<?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">zero</a> <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">z1</font>, <font color="Olive" title="b2">z2</font> being  <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/> for <font color="Olive" title="b3">d1</font>, <font color="Olive" title="b4">d2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a>  holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">z1</font> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Olive" title="b3">d1</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span> <a href="binarith.html#K7" title="BINARITH:func.7">+</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">z2</font> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Olive" title="b4">d2</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">z1</font> <a href="binarith.html#K7" title="BINARITH:func.7">+</a> <font color="Olive" title="b2">z2</font></span>)</span> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p1"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b3">d1</font> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <font color="Olive" title="b4">d2</font></span>)</span> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <span class="p3">(<span class="default"><a href="binarith.html#K8" title="BINARITH:func.8">add_ovfl</a> <font color="Olive" title="b1">z1</font>,<font color="Olive" title="b2">z2</font></span>)</span></span>)</span></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span><br/><b>let </b><font color="Maroon" title="c2">z1</font>, <font color="Maroon" title="c3">z2</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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">d1</font>, <font color="Olive" title="b2">d2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a>  holds  <span class="p1">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Olive" title="b1">d1</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span> <a href="binarith.html#K7" title="BINARITH:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">z2</font> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Olive" title="b2">d2</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="binarith.html#K7" title="BINARITH:func.7">+</a> <font color="Maroon" title="c3">z2</font></span>)</span> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p1"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">d1</font> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <font color="Olive" title="b2">d2</font></span>)</span> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <span class="p3">(<span class="default"><a href="binarith.html#K8" title="BINARITH:func.8">add_ovfl</a> <font color="Maroon" title="c2">z1</font>,<font color="Maroon" title="c3">z2</font></span>)</span></span>)</span></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span><br/><b>let </b><font color="Maroon" title="c4">d1</font>, <font color="Maroon" title="c5">d2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="margrel1.html#K6" title="MARGREL1:func.6">BOOLEAN</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c4">d1</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span> <a href="binarith.html#K7" title="BINARITH:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">z2</font> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c5">d2</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="binarith.html#K7" title="BINARITH:func.7">+</a> <font color="Maroon" title="c3">z2</font></span>)</span> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p1"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">d1</font> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <font color="Maroon" title="c5">d2</font></span>)</span> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <span class="p3">(<span class="default"><a href="binarith.html#K8" title="BINARITH:func.8">add_ovfl</a> <font color="Maroon" title="c2">z1</font>,<font color="Maroon" title="c3">z2</font></span>)</span></span>)</span></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span><br/>





<a NAME="E1:19"/>
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="binarith.html#K7" title="BINARITH:func.7">+</a> <font color="Maroon" title="c3">z2</font></span>)</span> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">d1</font> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <font color="Maroon" title="c5">d2</font></span>)</span> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <span class="p4">(<span class="default"><a href="binarith.html#K8" title="BINARITH:func.8">add_ovfl</a> <font color="Maroon" title="c2">z1</font>,<font color="Maroon" title="c3">z2</font></span>)</span></span>)</span></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p4"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c4">d1</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">z2</font> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p4"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c5">d2</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span></span>)</span> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binarith.html#K4" title="BINARITH:func.4">carry</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p4"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c4">d1</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">z2</font> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p4"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c5">d2</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">i</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c6">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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="binarith.html#K7" title="BINARITH:func.7">+</a> <font color="Maroon" title="c3">z2</font></span>)</span> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">d1</font> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <font color="Maroon" title="c5">d2</font></span>)</span> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <span class="p4">(<span class="default"><a href="binarith.html#K8" title="BINARITH:func.8">add_ovfl</a> <font color="Maroon" title="c2">z1</font>,<font color="Maroon" title="c3">z2</font></span>)</span></span>)</span></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p4"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c4">d1</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">i</font></span>)</span> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">z2</font> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p4"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c5">d2</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">i</font></span>)</span></span>)</span> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binarith.html#K4" title="BINARITH:func.4">carry</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p4"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c4">d1</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">z2</font> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p4"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c5">d2</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">i</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:19_1"/><i><font color="Green" title="E13">A1</font></i>: 
<font color="Maroon" title="c6">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> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="binarith.html#K7" title="BINARITH:func.7">+</a> <font color="Maroon" title="c3">z2</font></span>)</span> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">d1</font> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <font color="Maroon" title="c5">d2</font></span>)</span> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <span class="p4">(<span class="default"><a href="binarith.html#K8" title="BINARITH:func.8">add_ovfl</a> <font color="Maroon" title="c2">z1</font>,<font color="Maroon" title="c3">z2</font></span>)</span></span>)</span></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p4"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c4">d1</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">i</font></span>)</span> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">z2</font> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p4"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c5">d2</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">i</font></span>)</span></span>)</span> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binarith.html#K4" title="BINARITH:func.4">carry</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p4"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c4">d1</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">z2</font> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p4"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c5">d2</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">i</font></span>)</span></span><br/>

<a NAME="E2:19_1"/><i><font color="Green" title="E14">A2</font></i>: 
 <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T11" title="FINSEQ_1:th.11">FINSEQ_1:11</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:19_1_1"/>
( <font color="Maroon" title="c6">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> or <font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span> )
 </span><b>by </b><i><a class="txt" href="#E1:19_1"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="#E2:19_1"><i><font color="Green" title="E14">A2</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_1_1"><b>suppose </b></a><a NAME="E1:19_1_1_1"/><i><font color="Green" title="E15">A3</font></i>: 
<font color="Maroon" title="c6">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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="binarith.html#K7" title="BINARITH:func.7">+</a> <font color="Maroon" title="c3">z2</font></span>)</span> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">d1</font> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <font color="Maroon" title="c5">d2</font></span>)</span> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <span class="p4">(<span class="default"><a href="binarith.html#K8" title="BINARITH:func.8">add_ovfl</a> <font color="Maroon" title="c2">z1</font>,<font color="Maroon" title="c3">z2</font></span>)</span></span>)</span></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p4"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c4">d1</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">i</font></span>)</span> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">z2</font> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p4"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c5">d2</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">i</font></span>)</span></span>)</span> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binarith.html#K4" title="BINARITH:func.4">carry</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p4"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c4">d1</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">z2</font> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p4"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c5">d2</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">i</font></span>)</span></span><br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="binarith.html#K7" title="BINARITH:func.7">+</a> <font color="Maroon" title="c3">z2</font></span>)</span> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">d1</font> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <font color="Maroon" title="c5">d2</font></span>)</span> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <span class="p4">(<span class="default"><a href="binarith.html#K8" title="BINARITH:func.8">add_ovfl</a> <font color="Maroon" title="c2">z1</font>,<font color="Maroon" title="c3">z2</font></span>)</span></span>)</span></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="binarith.html#K7" title="BINARITH:func.7">+</a> <font color="Maroon" title="c3">z2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">i</font>
<b>by </b><i><a class="ref" href="binarith.html#T2" target="_self" title="BINARITH:th.2">Th2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">i</font></span>)</span> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">z2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">i</font></span>)</span></span>)</span> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binarith.html#K4" title="BINARITH:func.4">carry</a> <font color="Maroon" title="c2">z1</font>,<font color="Maroon" title="c3">z2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">i</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:19_1_1_1"><i><font color="Green" title="E15">A3</font></i></a>, <a class="ref" href="binarith.html#D8" target="_self" title="BINARITH:def.8">Def8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p4"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c4">d1</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">i</font></span>)</span> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">z2</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">i</font></span>)</span></span>)</span> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binarith.html#K4" title="BINARITH:func.4">carry</a> <font color="Maroon" title="c2">z1</font>,<font color="Maroon" title="c3">z2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">i</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:19_1_1_1"><i><font color="Green" title="E15">A3</font></i></a>, <a class="ref" href="binarith.html#T2" target="_self" title="BINARITH:th.2">Th2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p4"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c4">d1</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">i</font></span>)</span> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">z2</font> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p4"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c5">d2</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">i</font></span>)</span></span>)</span> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binarith.html#K4" title="BINARITH:func.4">carry</a> <font color="Maroon" title="c2">z1</font>,<font color="Maroon" title="c3">z2</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">i</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:19_1_1_1"><i><font color="Green" title="E15">A3</font></i></a>, <a class="ref" href="binarith.html#T2" target="_self" title="BINARITH:th.2">Th2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p4"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c4">d1</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">i</font></span>)</span> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">z2</font> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p4"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c5">d2</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">i</font></span>)</span></span>)</span> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binarith.html#K4" title="BINARITH:func.4">carry</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p4"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c4">d1</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">z2</font> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p4"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c5">d2</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">i</font></span>)</span>
<b>by </b><i><a class="txt" href="#E1:19_1_1_1"><i><font color="Green" title="E15">A3</font></i></a>, <a class="ref" href="binarith.html#T43" target="_self" title="BINARITH:th.43">Th43</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_1_1_2"><b>suppose </b></a><a NAME="E1:19_1_1_2"/>
<font color="Maroon" title="c6">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="setwiseo.html#K2" title="SETWISEO:func.2">{.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span><a href="setwiseo.html#K2" title="SETWISEO:func.2">.}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="binarith.html#K7" title="BINARITH:func.7">+</a> <font color="Maroon" title="c3">z2</font></span>)</span> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">d1</font> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <font color="Maroon" title="c5">d2</font></span>)</span> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <span class="p4">(<span class="default"><a href="binarith.html#K8" title="BINARITH:func.8">add_ovfl</a> <font color="Maroon" title="c2">z1</font>,<font color="Maroon" title="c3">z2</font></span>)</span></span>)</span></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p4"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c4">d1</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">i</font></span>)</span> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">z2</font> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p4"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c5">d2</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">i</font></span>)</span></span>)</span> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binarith.html#K4" title="BINARITH:func.4">carry</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p4"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c4">d1</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">z2</font> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p4"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c5">d2</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">i</font></span>)</span></span><br/><div class="add"><a NAME="E2:19_1_1_2"/><b>then </b><i><font color="Green" title="E15">A4</font></i>: 
<font color="Maroon" title="c6">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="binarith.html#K7" title="BINARITH:func.7">+</a> <font color="Maroon" title="c3">z2</font></span>)</span> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">d1</font> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <font color="Maroon" title="c5">d2</font></span>)</span> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <span class="p4">(<span class="default"><a href="binarith.html#K8" title="BINARITH:func.8">add_ovfl</a> <font color="Maroon" title="c2">z1</font>,<font color="Maroon" title="c3">z2</font></span>)</span></span>)</span></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">d1</font> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <font color="Maroon" title="c5">d2</font></span>)</span> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <span class="p1">(<span class="default"><a href="binarith.html#K8" title="BINARITH:func.8">add_ovfl</a> <font color="Maroon" title="c2">z1</font>,<font color="Maroon" title="c3">z2</font></span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#T3" target="_self" title="BINARITH:th.3">Th3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">d1</font> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <font color="Maroon" title="c5">d2</font></span>)</span> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binarith.html#K4" title="BINARITH:func.4">carry</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p4"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c4">d1</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">z2</font> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p4"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c5">d2</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">i</font></span>)</span>
<b>by </b><i><a class="txt" href="#E2:19_1_1_2"><i><font color="Green" title="E15">A4</font></i></a>, <a class="ref" href="binarith.html#T44" target="_self" title="BINARITH:th.44">Th44</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">d1</font> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">z2</font> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p4"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c5">d2</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">i</font></span>)</span></span>)</span> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binarith.html#K4" title="BINARITH:func.4">carry</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p4"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c4">d1</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">z2</font> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p4"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c5">d2</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">i</font></span>)</span>
<b>by </b><i><a class="txt" href="#E2:19_1_1_2"><i><font color="Green" title="E15">A4</font></i></a>, <a class="ref" href="binarith.html#T3" target="_self" title="BINARITH:th.3">Th3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p4"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c4">d1</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">i</font></span>)</span> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">z2</font> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p4"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c5">d2</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">i</font></span>)</span></span>)</span> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="binarith.html#K4" title="BINARITH:func.4">carry</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p4"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c4">d1</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c3">z2</font> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p4"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c5">d2</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">i</font></span>)</span>
<b>by </b><i><a class="txt" href="#E2:19_1_1_2"><i><font color="Green" title="E15">A4</font></i></a>, <a class="ref" href="binarith.html#T3" target="_self" title="BINARITH:th.3">Th3</a></i>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E2:19"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c4">d1</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span> <a href="binarith.html#K7" title="BINARITH:func.7">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">z2</font> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p2"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c5">d2</font></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">z1</font> <a href="binarith.html#K7" title="BINARITH:func.7">+</a> <font color="Maroon" title="c3">z2</font></span>)</span> <a href="binarith.html#K9" title="BINARITH:func.9">^</a> <span class="p1"><a href="binarith.html#K10" title="BINARITH:func.10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">d1</font> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <font color="Maroon" title="c5">d2</font></span>)</span> <a href="binarith.html#K2" title="BINARITH:func.2">'xor'</a> <span class="p3">(<span class="default"><a href="binarith.html#K8" title="BINARITH:func.8">add_ovfl</a> <font color="Maroon" title="c2">z1</font>,<font color="Maroon" title="c3">z2</font></span>)</span></span>)</span></span><a href="binarith.html#K10" title="BINARITH:func.10">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="binarith.html#D8" target="_self" title="BINARITH:def.8">Def8</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
