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

<b>let </b><font color="Maroon">n</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">z1</font> be   <a href="binarith.html#NM1">Tuple</a> of <font color="Maroon">n</font>,<a href="margrel1.html#K6">BOOLEAN</a> , <font color="Maroon">z2</font> be   <a href="binarith.html#NM1">Tuple</a> of <font color="Maroon">n</font>,<a href="margrel1.html#K6">BOOLEAN</a> ;<br/><b>let </b><font color="Maroon">d1</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="margrel1.html#K6">BOOLEAN</a> , <font color="Maroon">d2</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="margrel1.html#K6">BOOLEAN</a> ;<br/>





<a NAME="E1:23"/>
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">z1</font> <a href="binarith.html#K8" target="_self">+</a> <font color="Maroon">z2</font></span>)</span> <a href="binarith.html#K10" target="_self">^</a> <span class="p2"><a href="binarith.html#K11" target="_self">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">d1</font> <a href="binarith.html#K2" target="_self">'xor'</a> <font color="Maroon">d2</font></span>)</span> <a href="binarith.html#K2" target="_self">'xor'</a> <span class="p4">(<span class="default"><a href="binarith.html#K9" target="_self">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font></span>)</span></span>)</span></span><a href="binarith.html#K11" target="_self">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">z1</font> <a href="binarith.html#K10" target="_self">^</a> <span class="p4"><a href="binarith.html#K11" target="_self">&lt;*</a><span class="default"><font color="Maroon">d1</font></span><a href="binarith.html#K11" target="_self">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span> <a href="binarith.html#K2" target="_self">'xor'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">z2</font> <a href="binarith.html#K10" target="_self">^</a> <span class="p4"><a href="binarith.html#K11" target="_self">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11" target="_self">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span></span>)</span> <a href="binarith.html#K2" target="_self">'xor'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binarith.html#K5" target="_self">carry</a> <span class="p3">(<span class="default"><font color="Maroon">z1</font> <a href="binarith.html#K10" target="_self">^</a> <span class="p4"><a href="binarith.html#K11" target="_self">&lt;*</a><span class="default"><font color="Maroon">d1</font></span><a href="binarith.html#K11" target="_self">*&gt;</a></span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">z2</font> <a href="binarith.html#K10" target="_self">^</a> <span class="p4"><a href="binarith.html#K11" target="_self">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11" target="_self">*&gt;</a></span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:23_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span>
 ;<br/>

<a NAME="E2:23_1"/><i><font color="Green">E40</font></i>: 
 <a href="finseq_1.html#K2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="setwiseo.html#K2">{.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span><a href="setwiseo.html#K2">.}</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T11">FINSEQ_1:11</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:23_1_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> or <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="setwiseo.html#K2">{.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span><a href="setwiseo.html#K2">.}</a></span> )
 </span><b>by </b><i>, , <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_1"><b>suppose </b></a><a NAME="E1:23_1_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">z1</font> <a href="binarith.html#K8" target="_self">+</a> <font color="Maroon">z2</font></span>)</span> <a href="binarith.html#K10" target="_self">^</a> <span class="p2"><a href="binarith.html#K11" target="_self">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">d1</font> <a href="binarith.html#K2" target="_self">'xor'</a> <font color="Maroon">d2</font></span>)</span> <a href="binarith.html#K2" target="_self">'xor'</a> <span class="p4">(<span class="default"><a href="binarith.html#K9" target="_self">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font></span>)</span></span>)</span></span><a href="binarith.html#K11" target="_self">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon">z1</font> <a href="binarith.html#K8" target="_self">+</a> <font color="Maroon">z2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">z1</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="binarith.html#K2" target="_self">'xor'</a> <span class="p2">(<span class="default"><font color="Maroon">z2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="binarith.html#K2" target="_self">'xor'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binarith.html#K5" target="_self">carry</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">z1</font> <a href="binarith.html#K10" target="_self">^</a> <span class="p4"><a href="binarith.html#K11" target="_self">&lt;*</a><span class="default"><font color="Maroon">d1</font></span><a href="binarith.html#K11" target="_self">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="binarith.html#K2" target="_self">'xor'</a> <span class="p2">(<span class="default"><font color="Maroon">z2</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="binarith.html#K2" target="_self">'xor'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binarith.html#K5" target="_self">carry</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">z1</font> <a href="binarith.html#K10" target="_self">^</a> <span class="p4"><a href="binarith.html#K11" target="_self">&lt;*</a><span class="default"><font color="Maroon">d1</font></span><a href="binarith.html#K11" target="_self">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="binarith.html#K2" target="_self">'xor'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">z2</font> <a href="binarith.html#K10" target="_self">^</a> <span class="p4"><a href="binarith.html#K11" target="_self">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11" target="_self">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="binarith.html#K2" target="_self">'xor'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binarith.html#K5" target="_self">carry</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">z1</font> <a href="binarith.html#K10" target="_self">^</a> <span class="p4"><a href="binarith.html#K11" target="_self">&lt;*</a><span class="default"><font color="Maroon">d1</font></span><a href="binarith.html#K11" target="_self">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="binarith.html#K2" target="_self">'xor'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">z2</font> <a href="binarith.html#K10" target="_self">^</a> <span class="p4"><a href="binarith.html#K11" target="_self">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11" target="_self">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="binarith.html#K2" target="_self">'xor'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binarith.html#K5" target="_self">carry</a> <span class="p3">(<span class="default"><font color="Maroon">z1</font> <a href="binarith.html#K10" target="_self">^</a> <span class="p4"><a href="binarith.html#K11" target="_self">&lt;*</a><span class="default"><font color="Maroon">d1</font></span><a href="binarith.html#K11" target="_self">*&gt;</a></span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">z2</font> <a href="binarith.html#K10" target="_self">^</a> <span class="p4"><a href="binarith.html#K11" target="_self">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11" target="_self">*&gt;</a></span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, </i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="23_1_1_2"><b>suppose </b></a><a NAME="E1:23_1_1_2"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="setwiseo.html#K2">{.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1</span>)</span></span><a href="setwiseo.html#K2">.}</a></span>
 ;<br/><div class="add"><a NAME="E2:23_1_1_2"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="binop_2.html#K23">+</a> 1
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">z1</font> <a href="binarith.html#K8" target="_self">+</a> <font color="Maroon">z2</font></span>)</span> <a href="binarith.html#K10" target="_self">^</a> <span class="p2"><a href="binarith.html#K11" target="_self">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">d1</font> <a href="binarith.html#K2" target="_self">'xor'</a> <font color="Maroon">d2</font></span>)</span> <a href="binarith.html#K2" target="_self">'xor'</a> <span class="p4">(<span class="default"><a href="binarith.html#K9" target="_self">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font></span>)</span></span>)</span></span><a href="binarith.html#K11" target="_self">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon">d1</font> <a href="binarith.html#K2" target="_self">'xor'</a> <font color="Maroon">d2</font></span>)</span> <a href="binarith.html#K2" target="_self">'xor'</a> <span class="p1">(<span class="default"><a href="binarith.html#K9" target="_self">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">d1</font> <a href="binarith.html#K2" target="_self">'xor'</a> <font color="Maroon">d2</font></span>)</span> <a href="binarith.html#K2" target="_self">'xor'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binarith.html#K5" target="_self">carry</a> <span class="p3">(<span class="default"><font color="Maroon">z1</font> <a href="binarith.html#K10" target="_self">^</a> <span class="p4"><a href="binarith.html#K11" target="_self">&lt;*</a><span class="default"><font color="Maroon">d1</font></span><a href="binarith.html#K11" target="_self">*&gt;</a></span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">z2</font> <a href="binarith.html#K10" target="_self">^</a> <span class="p4"><a href="binarith.html#K11" target="_self">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11" target="_self">*&gt;</a></span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">d1</font> <a href="binarith.html#K2" target="_self">'xor'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">z2</font> <a href="binarith.html#K10" target="_self">^</a> <span class="p4"><a href="binarith.html#K11" target="_self">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11" target="_self">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="binarith.html#K2" target="_self">'xor'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binarith.html#K5" target="_self">carry</a> <span class="p3">(<span class="default"><font color="Maroon">z1</font> <a href="binarith.html#K10" target="_self">^</a> <span class="p4"><a href="binarith.html#K11" target="_self">&lt;*</a><span class="default"><font color="Maroon">d1</font></span><a href="binarith.html#K11" target="_self">*&gt;</a></span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">z2</font> <a href="binarith.html#K10" target="_self">^</a> <span class="p4"><a href="binarith.html#K11" target="_self">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11" target="_self">*&gt;</a></span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">z1</font> <a href="binarith.html#K10" target="_self">^</a> <span class="p4"><a href="binarith.html#K11" target="_self">&lt;*</a><span class="default"><font color="Maroon">d1</font></span><a href="binarith.html#K11" target="_self">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="binarith.html#K2" target="_self">'xor'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">z2</font> <a href="binarith.html#K10" target="_self">^</a> <span class="p4"><a href="binarith.html#K11" target="_self">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11" target="_self">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="binarith.html#K2" target="_self">'xor'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binarith.html#K5" target="_self">carry</a> <span class="p3">(<span class="default"><font color="Maroon">z1</font> <a href="binarith.html#K10" target="_self">^</a> <span class="p4"><a href="binarith.html#K11" target="_self">&lt;*</a><span class="default"><font color="Maroon">d1</font></span><a href="binarith.html#K11" target="_self">*&gt;</a></span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">z2</font> <a href="binarith.html#K10" target="_self">^</a> <span class="p4"><a href="binarith.html#K11" target="_self">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11" target="_self">*&gt;</a></span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, </i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E2:23"/>
<span class="p1">(<span class="default"><font color="Maroon">z1</font> <a href="binarith.html#K10" target="_self">^</a> <span class="p2"><a href="binarith.html#K11" target="_self">&lt;*</a><span class="default"><font color="Maroon">d1</font></span><a href="binarith.html#K11" target="_self">*&gt;</a></span></span>)</span> <a href="binarith.html#K8" target="_self">+</a> <span class="p1">(<span class="default"><font color="Maroon">z2</font> <a href="binarith.html#K10" target="_self">^</a> <span class="p2"><a href="binarith.html#K11" target="_self">&lt;*</a><span class="default"><font color="Maroon">d2</font></span><a href="binarith.html#K11" target="_self">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">z1</font> <a href="binarith.html#K8" target="_self">+</a> <font color="Maroon">z2</font></span>)</span> <a href="binarith.html#K10" target="_self">^</a> <span class="p1"><a href="binarith.html#K11" target="_self">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">d1</font> <a href="binarith.html#K2" target="_self">'xor'</a> <font color="Maroon">d2</font></span>)</span> <a href="binarith.html#K2" target="_self">'xor'</a> <span class="p3">(<span class="default"><a href="binarith.html#K9" target="_self">add_ovfl</a> <font color="Maroon">z1</font>,<font color="Maroon">z2</font></span>)</span></span>)</span></span><a href="binarith.html#K11" target="_self">*&gt;</a></span>
 <b>by </b><i/>;<br/>


</div>
