAME="E4:22"/><i><font color="Green" title="E19">A4</font></i>: 
<font color="Maroon" title="c2">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 
<b>by </b><i><a class="txt" href="binari_3.html#E1:22"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="binari_3.html#E2:22"><i><font color="Green" title="E17">A2</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/>
<a NAME="E5:22"/><i><font color="Green" title="E20">A5</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T109" title="FINSEQ_2:th.109">FINSEQ_2:109</a></i>;<br/>
<i><font color="Green" title="E21">A6</font></i>: <span class="p1">(<span class="default"><a href="binarith.html#K4" title="BINARITH:func.4">'not'</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c1">n</font> = 
 <a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c1">n</font></span>)</span>
<b>by </b><i><a class="txt" href="binari_3.html#E2:22"><i><font color="Green" title="E17">A2</font></i></a>, <a class="ref" href="binarith.html#D4" title="BINARITH:def.4">BINARITH:def 4</a></i>
<br/>.= 
 <a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
<b>by </b><i><a class="txt" href="binari_3.html#E3:22"><i><font color="Green" title="E18">A3</font></i></a>, <a class="txt" href="binari_3.html#E4:22"><i><font color="Green" title="E19">A4</font></i></a>, <a class="txt" href="binari_3.html#E5:22"><i><font color="Green" title="E20">A5</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>
<br/>.= 
 <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 

;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:22_2"/>
( <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_2_1"><b>suppose </b></a><a NAME="E1:22_2_1"/><i><font color="Green" title="E22">A7</font></i>: 
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">i</font> be    <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> ;<br/><b>assume </b><a NAME="E1:22_2_1_1"/><i><font color="Green" title="E23">A8</font></i>: 
<font color="Maroon" title="c3">i</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> <font color="Maroon" title="c1">n</font>
 ;<br/><a NAME="E2:22_2_1_1"/><b>then </b><i><font color="Green" title="E24">A9</font></i>: 
<font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="binari_3.html#E1:22_2_1"><i><font color="Green" title="E22">A7</font></i></a>, <a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E3:22_2_1_1"/><i><font color="Green" title="E25">A10</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="binarith.html#K4" title="BINARITH:func.4">'not'</a> <span class="p2">(<span class="default"><a href="binari_2.html#K2" title="BINARI_2:func.2">Bin1</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="ref" href="finseq_2.html#T109" title="FINSEQ_2:th.109">FINSEQ_2:109</a></i>;<br/><a NAME="E4:22_2_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="binarith.html#K5" title="BINARITH:func.5">carry</a> <span class="p2">(<span class="default"><a href="binarith.html#K4" title="BINARITH:func.4">'not'</a> <font color="Maroon" title="c2">y</font></span>)</span>,<span class="p2">(<span class="default"><a href="binari_2.html#K2" title="BINARI_2:func.2">Bin1</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font>
 <b>by </b><i><a class="ref" href="finseq_2.html#T109" title="FINSEQ_2:th.109">FINSEQ_2:109</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="binarith.html#K5" title="BINARITH:func.5">carry</a> <span class="p2">(<span class="default"><a href="binarith.html#K4" title="BINARITH:func.4">'not'</a> <font color="Maroon" title="c2">y</font></span>)</span>,<span class="p2">(<span class="default"><a href="binari_2.html#K2" title="BINARI_2:func.2">Bin1</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font> = 
<span class="p1">(<span class="default"><a href="binarith.html#K5" title="BINARITH:func.5">carry</a> <span class="p2">(<span class="default"><a href="binarith.html#K4" title="BINARITH:func.4">'not'</a> <font color="Maroon" title="c2">y</font></span>)</span>,<span class="p2">(<span class="default"><a href="binari_2.html#K2" title="BINARI_2:func.2">Bin1</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c3">i</font>
<b>by </b><i><a class="txt" href="binari_3.html#E1:22_2_1"><i><font color="Green" title="E22">A7</font></i></a>, <a class="txt" href="binari_3.html#E2:22_2_1_1"><i><font color="Green" title="E24">A9</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>
<br/>.= 
 <a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
<b>by </b><i><a class="txt" href="binari_3.html#E2:22_2_1_1"><i><font color="Green" title="E24">A9</font></i></a>, <a class="ref" href="binarith.html#D5" title="BINARITH:def.5">BINARITH:def 5</a></i>
<br/>.= 
 <a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binari_2.html#K2" title="BINARI_2:func.2">Bin1</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c3">i</font></span>)</span>
<b>by </b><i><a class="txt" href="binari_3.html#E1:22_2_1_1"><i><font color="Green" title="E23">A8</font></i></a>, <a class="txt" href="binari_3.html#E2:22_2_1_1"><i><font color="Green" title="E24">A9</font></i></a>, <a class="ref" href="binari_2.html#T7" title="BINARI_2:th.7">BINARI_2:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="binarith.html#K4" title="BINARITH:func.4">'not'</a> <span class="p2">(<span class="default"><a href="binari_2.html#K2" title="BINARI_2:func.2">Bin1</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c3">i</font>
<b>by </b><i><a class="txt" href="binari_3.html#E1:22_2_1_1"><i><font color="Green" title="E23">A8</font></i></a>, <a class="ref" href="binarith.html#D4" title="BINARITH:def.4">BINARITH:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="binarith.html#K4" title="BINARITH:func.4">'not'</a> <span class="p2">(<span class="default"><a href="binari_2.html#K2" title="BINARI_2:func.2">Bin1</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font>
<b>by </b><i><a class="txt" href="binari_3.html#E1:22_2_1"><i><font color="Green" title="E22">A7</font></i></a>, <a class="txt" href="binari_3.html#E2:22_2_1_1"><i><font color="Green" title="E24">A9</font></i></a>, <a class="txt" href="binari_3.html#E3:22_2_1_1"><i><font color="Green" title="E25">A10</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>
;<br/><br/></div><b>end;</b></div><b>hence </b><a NAME="E3:22_2_1"/>
 <a href="binarith.html#K5" title="BINARITH:func.5">carry</a> <span class="p1">(<span class="default"><a href="binarith.html#K4" title="BINARITH:func.4">'not'</a> <font color="Maroon" title="c2">y</font></span>)</span>,<span class="p1">(<span class="default"><a href="binari_2.html#K2" title="BINARI_2:func.2">Bin1</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binarith.html#K4" title="BINARITH:func.4">'not'</a> <span class="p1">(<span class="default"><a href="binari_2.html#K2" title="BINARI_2:func.2">Bin1</a> <font color="Maroon" title="c1">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_2.html#T139" title="FINSEQ_2:th.139">FINSEQ_2:139</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_2_2"><b>suppose </b></a><a NAME="E1:22_2_2"/>
<font color="Maroon" title="c1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:22_2_2"/><b>then </b><i><font color="Green" title="E22">A11</font></i>: 
not <font color="Maroon" title="c1">n</font> is <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a>
 <b>by </b><i><a class="ref" href="nat_2.html#D1" title="NAT_2:def.1">NAT_2:def 1</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <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>means </b>( $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> implies <span class="p1">(<span class="default"><a href="binarith.html#K5" title="BINARITH:func.5">carry</a> <span class="p2">(<span class="default"><a href="binarith.html#K4" title="BINARITH:func.4">'not'</a> <font color="Maroon" title="c2">y</font></span>)</span>,<span class="p2">(<span class="default"><a href="binari_2.html#K2" title="BINARI_2:func.2">Bin1</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>  );<br/><a NAME="E3:22_2_2"/><i><font color="Green" title="E23">A12</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[2]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:22_2_2_1"/>
2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 ;<br/>

<a NAME="E2:22_2_2_1"/><b>then </b>
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 
;<br/>
<a NAME="E3:22_2_2_1"/><b>then </b><i><font color="Green" title="E24">A13</font></i>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E4:22_2_2_1"/><b>then </b><i><font color="Green" title="E25">A14</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> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E5:22_2_2_1"/><b>then </b><i><font color="Green" title="E26">A15</font></i>: 
<font color="Maroon" title="c2">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <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="txt" href="binari_3.html#E1:22"><i><font color="Green" title="E16">A1</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/>
<span class="p1">(<span class="default"><a href="binarith.html#K4" title="BINARITH:func.4">'not'</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1 = 
 <a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="binari_3.html#E4:22_2_2_1"><i><font color="Green" title="E25">A14</font></i></a>, <a class="ref" href="binarith.html#D4" title="BINARITH:def.4">BINARITH:def 4</a></i>
<br/>.= 
 <a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
<b>by </b><i><a class="txt" href="binari_3.html#E5:22"><i><font color="Green" title="E20">A5</font></i></a>, <a class="txt" href="binari_3.html#E3:22_2_2_1"><i><font color="Green" title="E24">A13</font></i></a>, <a class="txt" href="binari_3.html#E5:22_2_2_1"><i><font color="Green" title="E26">A15</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>
<br/>.= 
 <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 

;<br/>
<a NAME="E7:22_2_2_1"/><b>then </b><i><font color="Green" title="E27">A16</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binarith.html#K4" title="BINARITH:func.4">'not'</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1</span>)</span> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binari_2.html#K2" title="BINARI_2:func.2">Bin1</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 
<b>by </b><i><a class="txt" href="binari_3.html#E4:22_2_2_1"><i><font color="Green" title="E25">A14</font></i></a>, <a class="ref" href="binari_2.html#T7" title="BINARI_2:th.7">BINARI_2:7</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><a href="binarith.html#K5" title="BINARITH:func.5">carry</a> <span class="p2">(<span class="default"><a href="binarith.html#K4" title="BINARITH:func.4">'not'</a> <font color="Maroon" title="c2">y</font></span>)</span>,<span class="p2">(<span class="default"><a href="binari_2.html#K2" title="BINARI_2:func.2">Bin1</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 2 = 
<span class="p1">(<span class="default"><a href="binarith.html#K5" title="BINARITH:func.5">carry</a> <span class="p2">(<span class="default"><a href="binarith.html#K4" title="BINARITH:func.4">'not'</a> <font color="Maroon" title="c2">y</font></span>)</span>,<span class="p2">(<span class="default"><a href="binari_2.html#K2" title="BINARI_2:func.2">Bin1</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="binarith.html#K4" title="BINARITH:func.4">'not'</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1</span>)</span> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="binari_2.html#K2" title="BINARI_2:func.2">Bin1</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1</span>)</span></span>)</span> <a href="binarith.html#K1" title="BINARITH:func.1">'or'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="binarith.html#K4" title="BINARITH:func.4">'not'</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1</span>)</span> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="binarith.html#K5" title="BINARITH:func.5">carry</a> <span class="p5">(<span class="default"><a href="binarith.html#K4" title="BINARITH:func.4">'not'</a> <font color="Maroon" title="c2">y</font></span>)</span>,<span class="p5">(<span class="default"><a href="binari_2.html#K2" title="BINARI_2:func.2">Bin1</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1</span>)</span></span>)</span></span>)</span> <a href="binarith.html#K1" title="BINARITH:func.1">'or'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binari_2.html#K2" title="BINARI_2:func.2">Bin1</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1</span>)</span> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binarith.html#K5" title="BINARITH:func.5">carry</a> <span class="p4">(<span class="default"><a href="binarith.html#K4" title="BINARITH:func.4">'not'</a> <font color="Maroon" title="c2">y</font></span>)</span>,<span class="p4">(<span class="default"><a href="binari_2.html#K2" title="BINARI_2:func.2">Bin1</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="binari_3.html#E3:22_2_2_1"><i><font color="Green" title="E24">A13</font></i></a>, <a class="ref" href="binarith.html#D5" title="BINARITH:def.5">BINARITH:def 5</a></i>
<br/>.= 
 <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
<b>by </b><i><a class="txt" href="binari_3.html#E7:22_2_2_1"><i><font color="Green" title="E27">A16</font></i></a></i>
;<br/>


</div><b>end;</b></div><a NAME="E4:22_2_2"/><i><font color="Green" title="E24">A17</font></i>: 
for <font color="Olive" title="b1">i</font> being non <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a>  <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>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">i</font> be  non <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a>  <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> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:22_2_2_2"/><i><font color="Green" title="E25">A18</font></i>: 
( <font color="Maroon" title="c3">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font> implies <span class="p1">(<span class="default"><a href="binarith.html#K5" title="BINARITH:func.5">carry</a> <span class="p2">(<span class="default"><a href="binarith.html#K4" title="BINARITH:func.4">'not'</a> <font color="Maroon" title="c2">y</font></span>)</span>,<span class="p2">(<span class="default"><a href="binari_2.html#K2" title="BINARI_2:func.2">Bin1</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a>  )
 <b>and </b><br/><a NAME="E2:22_2_2_2"/><i><font color="Green" title="E26">A19</font></i>: 
<font color="Maroon" title="c3">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">n</font>
 ;<br/>

<a NAME="E3:22_2_2_2"/><i><font color="Green" title="E27">A20</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">i</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/>
<a NAME="E4:22_2_2_2"/><i><font color="Green" title="E28">A21</font></i>: 
<font color="Maroon" title="c3">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="txt" href="binari_3.html#E2:22_2_2_2"><i><font color="Green" title="E26">A19</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E5:22_2_2_2"/><b>then </b><i><font color="Green" title="E29">A22</font></i>: 
<font color="Maroon" title="c3">i</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> <font color="Maroon" title="c1">n</font>
 
<b>by </b><i><a class="txt" href="binari_3.html#E3:22_2_2_2"><i><font color="Green" title="E27">A20</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E6:22_2_2_2"/><b>then </b><i><font color="Green" title="E30">A23</font></i>: 
<font color="Maroon" title="c2">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c3">i</font> <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="txt" href="binari_3.html#E1:22"><i><font color="Green" title="E16">A1</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>;<br/>
<i><font color="Green" title="E31">A24</font></i>: <span class="p1">(<span class="default"><a href="binarith.html#K4" title="BINARITH:func.4">'not'</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c3">i</font> = 
 <a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">y</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c3">i</font></span>)</span>
<b>by </b><i><a class="txt" href="binari_3.html#E5:22_2_2_2"><i><font color="Green" title="E29">A22</font></i></a>, <a class="ref" href="binarith.html#D4" title="BINARITH:def.4">BINARITH:def 4</a></i>
<br/>.= 
 <a href="margrel1.html#K9" title="MARGREL1:func.9">'not'</a> <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a> 
<b>by </b><i><a class="txt" href="binari_3.html#E5:22"><i><font color="Green" title="E20">A5</font></i></a>, <a class="txt" href="binari_3.html#E3:22_2_2_2"><i><font color="Green" title="E27">A20</font></i></a>, <a class="txt" href="binari_3.html#E4:22_2_2_2"><i><font color="Green" title="E28">A21</font></i></a>, <a class="txt" href="binari_3.html#E6:22_2_2_2"><i><font color="Green" title="E30">A23</font></i></a>, <a class="ref" href="finseq_4.html#T24" title="FINSEQ_4:th.24">FINSEQ_4:24</a></i>
<br/>.= 
 <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 

;<br/>
<a NAME="E8:22_2_2_2"/>
<font color="Maroon" title="c3">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 
<b>by </b><i><a class="ref" href="nat_2.html#D1" title="NAT_2:def.1">NAT_2:def 1</a></i>;<br/>
<a NAME="E9:22_2_2_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="binari_2.html#K2" title="BINARI_2:func.2">Bin1</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c3">i</font> <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="txt" href="binari_3.html#E5:22_2_2_2"><i><font color="Green" title="E29">A22</font></i></a>, <a class="ref" href="binari_2.html#T8" title="BINARI_2:th.8">BINARI_2:8</a></i>;<br/>
<a NAME="E10:22_2_2_2"/><b>then </b>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binarith.html#K4" title="BINARITH:func.4">'not'</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c3">i</font></span>)</span> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binari_2.html#K2" title="BINARI_2:func.2">Bin1</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c3">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binari_2.html#K2" title="BINARI_2:func.2">Bin1</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c3">i</font></span>)</span> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binarith.html#K5" title="BINARITH:func.5">carry</a> <span class="p3">(<span class="default"><a href="binarith.html#K4" title="BINARITH:func.4">'not'</a> <font color="Maroon" title="c2">y</font></span>)</span>,<span class="p3">(<span class="default"><a href="binari_2.html#K2" title="BINARI_2:func.2">Bin1</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c3">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K7" title="MARGREL1:func.7">FALSE</a>  )
 
;<br/>
<b>hence </b><span class="p1">(<span class="default"><a href="binarith.html#K5" title="BINARITH:func.5">carry</a> <span class="p2">(<span class="default"><a href="binarith.html#K4" title="BINARITH:func.4">'not'</a> <font color="Maroon" title="c2">y</font></span>)</span>,<span class="p2">(<span class="default"><a href="binari_2.html#K2" title="BINARI_2:func.2">Bin1</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binarith.html#K4" title="BINARITH:func.4">'not'</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c3">i</font></span>)</span> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binari_2.html#K2" title="BINARI_2:func.2">Bin1</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c3">i</font></span>)</span></span>)</span> <a href="binarith.html#K1" title="BINARITH:func.1">'or'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binarith.html#K4" title="BINARITH:func.4">'not'</a> <font color="Maroon" title="c2">y</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c3">i</font></span>)</span> <a href="margrel1.html#K10" title="MARGREL1:func.10">'&amp;'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binarith.html#K5" title="BINARITH:func.5">carry</a> <span class="p4">(<span class="default"><a href="binarith.html#K4" title="BINARITH:func.4">'not'</a> <font color="Maroon" title="c2">y</font></span>)</span>,<span class="p4">(<span class="default"><a href="binari_2.html#K2" title="BINARI_2:func.2">Bin1</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c3">i</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="binari_3.html#E3:22_2_2_2"><i><font color="Green" title="E27">A20</font></i></a>, <a class="txt" href="binari_3.html#E4:22_2_2_2"><i><font color="Green" title="E28">A21</font></i></a>, <a class="ref" href="binarith.html#D5" title="BINARITH:def.5">BINARITH:def 5</a></i>
<br/>.= 
 <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
<b>by </b><i><a class="txt" href="binari_3.html#E1:22_2_2_2"><i><font color="Green" title="E25">A18</font></i></a>, <a class="txt" href="binari_3.html#E2:22_2_2_2"><i><font color="Green" title="E26">A19</font></i></a>, <a class="txt" href="binari_3.html#E7:22_2_2_2"><i><font color="Green" title="E31">A24</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>
;<br/><br/>


</div><b>end;</b></div><a NAME="E5:22_2_2"/>
for <font color="Olive" title="b1">i</font> being non <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a>  <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>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>]
 <b>from </b><i><a class="ref" href="nat_2.html#S2" title="NAT_2:sch.2">NAT_2:sch 2</a>(<a class="txt" href="binari_3.html#E3:22_2_2"><i><font color="Green" title="E23">A12</font></i></a>, <a class="txt" href="binari_3.html#E4:22_2_2"><i><font color="Green" title="E24">A17</font></i></a>);<br/></i><a NAME="E6:22_2_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="binarith.html#K5" title="BINARITH:func.5">carry</a> <span class="p2">(<span class="default"><a href="binarith.html#K4" title="BINARITH:func.4">'not'</a> <font color="Maroon" title="c2">y</font></span>)</span>,<span class="p2">(<span class="default"><a href="binari_2.html#K2" title="BINARI_2:func.2">Bin1</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> 
 <b>by </b><i><a class="txt" href="binari_3.html#E2:22_2_2"><i><font color="Green" title="E22">A11</font></i></a></i>;<br/><b>hence </b><a NAME="E7:22_2_2"/>
 <a href="binarith.html#K5" title="BINARITH:func.5">carry</a> <span class="p1">(<span class="default"><a href="binarith.html#K4" title="BINARITH:func.4">'not'</a> <font color="Maroon" title="c2">y</font></span>)</span>,<span class="p1">(<span class="default"><a href="binari_2.html#K2" title="BINARI_2:func.2">Bin1</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="binarith.html#K4" title="BINARITH:func.4">'not'</a> <span class="p1">(<span class="default"><a href="binari_2.html#K2" title="BINARI_2:func.2">Bin1</a> <font color="Maroon" title="c1">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="binari_3.html#E6:22"><i><font color="Green" title="E21">A6</font></i></a>, <a class="ref" href="binari_3.html#T21" target="_self" title="BINARI_3:th.21">Th21</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
