<?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>, <font color="Maroon">c<sub>3</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="E1:5"/><i><font color="Green">E5</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> &amp; <font color="Maroon">c<sub>3</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="E2:5"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="binarith.html#K7">carry</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></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="E3:5"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="binarith.html#K7">carry</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/>
<b>then </b><i><font color="Green">E6</font></i>: <span class="p1">(<span class="default"><a href="binarith.html#K7">carry</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> 1 = 
<span class="p1">(<span class="default"><a href="binarith.html#K7">carry</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1
<b>by </b><i><a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>
<br/>.= 
0
<b>by </b><i><a class="ref" href="binarith.html#D5">BINARITH:def 5</a>, <a class="ref" href="margrel1.html#T36">MARGREL1:36</a></i>
;<br/>
<a NAME="E5:5"/><i><font color="Green">E7</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><span class="p1">(<span class="default"><a href="binarith.html#K7">carry</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_2"><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="E1:5_2"/><i><font color="Green">E8</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="real_1.html#K5">-</a> 1 as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="binari_4.html#E1:5_2"><i><font color="Green">E8</font></i></a>, <a class="ref" href="int_1.html#T18">INT_1:18</a></i>;<br/>
<a NAME="E3:5_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 
;<br/>
<a NAME="E4:5_2"/><b>then </b><i><font color="Green">E9</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i><a class="txt" href="binari_4.html#E1:5_2"><i><font color="Green">E8</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E5:5_2"/><b>then </b><i><font color="Green">E10</font></i>: 
( <font color="Maroon">c<sub>5</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> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a>, <a class="ref" href="finseq_2.html#T109">FINSEQ_2:109</a></i>;<br/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> = 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="finseqop.html#K4">|-&gt;</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#D4">EUCLID:def 4</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font>
<b>by </b><i><a class="ref" href="finseq_2.html#T69">FINSEQ_2:69</a></i>
;<br/>
<b>then </b><i><font color="Green">E11</font></i>: <font color="Maroon">c<sub>2</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>5</sub></font> = 
<span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="binari_4.html#E1:5"><i><font color="Green">E5</font></i></a>, <a class="txt" href="binari_4.html#E4:5_2"><i><font color="Green">E9</font></i></a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>
<br/>.= 
<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>5</sub></font>
<b>by </b><i><a class="ref" href="euclid.html#D4">EUCLID:def 4</a></i>
<br/>.= 
 <a href="margrel1.html#K7">FALSE</a> 
<b>by </b><i><a class="txt" href="binari_4.html#E5:5_2"><i><font color="Green">E10</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/>
<a NAME="E8:5_2"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="binarith.html#K7">carry</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></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>then </b><span class="p1">(<span class="default"><a href="binarith.html#K7">carry</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> = 
<span class="p1">(<span class="default"><a href="binarith.html#K7">carry</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</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="txt" href="binari_4.html#E1:5_2"><i><font color="Green">E8</font></i></a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="margrel1.html#K7">FALSE</a>  <a href="margrel1.html#K12">'&amp;'</a> <a href="margrel1.html#K7">FALSE</a> </span>)</span> <a href="binarith.html#K3">'or'</a> <span class="p2">(<span class="default"><a href="margrel1.html#K7">FALSE</a>  <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> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K3">'or'</a> <span class="p1">(<span class="default"><a href="margrel1.html#K7">FALSE</a>  <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> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="binari_4.html#E1:5"><i><font color="Green">E5</font></i></a>, <a class="txt" href="binari_4.html#E4:5_2"><i><font color="Green">E9</font></i></a>, <a class="txt" href="binari_4.html#E7:5_2"><i><font color="Green">E11</font></i></a>, <a class="ref" href="binarith.html#D5">BINARITH:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="margrel1.html#K7">FALSE</a>  <a href="binarith.html#K3">'or'</a> <span class="p2">(<span class="default"><a href="margrel1.html#K7">FALSE</a>  <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> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span></span>)</span> <a href="binarith.html#K3">'or'</a> <span class="p1">(<span class="default"><a href="margrel1.html#K7">FALSE</a>  <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> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="margrel1.html#T49">MARGREL1:49</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="margrel1.html#K7">FALSE</a>  <a href="binarith.html#K3">'or'</a> <a href="margrel1.html#K7">FALSE</a> </span>)</span> <a href="binarith.html#K3">'or'</a> <span class="p1">(<span class="default"><a href="margrel1.html#K7">FALSE</a>  <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> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="margrel1.html#T49">MARGREL1:49</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="margrel1.html#K7">FALSE</a>  <a href="binarith.html#K3">'or'</a> <a href="margrel1.html#K7">FALSE</a> </span>)</span> <a href="binarith.html#K3">'or'</a> <a href="margrel1.html#K7">FALSE</a> 
<b>by </b><i><a class="ref" href="margrel1.html#T49">MARGREL1:49</a></i>
<br/>.= 
<a href="margrel1.html#K7">FALSE</a>  <a href="binarith.html#K3">'or'</a> <a href="margrel1.html#K7">FALSE</a> 
<b>by </b><i><a class="ref" href="binarith.html#T7">BINARITH:7</a></i>
<br/>.= 
 <a href="margrel1.html#K7">FALSE</a> 
<b>by </b><i><a class="ref" href="binarith.html#T7">BINARITH:7</a></i>
;<br/>
<b>hence </b><a NAME="E10:5_2"/>
<span class="p1">(<span class="default"><a href="binarith.html#K7">carry</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="margrel1.html#T36">MARGREL1:36</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:5"/>
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> <font color="Maroon">c<sub>1</sub></font> holds <br/><span class="p1">(<span class="default"><a href="binarith.html#K7">carry</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_3"><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="E1:5_3"/><i><font color="Green">E8</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/>

<a NAME="E2:5_3"/><i><font color="Green">E9</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="binari_4.html#E1:5_3"><i><font color="Green">E8</font></i></a>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<i><font color="Green">E10</font></i>: <span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</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>4</sub></font>
<b>by </b><i><a class="ref" href="euclid.html#D4">EUCLID:def 4</a></i>
<br/>.= 
0
<b>by </b><i><a class="txt" href="binari_4.html#E1:5_3"><i><font color="Green">E8</font></i></a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_3_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_3_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:5_3_2_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 1 or ( 1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> ) )
 </span><b>by </b><i><a class="txt" href="binari_4.html#E2:5_3"><i><font color="Green">E9</font></i></a>, <a class="ref" href="binari_4.html#T4" target="_self">Th4</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_3_2_1_1"><b>suppose </b></a><a NAME="E1:5_3_2_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:5_3_2_1_1"/>
<span class="p1">(<span class="default"><a href="binarith.html#K7">carry</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="binari_4.html#E4:5"><i><font color="Green">E6</font></i></a>, <a class="txt" href="binari_4.html#E3:5_3"><i><font color="Green">E10</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_3_2_1_2"><b>suppose </b></a><a NAME="E1:5_3_2_1_2"/>
( 1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/><div class="add"><a NAME="E2:5_3_2_1_2"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&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/><b>hence </b><a NAME="E3:5_3_2_1_2"/>
<span class="p1">(<span class="default"><a href="binarith.html#K7">carry</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="binari_4.html#E5:5"><i><font color="Green">E7</font></i></a>, <a class="txt" href="binari_4.html#E3:5_3"><i><font color="Green">E10</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E5:5_3"/>
<span class="p1">(<span class="default"><a href="binarith.html#K7">carry</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="euclid.html#K5">0*</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:5"/>
 <a href="binarith.html#K7">carry</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K5">0*</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="binari_4.html#E1:5"><i><font color="Green">E5</font></i></a>, <a class="ref" href="finseq_2.html#T139">FINSEQ_2:139</a></i>;<br/>


</div>
