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

<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>assume </b><a NAME="E1:22"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K5">0*</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E2:22"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>1</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>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T5">FINSEQ_1:5</a></i>;<br/>
<a NAME="E3:22"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/>
<i><font color="Green">E20</font></i>: <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseqop.html#K4">|-&gt;</a> 0</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="txt" href="binari_3.html#E1:22"><i><font color="Green">E17</font></i></a></i>
<br/>.= 
0
<b>by </b><i><a class="txt" href="binari_3.html#E2:22"><i><font color="Green">E18</font></i></a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
;<br/>
<a NAME="E5:22"/><i><font color="Green">E21</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/>
<i><font color="Green">E22</font></i>: <span class="p1">(<span class="default"><a href="binarith.html#K6">'not'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>1</sub></font> = 
 <a href="margrel1.html#K11">'not'</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>1</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="binari_3.html#E2:22"><i><font color="Green">E18</font></i></a>, <a class="ref" href="binarith.html#D4">BINARITH:def 4</a></i>
<br/>.= 
 <a href="margrel1.html#K11">'not'</a> <a href="margrel1.html#K7">FALSE</a> 
<b>by </b><i><a class="txt" href="binari_3.html#E3:22"><i><font color="Green">E19</font></i></a>, <a class="txt" href="binari_3.html#E4:22"><i><font color="Green">E20</font></i></a>, <a class="txt" href="binari_3.html#E5:22"><i><font color="Green">E21</font></i></a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a>, <a class="ref" href="margrel1.html#T36">MARGREL1:36</a></i>
<br/>.= 
 <a href="margrel1.html#K8">TRUE</a> 
<b>by </b><i><a class="ref" href="margrel1.html#T41">MARGREL1:41</a></i>
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:22_3"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_3_1"><b>suppose </b></a><a NAME="E1:22_3_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_3_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:22_3_1_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>3</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/><a NAME="E2:22_3_1_1"/><b>then </b><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="binari_3.html#E1:22_3_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:22_3_1_1"/><i><font color="Green">E26</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="binarith.html#K6">'not'</a> <span class="p2">(<span class="default"><a href="binari_2.html#K2">Bin1</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/><a NAME="E4:22_3_1_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="binarith.html#K7">carry</a> <span class="p2">(<span class="default"><a href="binarith.html#K6">'not'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="binari_2.html#K2">Bin1</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="binarith.html#K7">carry</a> <span class="p2">(<span class="default"><a href="binarith.html#K6">'not'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="binari_2.html#K2">Bin1</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> = 
<span class="p1">(<span class="default"><a href="binarith.html#K7">carry</a> <span class="p2">(<span class="default"><a href="binarith.html#K6">'not'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="binari_2.html#K2">Bin1</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="binari_3.html#E1:22_3_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="binari_3.html#E2:22_3_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>
<br/>.= 
 <a href="margrel1.html#K7">FALSE</a> 
<b>by </b><i><a class="txt" href="binari_3.html#E2:22_3_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="binarith.html#D5">BINARITH:def 5</a></i>
<br/>.= 
 <a href="margrel1.html#K11">'not'</a> <a href="margrel1.html#K8">TRUE</a> 
<b>by </b><i><a class="ref" href="margrel1.html#T41">MARGREL1:41</a></i>
<br/>.= 
 <a href="margrel1.html#K11">'not'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binari_2.html#K2">Bin1</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="binari_3.html#E1:22_3_1_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="binari_3.html#E2:22_3_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="binari_2.html#T7">BINARI_2:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="binarith.html#K6">'not'</a> <span class="p2">(<span class="default"><a href="binari_2.html#K2">Bin1</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="binari_3.html#E1:22_3_1_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="binarith.html#D4">BINARITH:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="binarith.html#K6">'not'</a> <span class="p2">(<span class="default"><a href="binari_2.html#K2">Bin1</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="binari_3.html#E1:22_3_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="binari_3.html#E2:22_3_1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="binari_3.html#E3:22_3_1_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>
;<br/><br/></div><b>end;</b></div><b>hence </b><a NAME="E3:22_3_1"/>
 <a href="binarith.html#K7">carry</a> <span class="p1">(<span class="default"><a href="binarith.html#K6">'not'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="binari_2.html#K2">Bin1</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="binarith.html#K6">'not'</a> <span class="p1">(<span class="default"><a href="binari_2.html#K2">Bin1</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_2.html#T139">FINSEQ_2:139</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="22_3_2"><b>suppose </b></a><a NAME="E1:22_3_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:22_3_2"/><b>then </b><i><font color="Green">E23</font></i>: 
not <font color="Maroon">c<sub>1</sub></font> is <a href="realset1.html#V1">trivial</a>
 <b>by </b><i><a class="ref" href="nat_2.html#D1">NAT_2:def 1</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means ( <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> implies <span class="p1">(<span class="default"><a href="binarith.html#K7">carry</a> <span class="p2">(<span class="default"><a href="binarith.html#K6">'not'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="binari_2.html#K2">Bin1</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  );<br/><a NAME="E3:22_3_2"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[2]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_3_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:22_3_2_1"/>
2 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E2:22_3_2_1"/><b>then </b>
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font>
 
;<br/>
<a NAME="E3:22_3_2_1"/><b>then </b><i><font color="Green">E25</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E4:22_3_2_1"/><b>then </b><i><font color="Green">E26</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E5:22_3_2_1"/><b>then </b><i><font color="Green">E27</font></i>: 
<span class="p1">(<span class="default"><a href="binari_2.html#K2">Bin1</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 
<b>by </b><i><a class="ref" href="binari_2.html#T7">BINARI_2:7</a></i>;<br/>
<i><font color="Green">E28</font></i>: <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseqop.html#K4">|-&gt;</a> 0</span>)</span> <a href="funct_1.html#K1">.</a> 1
<b>by </b><i><a class="txt" href="binari_3.html#E1:22"><i><font color="Green">E17</font></i></a></i>
<br/>.= 
 <a href="margrel1.html#K7">FALSE</a> 
<b>by </b><i><a class="txt" href="binari_3.html#E4:22_3_2_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a>, <a class="ref" href="margrel1.html#T36">MARGREL1:36</a></i>
;<br/>
<span class="p1">(<span class="default"><a href="binarith.html#K6">'not'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 = 
 <a href="margrel1.html#K11">'not'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="binari_3.html#E4:22_3_2_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="binarith.html#D4">BINARITH:def 4</a></i>
<br/>.= 
 <a href="margrel1.html#K11">'not'</a> <a href="margrel1.html#K7">FALSE</a> 
<b>by </b><i><a class="txt" href="binari_3.html#E5:22"><i><font color="Green">E21</font></i></a>, <a class="txt" href="binari_3.html#E3:22_3_2_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="binari_3.html#E6:22_3_2_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>
<br/>.= 
 <a href="margrel1.html#K8">TRUE</a> 
<b>by </b><i><a class="ref" href="margrel1.html#T41">MARGREL1:41</a></i>
;<br/>
<a NAME="E8:22_3_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binarith.html#K6">'not'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="margrel1.html#K12">'&amp;'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binari_2.html#K2">Bin1</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 
<b>by </b><i><a class="txt" href="binari_3.html#E5:22_3_2_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="margrel1.html#T45">MARGREL1:45</a></i>;<br/>
<a NAME="E9:22_3_2_1"/><b>then </b><i><font color="Green">E29</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binarith.html#K6">'not'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="margrel1.html#K12">'&amp;'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binari_2.html#K2">Bin1</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span> <a href="binarith.html#K3">'or'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binarith.html#K6">'not'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="margrel1.html#K12">'&amp;'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binarith.html#K7">carry</a> <span class="p4">(<span class="default"><a href="binarith.html#K6">'not'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p4">(<span class="default"><a href="binari_2.html#K2">Bin1</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 
<b>by </b><i><a class="ref" href="binarith.html#T19">BINARITH:19</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><a href="binarith.html#K7">carry</a> <span class="p2">(<span class="default"><a href="binarith.html#K6">'not'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="binari_2.html#K2">Bin1</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 2 = 
<span class="p1">(<span class="default"><a href="binarith.html#K7">carry</a> <span class="p2">(<span class="default"><a href="binarith.html#K6">'not'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="binari_2.html#K2">Bin1</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</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#K6">'not'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="margrel1.html#K12">'&amp;'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="binari_2.html#K2">Bin1</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span> <a href="binarith.html#K3">'or'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="binarith.html#K6">'not'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="margrel1.html#K12">'&amp;'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="binarith.html#K7">carry</a> <span class="p5">(<span class="default"><a href="binarith.html#K6">'not'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p5">(<span class="default"><a href="binari_2.html#K2">Bin1</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span></span>)</span> <a href="binarith.html#K3">'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">Bin1</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span> <a href="margrel1.html#K12">'&amp;'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binarith.html#K7">carry</a> <span class="p4">(<span class="default"><a href="binarith.html#K6">'not'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p4">(<span class="default"><a href="binari_2.html#K2">Bin1</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="binari_3.html#E3:22_3_2_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="binarith.html#D5">BINARITH:def 5</a></i>
<br/>.= 
 <a href="margrel1.html#K8">TRUE</a> 
<b>by </b><i><a class="txt" href="binari_3.html#E9:22_3_2_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="binarith.html#T19">BINARITH:19</a></i>
;<br/>


</div><b>end;</b></div><a NAME="E4:22_3_2"/><i><font color="Green">E25</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being non <a href="realset1.html#V1">trivial</a> <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_3_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be  non <a href="realset1.html#V1">trivial</a> <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:22_3_2_2"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> implies <span class="p1">(<span class="default"><a href="binarith.html#K7">carry</a> <span class="p2">(<span class="default"><a href="binarith.html#K6">'not'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="binari_2.html#K2">Bin1</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  )
 <b>and </b><br/><a NAME="E2:22_3_2_2"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E3:22_3_2_2"/><i><font color="Green">E28</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/>
<a NAME="E4:22_3_2_2"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="binari_3.html#E2:22_3_2_2"><i><font color="Green">E27</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E5:22_3_2_2"/><b>then </b><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>3</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>
 
<b>by </b><i><a class="txt" href="binari_3.html#E3:22_3_2_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<i><font color="Green">E31</font></i>: <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseqop.html#K4">|-&gt;</a> 0</span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="binari_3.html#E1:22"><i><font color="Green">E17</font></i></a></i>
<br/>.= 
 <a href="margrel1.html#K7">FALSE</a> 
<b>by </b><i><a class="txt" href="binari_3.html#E5:22_3_2_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a>, <a class="ref" href="margrel1.html#T36">MARGREL1:36</a></i>
;<br/>
<i><font color="Green">E32</font></i>: <span class="p1">(<span class="default"><a href="binarith.html#K6">'not'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>3</sub></font> = 
 <a href="margrel1.html#K11">'not'</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>3</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="binari_3.html#E5:22_3_2_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="binarith.html#D4">BINARITH:def 4</a></i>
<br/>.= 
 <a href="margrel1.html#K11">'not'</a> <a href="margrel1.html#K7">FALSE</a> 
<b>by </b><i><a class="txt" href="binari_3.html#E5:22"><i><font color="Green">E21</font></i></a>, <a class="txt" href="binari_3.html#E3:22_3_2_2"><i><font color="Green">E28</font></i></a>, <a class="txt" href="binari_3.html#E4:22_3_2_2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="binari_3.html#E6:22_3_2_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>
<br/>.= 
 <a href="margrel1.html#K8">TRUE</a> 
<b>by </b><i><a class="ref" href="margrel1.html#T41">MARGREL1:41</a></i>
;<br/>
<a NAME="E8:22_3_2_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 
<b>by </b><i><a class="ref" href="nat_2.html#D1">NAT_2:def 1</a></i>;<br/>
<a NAME="E9:22_3_2_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="binari_2.html#K2">Bin1</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 
<b>by </b><i><a class="txt" href="binari_3.html#E5:22_3_2_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="binari_2.html#T8">BINARI_2:8</a></i>;<br/>
<a NAME="E10:22_3_2_2"/><b>then </b><i><font color="Green">E33</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binarith.html#K6">'not'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="margrel1.html#K12">'&amp;'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binari_2.html#K2">Bin1</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binari_2.html#K2">Bin1</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="margrel1.html#K12">'&amp;'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binarith.html#K7">carry</a> <span class="p3">(<span class="default"><a href="binarith.html#K6">'not'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="binari_2.html#K2">Bin1</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  )
 
<b>by </b><i><a class="ref" href="margrel1.html#T45">MARGREL1:45</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><a href="binarith.html#K7">carry</a> <span class="p2">(<span class="default"><a href="binarith.html#K6">'not'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="binari_2.html#K2">Bin1</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<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#K6">'not'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="margrel1.html#K12">'&amp;'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="binari_2.html#K2">Bin1</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="binarith.html#K3">'or'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="binarith.html#K6">'not'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="margrel1.html#K12">'&amp;'</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="binarith.html#K7">carry</a> <span class="p5">(<span class="default"><a href="binarith.html#K6">'not'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p5">(<span class="default"><a href="binari_2.html#K2">Bin1</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K3">'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">Bin1</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="margrel1.html#K12">'&amp;'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binarith.html#K7">carry</a> <span class="p4">(<span class="default"><a href="binarith.html#K6">'not'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p4">(<span class="default"><a href="binari_2.html#K2">Bin1</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="binari_3.html#E3:22_3_2_2"><i><font color="Green">E28</font></i></a>, <a class="txt" href="binari_3.html#E4:22_3_2_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="binarith.html#D5">BINARITH:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binarith.html#K6">'not'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="margrel1.html#K12">'&amp;'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binari_2.html#K2">Bin1</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="binarith.html#K3">'or'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binarith.html#K6">'not'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="margrel1.html#K12">'&amp;'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="binarith.html#K7">carry</a> <span class="p4">(<span class="default"><a href="binarith.html#K6">'not'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p4">(<span class="default"><a href="binari_2.html#K2">Bin1</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="binari_3.html#E10:22_3_2_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="binarith.html#T7">BINARITH:7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binarith.html#K6">'not'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="margrel1.html#K12">'&amp;'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binarith.html#K7">carry</a> <span class="p3">(<span class="default"><a href="binarith.html#K6">'not'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="binari_2.html#K2">Bin1</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="binari_3.html#E10:22_3_2_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="binarith.html#T7">BINARITH:7</a></i>
<br/>.= 
 <a href="margrel1.html#K8">TRUE</a> 
<b>by </b><i><a class="txt" href="binari_3.html#E1:22_3_2_2"><i><font color="Green">E26</font></i></a>, <a class="txt" href="binari_3.html#E2:22_3_2_2"><i><font color="Green">E27</font></i></a>, <a class="txt" href="binari_3.html#E7:22_3_2_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="margrel1.html#T45">MARGREL1:45</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>
;<br/>


</div><b>end;</b></div><a NAME="E5:22_3_2"/>
for <font color="Olive">b<sub>1</sub></font> being non <a href="realset1.html#V1">trivial</a> <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="nat_2.html#S2">NAT_2:sch 2</a>(<a class="txt" href="binari_3.html#E3:22_3_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="binari_3.html#E4:22_3_2"><i><font color="Green">E25</font></i></a>);<br/></i><a NAME="E6:22_3_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="binarith.html#K7">carry</a> <span class="p2">(<span class="default"><a href="binarith.html#K6">'not'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="binari_2.html#K2">Bin1</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i><a class="txt" href="binari_3.html#E2:22_3_2"><i><font color="Green">E23</font></i></a></i>;<br/><b>hence </b><a NAME="E7:22_3_2"/>
 <a href="binarith.html#K7">carry</a> <span class="p1">(<span class="default"><a href="binarith.html#K6">'not'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="binari_2.html#K2">Bin1</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="binarith.html#K6">'not'</a> <span class="p1">(<span class="default"><a href="binari_2.html#K2">Bin1</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="binari_3.html#E6:22"><i><font color="Green">E22</font></i></a>, <a class="ref" href="binari_3.html#T21" target="_self">Th21</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
